{"text": "[GOAL]\n\u03b1 : Type u\n\u22a2 Inhabited (FreeRing \u03b1)\n[PROOFSTEP]\ndsimp only [FreeRing]\n[GOAL]\n\u03b1 : Type u\n\u22a2 Inhabited (FreeAbelianGroup (FreeMonoid \u03b1))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\nC : FreeRing \u03b1 \u2192 Prop\nz : FreeRing \u03b1\nhn1 : C (-1)\nhb : \u2200 (b : \u03b1), C (of b)\nha : \u2200 (x y : FreeRing \u03b1), C x \u2192 C y \u2192 C (x + y)\nhm : \u2200 (x y : FreeRing \u03b1), C x \u2192 C y \u2192 C (x * y)\nhn : \u2200 (x : FreeRing \u03b1), C x \u2192 C (-x)\nh1 : C 1\nm\u271d : FreeMonoid \u03b1\na : \u03b1\nm : List \u03b1\nih : C (FreeAbelianGroup.of m)\n\u22a2 C (FreeAbelianGroup.of (a :: m))\n[PROOFSTEP]\nconvert hm _ _ (hb a) ih\n[GOAL]\ncase h.e'_1\n\u03b1 : Type u\nC : FreeRing \u03b1 \u2192 Prop\nz : FreeRing \u03b1\nhn1 : C (-1)\nhb : \u2200 (b : \u03b1), C (of b)\nha : \u2200 (x y : FreeRing \u03b1), C x \u2192 C y \u2192 C (x + y)\nhm : \u2200 (x y : FreeRing \u03b1), C x \u2192 C y \u2192 C (x * y)\nhn : \u2200 (x : FreeRing \u03b1), C x \u2192 C (-x)\nh1 : C 1\nm\u271d : FreeMonoid \u03b1\na : \u03b1\nm : List \u03b1\nih : C (FreeAbelianGroup.of m)\n\u22a2 FreeAbelianGroup.of (a :: m) = of a * FreeAbelianGroup.of m\n[PROOFSTEP]\nrw [of, \u2190 FreeAbelianGroup.of_mul]\n[GOAL]\ncase h.e'_1\n\u03b1 : Type u\nC : FreeRing \u03b1 \u2192 Prop\nz : FreeRing \u03b1\nhn1 : C (-1)\nhb : \u2200 (b : \u03b1), C (of b)\nha : \u2200 (x y : FreeRing \u03b1), C x \u2192 C y \u2192 C (x + y)\nhm : \u2200 (x y : FreeRing \u03b1), C x \u2192 C y \u2192 C (x * y)\nhn : \u2200 (x : FreeRing \u03b1), C x \u2192 C (-x)\nh1 : C 1\nm\u271d : FreeMonoid \u03b1\na : \u03b1\nm : List \u03b1\nih : C (FreeAbelianGroup.of m)\n\u22a2 FreeAbelianGroup.of (a :: m) = FreeAbelianGroup.of (FreeMonoid.of a * m)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.FreeRing", "llama_tokens": 698, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971175657575, "lm_q2_score": 0.7057850216484838, "lm_q1q2_score": 0.5998969246120688}}
{"text": "[GOAL]\nR\u271d : Type u_1\nS : Type u_2\nA\u271d : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\u271d\ninst\u271d\u00b9\u00b9 : CommSemiring S\ninst\u271d\u00b9\u2070 : Semiring A\u271d\ninst\u271d\u2079 : Semiring B\ninst\u271d\u2078 : Algebra R\u271d S\ninst\u271d\u2077 : Algebra R\u271d A\u271d\ninst\u271d\u2076 : Algebra R\u271d B\ninst\u271d\u2075 : Algebra S A\u271d\ninst\u271d\u2074 : SMulCommClass R\u271d S A\u271d\ninst\u271d\u00b3 : IsScalarTower R\u271d S A\u271d\nR : Type ?u.2035\nA : Type ?u.2038\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nr : R\nx : A\n\u22a2 \u2191(RingHom.toOpposite (algebraMap R A)\n            (_ :\n              \u2200 (x y : R),\n                \u2191(algebraMap R ((fun x => A) x)) x * \u2191(algebraMap R A) y =\n                  \u2191(algebraMap R A) y * \u2191(algebraMap R ((fun x => A) x)) x))\n        r *\n      op x =\n    op x *\n      \u2191(RingHom.toOpposite (algebraMap R A)\n            (_ :\n              \u2200 (x y : R),\n                \u2191(algebraMap R ((fun x => A) x)) x * \u2191(algebraMap R A) y =\n                  \u2191(algebraMap R A) y * \u2191(algebraMap R ((fun x => A) x)) x))\n        r\n[PROOFSTEP]\nsimp only [RingHom.toOpposite_apply, Function.comp_apply, \u2190 op_mul, Algebra.commutes]\n[GOAL]\nR\u271d : Type u_1\nS : Type u_2\nA\u271d : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\u271d\ninst\u271d\u00b9\u00b9 : CommSemiring S\ninst\u271d\u00b9\u2070 : Semiring A\u271d\ninst\u271d\u2079 : Semiring B\ninst\u271d\u2078 : Algebra R\u271d S\ninst\u271d\u2077 : Algebra R\u271d A\u271d\ninst\u271d\u2076 : Algebra R\u271d B\ninst\u271d\u2075 : Algebra S A\u271d\ninst\u271d\u2074 : SMulCommClass R\u271d S A\u271d\ninst\u271d\u00b3 : IsScalarTower R\u271d S A\u271d\nR : Type ?u.2035\nA : Type ?u.2038\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nc : R\nx : (fun x => A\u1d50\u1d52\u1d56) c\n\u22a2 unop (c \u2022 x) =\n    unop\n      (\u2191(RingHom.toOpposite (algebraMap R A)\n              (_ :\n                \u2200 (x y : R),\n                  \u2191(algebraMap R ((fun x => A) x)) x * \u2191(algebraMap R A) y =\n                    \u2191(algebraMap R A) y * \u2191(algebraMap R ((fun x => A) x)) x))\n          c *\n        x)\n[PROOFSTEP]\nsimp only [unop_smul, RingHom.toOpposite_apply, Function.comp_apply, unop_mul, op_mul, Algebra.smul_def,\n  Algebra.commutes, op_unop, unop_op]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Algebra.Opposite", "llama_tokens": 930, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424411924674, "lm_q2_score": 0.7090191337850933, "lm_q1q2_score": 0.5997893768863305}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : ConditionallyCompleteLattice G\ninst\u271d\u00b9 : CovariantClass G G (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 G\nhf : BddAbove (range f)\na : G\n\u22a2 (\u2a06 (i : \u03b9), f i) / a = \u2a06 (i : \u03b9), f i / a\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, ciSup_mul hf]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Bounds", "llama_tokens": 172, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8962513675912912, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.5994848540281341}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nsrc\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop (WithTop \u03b1) := linearOrderedAddCommMonoidWithTop\nsrc\u271d : Nontrivial (Option \u03b1) := Option.nontrivial\n\u22a2 \u2200 (a : WithTop \u03b1), a \u2260 \u22a4 \u2192 a + -a = 0\n[PROOFSTEP]\nrintro (a | a) ha\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nsrc\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop (WithTop \u03b1) := linearOrderedAddCommMonoidWithTop\nsrc\u271d : Nontrivial (Option \u03b1) := Option.nontrivial\nha : none \u2260 \u22a4\n\u22a2 none + -none = 0\n[PROOFSTEP]\nexact (ha rfl).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b c d : \u03b1\nsrc\u271d\u00b9 : LinearOrderedAddCommMonoidWithTop (WithTop \u03b1) := linearOrderedAddCommMonoidWithTop\nsrc\u271d : Nontrivial (Option \u03b1) := Option.nontrivial\na : \u03b1\nha : Option.some a \u2260 \u22a4\n\u22a2 Option.some a + -Option.some a = 0\n[PROOFSTEP]\nexact WithTop.coe_add.symm.trans (WithTop.coe_eq_coe.2 (add_neg_self a))\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Group.WithTop", "llama_tokens": 408, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240860523328, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.5992905268192273}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : IsPeriodicPt f n x\nhm : IsPeriodicPt f m x\n\u22a2 IsPeriodicPt f (n + m) x\n[PROOFSTEP]\nrw [IsPeriodicPt, iterate_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : IsPeriodicPt f n x\nhm : IsPeriodicPt f m x\n\u22a2 IsFixedPt (f^[n] \u2218 f^[m]) x\n[PROOFSTEP]\nexact hn.comp hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : IsPeriodicPt f (n + m) x\nhm : IsPeriodicPt f m x\n\u22a2 IsPeriodicPt f n x\n[PROOFSTEP]\nrw [IsPeriodicPt, iterate_add] at hn \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : IsFixedPt (f^[n] \u2218 f^[m]) x\nhm : IsPeriodicPt f m x\n\u22a2 IsPeriodicPt f n x\n[PROOFSTEP]\nexact hn.left_of_comp hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : IsPeriodicPt f (n + m) x\nhm : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f m x\n[PROOFSTEP]\nrw [add_comm] at hn \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : IsPeriodicPt f (m + n) x\nhm : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f m x\n[PROOFSTEP]\nexact hn.left_of_add hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (m - n) x\n[PROOFSTEP]\ncases' le_total n m with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\nh : n \u2264 m\n\u22a2 IsPeriodicPt f (m - n) x\n[PROOFSTEP]\nrefine' left_of_add _ hn\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\nh : n \u2264 m\n\u22a2 IsPeriodicPt f (m - n + n) x\n[PROOFSTEP]\nrwa [tsub_add_cancel_of_le h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\nh : m \u2264 n\n\u22a2 IsPeriodicPt f (m - n) x\n[PROOFSTEP]\nrw [tsub_eq_zero_iff_le.mpr h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\nh : m \u2264 n\n\u22a2 IsPeriodicPt f 0 x\n[PROOFSTEP]\napply isPeriodicPt_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhm : IsPeriodicPt f m x\nn : \u2115\n\u22a2 IsPeriodicPt f (m * n) x\n[PROOFSTEP]\nsimp only [IsPeriodicPt, iterate_mul, hm.isFixedPt.iterate n]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhm : IsPeriodicPt f m x\nn : \u2115\n\u22a2 IsPeriodicPt f (n * m) x\n[PROOFSTEP]\nsimp only [mul_comm n, hm.mul_const n]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n : \u2115\nhf : IsPeriodicPt f n x\nm : \u2115\n\u22a2 IsPeriodicPt f^[m] n x\n[PROOFSTEP]\nrw [IsPeriodicPt, \u2190 iterate_mul, mul_comm, iterate_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n : \u2115\nhf : IsPeriodicPt f n x\nm : \u2115\n\u22a2 IsFixedPt f^[n]^[m] x\n[PROOFSTEP]\nexact hf.isFixedPt.iterate m\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nhco : Function.Commute f g\nhf : IsPeriodicPt f n x\nhg : IsPeriodicPt g n x\n\u22a2 IsPeriodicPt (f \u2218 g) n x\n[PROOFSTEP]\nrw [IsPeriodicPt, hco.comp_iterate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nhco : Function.Commute f g\nhf : IsPeriodicPt f n x\nhg : IsPeriodicPt g n x\n\u22a2 IsFixedPt (f^[n] \u2218 g^[n]) x\n[PROOFSTEP]\nexact IsFixedPt.comp hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nhco : Function.Commute f g\nhfg : IsPeriodicPt (f \u2218 g) n x\nhg : IsPeriodicPt g n x\n\u22a2 IsPeriodicPt f n x\n[PROOFSTEP]\nrw [IsPeriodicPt, hco.comp_iterate] at hfg \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nhco : Function.Commute f g\nhfg : IsFixedPt (f^[n] \u2218 g^[n]) x\nhg : IsPeriodicPt g n x\n\u22a2 IsPeriodicPt f n x\n[PROOFSTEP]\nexact hfg.left_of_comp hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n : \u2115\nh : IsPeriodicPt f n x\nm : \u2115\n\u22a2 f^[m % n] x = f^[m] x\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Nat.mod_add_div m n, iterate_add_apply, (h.mul_const _).eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n : \u2115\nh : IsPeriodicPt f n x\nm : \u2115\n| f^[m] x\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div m n, iterate_add_apply, (h.mul_const _).eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n : \u2115\nh : IsPeriodicPt f n x\nm : \u2115\n| f^[m] x\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div m n, iterate_add_apply, (h.mul_const _).eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n : \u2115\nh : IsPeriodicPt f n x\nm : \u2115\n| f^[m] x\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div m n, iterate_add_apply, (h.mul_const _).eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (Nat.gcd m n) x\n[PROOFSTEP]\nrevert hm hn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 IsPeriodicPt f m x \u2192 IsPeriodicPt f n x \u2192 IsPeriodicPt f (Nat.gcd m n) x\n[PROOFSTEP]\nrefine' Nat.gcd.induction m n (fun n _ hn => _) fun m n _ ih hm hn => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d n : \u2115\nx\u271d : IsPeriodicPt f 0 x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (Nat.gcd 0 n) x\n[PROOFSTEP]\nrwa [Nat.gcd_zero_left]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nx\u271d : 0 < m\nih : IsPeriodicPt f (n % m) x \u2192 IsPeriodicPt f m x \u2192 IsPeriodicPt f (Nat.gcd (n % m) m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (Nat.gcd m n) x\n[PROOFSTEP]\nrw [Nat.gcd_rec]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nx\u271d : 0 < m\nih : IsPeriodicPt f (n % m) x \u2192 IsPeriodicPt f m x \u2192 IsPeriodicPt f (Nat.gcd (n % m) m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (Nat.gcd (n % m) m) x\n[PROOFSTEP]\nexact ih (hn.mod hm) hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : IsPeriodicPt f n x\nhy : IsPeriodicPt f n y\nhn : 0 < n\nh : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 hx.eq, \u2190 hy.eq, \u2190 iterate_pred_comp_of_pos f hn, comp_apply, comp_apply, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n\u271d : \u2115\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nhn : 0 < n\nx : \u03b1\nhx : x \u2208 ptsOfPeriod f n\n\u22a2 f (f^[Nat.pred n] x) = x\n[PROOFSTEP]\nrw [\u2190 comp_apply (f := f), comp_iterate_pred_of_pos f hn, hx.eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : x \u2208 periodicPts f\nhm : IsPeriodicPt f m (f^[n] x)\n\u22a2 IsPeriodicPt f m x\n[PROOFSTEP]\nrcases hx with \u27e8r, hr, hr'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\n\u22a2 IsPeriodicPt f m x\n[PROOFSTEP]\nsuffices n \u2264 (n / r + 1) * r by\n  -- porting note: convert used to unfold IsPeriodicPt\n  change _ = _\n  convert (hm.apply_iterate ((n / r + 1) * r - n)).eq <;>\n    rw [\u2190 iterate_add_apply, Nat.sub_add_cancel this, iterate_mul, (hr'.iterate _).eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n \u2264 (n / r + 1) * r\n\u22a2 IsPeriodicPt f m x\n[PROOFSTEP]\nchange _ = _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n \u2264 (n / r + 1) * r\n\u22a2 f^[m] x = x\n[PROOFSTEP]\nconvert (hm.apply_iterate ((n / r + 1) * r - n)).eq\n[GOAL]\ncase h.e'_2.h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n \u2264 (n / r + 1) * r\n\u22a2 x = f^[(n / r + 1) * r - n] (f^[n] x)\n[PROOFSTEP]\nrw [\u2190 iterate_add_apply, Nat.sub_add_cancel this, iterate_mul, (hr'.iterate _).eq]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\nthis : n \u2264 (n / r + 1) * r\n\u22a2 x = f^[(n / r + 1) * r - n] (f^[n] x)\n[PROOFSTEP]\nrw [\u2190 iterate_add_apply, Nat.sub_add_cancel this, iterate_mul, (hr'.iterate _).eq]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\n\u22a2 n \u2264 (n / r + 1) * r\n[PROOFSTEP]\nrw [add_mul, one_mul]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m (f^[n] x)\nr : \u2115\nhr : r > 0\nhr' : IsPeriodicPt f r x\n\u22a2 n \u2264 n / r * r + r\n[PROOFSTEP]\nexact (Nat.lt_div_mul_add hr).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nx : \u03b1\n\u22a2 x \u2208 \u22c3 (n : \u2115) (_ : n > 0), ptsOfPeriod f n \u2194 x \u2208 periodicPts f\n[PROOFSTEP]\nsimp [mem_periodicPts]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\n\u22a2 IsPeriodicPt f (minimalPeriod f x) x\n[PROOFSTEP]\ndelta minimalPeriod\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\n\u22a2 IsPeriodicPt f (if h : x \u2208 periodicPts f then Nat.find h else 0) x\n[PROOFSTEP]\nsplit_ifs with hx\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\n\u22a2 IsPeriodicPt f (Nat.find hx) x\n[PROOFSTEP]\nexact (Nat.find_spec hx).2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : \u00acx \u2208 periodicPts f\n\u22a2 IsPeriodicPt f 0 x\n[PROOFSTEP]\nexact isPeriodicPt_zero f x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 f^[n + minimalPeriod f x] x = f^[n] x\n[PROOFSTEP]\nrw [iterate_add_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 f^[n] (f^[minimalPeriod f x] x) = f^[n] x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 f^[minimalPeriod f x] x = x\n[PROOFSTEP]\nexact isPeriodicPt_minimalPeriod f x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : x \u2208 periodicPts f\n\u22a2 0 < minimalPeriod f x\n[PROOFSTEP]\nsimp only [minimalPeriod, dif_pos hx, (Nat.find_spec hx).1.lt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : \u00acx \u2208 periodicPts f\n\u22a2 minimalPeriod f x = 0\n[PROOFSTEP]\nsimp only [minimalPeriod, dif_neg hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : \u00acx \u2208 periodicPts f\n\u22a2 \u00ac0 < minimalPeriod f x\n[PROOFSTEP]\nsimp only [minimalPeriod, dif_neg h, lt_irrefl 0, not_false_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 minimalPeriod f x = 0 \u2194 \u00acx \u2208 periodicPts f\n[PROOFSTEP]\nrw [\u2190 minimalPeriod_pos_iff_mem_periodicPts, not_lt, nonpos_iff_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : 0 < n\nhx : IsPeriodicPt f n x\n\u22a2 minimalPeriod f x \u2264 n\n[PROOFSTEP]\nrw [minimalPeriod, dif_pos (mk_mem_periodicPts hn hx)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhn : 0 < n\nhx : IsPeriodicPt f n x\n\u22a2 Nat.find (_ : x \u2208 periodicPts f) \u2264 n\n[PROOFSTEP]\nexact Nat.find_min' (mk_mem_periodicPts hn hx) \u27e8hn, hx\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 minimalPeriod f (f^[n] x) = minimalPeriod f x\n[PROOFSTEP]\napply\n  (IsPeriodicPt.minimalPeriod_le (minimalPeriod_pos_of_mem_periodicPts hx) _).antisymm\n    ((isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate hx (isPeriodicPt_minimalPeriod f _)).minimalPeriod_le\n      (minimalPeriod_pos_of_mem_periodicPts _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 IsPeriodicPt f (minimalPeriod f x) (f^[n] x)\n[PROOFSTEP]\nexact (isPeriodicPt_minimalPeriod f x).apply_iterate n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 f^[n] x \u2208 periodicPts f\n[PROOFSTEP]\nrcases hx with \u27e8m, hm, hx\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d n m : \u2115\nhm : m > 0\nhx : IsPeriodicPt f m x\n\u22a2 f^[n] x \u2208 periodicPts f\n[PROOFSTEP]\nexact \u27e8m, hm, hx.apply_iterate n\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nhm : m < minimalPeriod f x\nhmn : f^[m] x = f^[n] x\n\u22a2 m \u2264 n\n[PROOFSTEP]\nby_contra' hmn'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nhm : m < minimalPeriod f x\nhmn : f^[m] x = f^[n] x\nhmn' : n < m\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Nat.add_sub_of_le hmn'.le, add_comm, iterate_add_apply] at hmn \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nhm : m < minimalPeriod f x\nhmn : f^[m - n] (f^[n] x) = f^[n] x\nhmn' : n < m\n\u22a2 False\n[PROOFSTEP]\nexact\n  ((IsPeriodicPt.minimalPeriod_le (tsub_pos_of_lt hmn')\n            (isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate\n              (minimalPeriod_pos_iff_mem_periodicPts.1 ((zero_le m).trans_lt hm)) hmn)).trans\n        (Nat.sub_le m n)).not_lt\n    hm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 minimalPeriod f x = 1 \u2194 IsFixedPt f x\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : minimalPeriod f x = 1\n\u22a2 IsFixedPt f x\n[PROOFSTEP]\nrw [\u2190 iterate_one f]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : minimalPeriod f x = 1\n\u22a2 IsFixedPt f^[1] x\n[PROOFSTEP]\nrefine' Function.IsPeriodicPt.isFixedPt _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : minimalPeriod f x = 1\n\u22a2 IsPeriodicPt f 1 x\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : minimalPeriod f x = 1\n\u22a2 IsPeriodicPt f (minimalPeriod f x) x\n[PROOFSTEP]\nexact isPeriodicPt_minimalPeriod f x\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : IsFixedPt f x\n\u22a2 minimalPeriod f x = 1\n[PROOFSTEP]\nexact\n  ((h.isPeriodicPt 1).minimalPeriod_le Nat.one_pos).antisymm\n    (Nat.succ_le_of_lt ((h.isPeriodicPt 1).minimalPeriod_pos Nat.one_pos))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y\u271d : \u03b1\nm n : \u2115\ng : \u03b2 \u2192 \u03b2\ny : \u03b2\n\u22a2 minimalPeriod f x = minimalPeriod g y \u2194 \u2200 (n : \u2115), IsPeriodicPt f n x \u2194 IsPeriodicPt g n y\n[PROOFSTEP]\nsimp_rw [isPeriodicPt_iff_minimalPeriod_dvd, dvd_right_iff_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n p k : \u2115\nhp : Fact (Nat.Prime p)\nhk : \u00acIsPeriodicPt f (p ^ k) x\nhk1 : IsPeriodicPt f (p ^ (k + 1)) x\n\u22a2 minimalPeriod f x = p ^ (k + 1)\n[PROOFSTEP]\napply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n p k : \u2115\nhp : Fact (Nat.Prime p)\nhk : \u00acIsPeriodicPt f (p ^ k) x\nhk1 : IsPeriodicPt f (p ^ (k + 1)) x\n\u22a2 \u00acminimalPeriod f x \u2223 p ^ k\n[PROOFSTEP]\nrwa [\u2190 isPeriodicPt_iff_minimalPeriod_dvd]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n p k : \u2115\nhp : Fact (Nat.Prime p)\nhk : \u00acIsPeriodicPt f (p ^ k) x\nhk1 : IsPeriodicPt f (p ^ (k + 1)) x\n\u22a2 minimalPeriod f x \u2223 p ^ (k + 1)\n[PROOFSTEP]\nrwa [\u2190 isPeriodicPt_iff_minimalPeriod_dvd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\n\u22a2 minimalPeriod (f \u2218 g) x \u2223 lcm (minimalPeriod f x) (minimalPeriod g x)\n[PROOFSTEP]\nrw [\u2190 isPeriodicPt_iff_minimalPeriod_dvd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\n\u22a2 IsPeriodicPt (f \u2218 g) (lcm (minimalPeriod f x) (minimalPeriod g x)) x\n[PROOFSTEP]\nexact (isPeriodicPt_minimalPeriod f x).comp_lcm h (isPeriodicPt_minimalPeriod g x)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\n\u22a2 minimalPeriod (f \u2218 g) x = minimalPeriod f x * minimalPeriod g x\n[PROOFSTEP]\napply h.minimalPeriod_of_comp_dvd_mul.antisymm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\n\u22a2 minimalPeriod f x * minimalPeriod g x \u2223 minimalPeriod (f \u2218 g) x\n[PROOFSTEP]\nsuffices :\n  \u2200 {f g : \u03b1 \u2192 \u03b1},\n    Commute f g \u2192 coprime (minimalPeriod f x) (minimalPeriod g x) \u2192 minimalPeriod f x \u2223 minimalPeriod (f \u2218 g) x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\nthis :\n  \u2200 {f g : \u03b1 \u2192 \u03b1},\n    Function.Commute f g \u2192 coprime (minimalPeriod f x) (minimalPeriod g x) \u2192 minimalPeriod f x \u2223 minimalPeriod (f \u2218 g) x\n\u22a2 minimalPeriod f x * minimalPeriod g x \u2223 minimalPeriod (f \u2218 g) x\n[PROOFSTEP]\nexact hco.mul_dvd_of_dvd_of_dvd (this h hco) (h.comp_eq.symm \u25b8 this h.symm hco.symm)\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\n\u22a2 \u2200 {f g : \u03b1 \u2192 \u03b1},\n    Function.Commute f g \u2192 coprime (minimalPeriod f x) (minimalPeriod g x) \u2192 minimalPeriod f x \u2223 minimalPeriod (f \u2218 g) x\n[PROOFSTEP]\nintro f g h hco\n[GOAL]\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng\u271d : \u03b1 \u2192 \u03b1\nh\u271d : Function.Commute f\u271d g\u271d\nhco\u271d : coprime (minimalPeriod f\u271d x) (minimalPeriod g\u271d x)\nf g : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\n\u22a2 minimalPeriod f x \u2223 minimalPeriod (f \u2218 g) x\n[PROOFSTEP]\nrefine' hco.dvd_of_dvd_mul_left (IsPeriodicPt.left_of_comp h _ _).minimalPeriod_dvd\n[GOAL]\ncase this.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng\u271d : \u03b1 \u2192 \u03b1\nh\u271d : Function.Commute f\u271d g\u271d\nhco\u271d : coprime (minimalPeriod f\u271d x) (minimalPeriod g\u271d x)\nf g : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\n\u22a2 IsPeriodicPt (f \u2218 g) (minimalPeriod g x * minimalPeriod (f \u2218 g) x) x\n[PROOFSTEP]\nexact (isPeriodicPt_minimalPeriod _ _).const_mul _\n[GOAL]\ncase this.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\ng\u271d : \u03b1 \u2192 \u03b1\nh\u271d : Function.Commute f\u271d g\u271d\nhco\u271d : coprime (minimalPeriod f\u271d x) (minimalPeriod g\u271d x)\nf g : \u03b1 \u2192 \u03b1\nh : Function.Commute f g\nhco : coprime (minimalPeriod f x) (minimalPeriod g x)\n\u22a2 IsPeriodicPt g (minimalPeriod g x * minimalPeriod (f \u2218 g) x) x\n[PROOFSTEP]\nexact (isPeriodicPt_minimalPeriod _ _).mul_const _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 minimalPeriod f^[n] x = minimalPeriod f x / gcd (minimalPeriod f x) n\n[PROOFSTEP]\napply Nat.dvd_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 minimalPeriod f^[n] x \u2223 minimalPeriod f x / gcd (minimalPeriod f x) n\n[PROOFSTEP]\napply IsPeriodicPt.minimalPeriod_dvd\n[GOAL]\ncase a.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 IsPeriodicPt f^[n] (minimalPeriod f x / gcd (minimalPeriod f x) n) x\n[PROOFSTEP]\nrw [IsPeriodicPt, IsFixedPt, \u2190 iterate_mul, \u2190 Nat.mul_div_assoc _ (gcd_dvd_left _ _), mul_comm,\n  Nat.mul_div_assoc _ (gcd_dvd_right _ _), mul_comm, iterate_mul]\n[GOAL]\ncase a.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 f^[n / gcd (minimalPeriod f x) n]^[minimalPeriod f x] x = x\n[PROOFSTEP]\nexact (isPeriodicPt_minimalPeriod f x).iterate _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 minimalPeriod f x / gcd (minimalPeriod f x) n \u2223 minimalPeriod f^[n] x\n[PROOFSTEP]\napply coprime.dvd_of_dvd_mul_right (coprime_div_gcd_div_gcd h)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 minimalPeriod f x / gcd (minimalPeriod f x) n \u2223 minimalPeriod f^[n] x * (n / gcd (minimalPeriod f x) n)\n[PROOFSTEP]\napply Nat.dvd_of_mul_dvd_mul_right h\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 minimalPeriod f x / gcd (minimalPeriod f x) n * gcd (minimalPeriod f x) n \u2223\n    minimalPeriod f^[n] x * (n / gcd (minimalPeriod f x) n) * gcd (minimalPeriod f x) n\n[PROOFSTEP]\nrw [Nat.div_mul_cancel (gcd_dvd_left _ _), mul_assoc, Nat.div_mul_cancel (gcd_dvd_right _ _), mul_comm]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 minimalPeriod f x \u2223 n * minimalPeriod f^[n] x\n[PROOFSTEP]\napply IsPeriodicPt.minimalPeriod_dvd\n[GOAL]\ncase a.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 IsPeriodicPt f (n * minimalPeriod f^[n] x) x\n[PROOFSTEP]\nrw [IsPeriodicPt, IsFixedPt, iterate_mul]\n[GOAL]\ncase a.hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nh : 0 < gcd (minimalPeriod f x) n\n\u22a2 f^[n]^[minimalPeriod f^[n] x] x = x\n[PROOFSTEP]\nexact isPeriodicPt_minimalPeriod _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 Cycle.length (periodicOrbit f x) = minimalPeriod f x\n[PROOFSTEP]\nrw [periodicOrbit, Cycle.length_coe, List.length_map, List.length_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 periodicOrbit f x = Cycle.nil \u2194 \u00acx \u2208 periodicPts f\n[PROOFSTEP]\nsimp [periodicOrbit]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 minimalPeriod f x = 0 \u2194 \u00acx \u2208 periodicPts f\n[PROOFSTEP]\nexact minimalPeriod_eq_zero_iff_nmem_periodicPts\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : x \u2208 periodicPts f\n\u22a2 y \u2208 periodicOrbit f x \u2194 \u2203 n, f^[n] x = y\n[PROOFSTEP]\nsimp only [periodicOrbit, Cycle.mem_coe_iff, List.mem_map, List.mem_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : x \u2208 periodicPts f\n\u22a2 (\u2203 a, a < minimalPeriod f x \u2227 f^[a] x = y) \u2194 \u2203 n, f^[n] x = y\n[PROOFSTEP]\nuse fun \u27e8a, _, ha'\u27e9 => \u27e8a, ha'\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : x \u2208 periodicPts f\n\u22a2 (\u2203 n, f^[n] x = y) \u2192 \u2203 a, a < minimalPeriod f x \u2227 f^[a] x = y\n[PROOFSTEP]\nrintro \u27e8n, rfl\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 \u2203 a, a < minimalPeriod f x \u2227 f^[a] x = f^[n] x\n[PROOFSTEP]\nuse n % minimalPeriod f x, mod_lt _ (minimalPeriod_pos_of_mem_periodicPts hx)\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 f^[n % minimalPeriod f x] x = f^[n] x\n[PROOFSTEP]\nrw [iterate_mod_minimalPeriod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 Cycle.Nodup (periodicOrbit f x)\n[PROOFSTEP]\nrw [periodicOrbit, Cycle.nodup_coe_iff, List.nodup_map_iff_inj_on (List.nodup_range _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 \u2200 (x_1 : \u2115),\n    x_1 \u2208 List.range (minimalPeriod f x) \u2192 \u2200 (y : \u2115), y \u2208 List.range (minimalPeriod f x) \u2192 f^[x_1] x = f^[y] x \u2192 x_1 = y\n[PROOFSTEP]\nintro m hm n hn hmn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m : \u2115\nhm : m \u2208 List.range (minimalPeriod f x)\nn : \u2115\nhn : n \u2208 List.range (minimalPeriod f x)\nhmn : f^[m] x = f^[n] x\n\u22a2 m = n\n[PROOFSTEP]\nrw [List.mem_range] at hm hn \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m : \u2115\nhm : m < minimalPeriod f x\nn : \u2115\nhn : n < minimalPeriod f x\nhmn : f^[m] x = f^[n] x\n\u22a2 m = n\n[PROOFSTEP]\nrwa [eq_iff_lt_minimalPeriod_of_iterate_eq hm hn] at hmn \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 List.rotate (List.map (fun n => f^[n] x) (List.range (minimalPeriod f x))) n =\n    List.map (fun n_1 => f^[n_1] (f^[n] x)) (List.range (minimalPeriod f (f^[n] x)))\n[PROOFSTEP]\napply List.ext_nthLe _ fun m _ _ => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 List.length (List.rotate (List.map (fun n => f^[n] x) (List.range (minimalPeriod f x))) n) =\n    List.length (List.map (fun n_1 => f^[n_1] (f^[n] x)) (List.range (minimalPeriod f (f^[n] x))))\n[PROOFSTEP]\nsimp [minimalPeriod_apply_iterate hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d : \u2115\nhx : x \u2208 periodicPts f\nn : \u2115\n\u22a2 \u2200 (m : \u2115) (x_1 : m < List.length (List.rotate (List.map (fun n => f^[n] x) (List.range (minimalPeriod f x))) n))\n    (x_2 : m < List.length (List.map (fun n_1 => f^[n_1] (f^[n] x)) (List.range (minimalPeriod f (f^[n] x))))),\n    List.nthLe (List.rotate (List.map (fun n => f^[n] x) (List.range (minimalPeriod f x))) n) m x_1 =\n      List.nthLe (List.map (fun n_1 => f^[n_1] (f^[n] x)) (List.range (minimalPeriod f (f^[n] x)))) m x_2\n[PROOFSTEP]\nsimp [List.nthLe_rotate, iterate_add_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\n\u22a2 Cycle.Chain r (periodicOrbit f x) \u2194 \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nby_cases hx : x \u2208 periodicPts f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\n\u22a2 Cycle.Chain r (periodicOrbit f x) \u2194 \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nhave hx' := minimalPeriod_pos_of_mem_periodicPts hx\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\n\u22a2 Cycle.Chain r (periodicOrbit f x) \u2194 \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nhave hM := Nat.sub_add_cancel (succ_le_iff.2 hx')\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\n\u22a2 Cycle.Chain r (periodicOrbit f x) \u2194 \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrw [periodicOrbit, \u2190 Cycle.map_coe, Cycle.chain_map, \u2190 hM, Cycle.chain_range_succ]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\n\u22a2 (r (f^[minimalPeriod f x - succ 0] x) (f^[0] x) \u2227\n      \u2200 (m : \u2115), m < minimalPeriod f x - succ 0 \u2192 r (f^[m] x) (f^[succ m] x)) \u2194\n    \u2200 (n : \u2115), n < minimalPeriod f x - succ 0 + succ 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrefine' \u27e8_, fun H => \u27e8_, fun m hm => H _ (hm.trans (Nat.lt_succ_self _))\u27e9\u27e9\n[GOAL]\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\n\u22a2 (r (f^[minimalPeriod f x - succ 0] x) (f^[0] x) \u2227\n      \u2200 (m : \u2115), m < minimalPeriod f x - succ 0 \u2192 r (f^[m] x) (f^[succ m] x)) \u2192\n    \u2200 (n : \u2115), n < minimalPeriod f x - succ 0 + succ 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrintro \u27e8hr, H\u27e9 n hn\n[GOAL]\ncase pos.refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n\u271d : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nhr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)\nH : \u2200 (m : \u2115), m < minimalPeriod f x - succ 0 \u2192 r (f^[m] x) (f^[succ m] x)\nn : \u2115\nhn : n < minimalPeriod f x - succ 0 + succ 0\n\u22a2 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\ncases' eq_or_lt_of_le (lt_succ_iff.1 hn) with hM' hM'\n[GOAL]\ncase pos.refine'_1.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n\u271d : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nhr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)\nH : \u2200 (m : \u2115), m < minimalPeriod f x - succ 0 \u2192 r (f^[m] x) (f^[succ m] x)\nn : \u2115\nhn : n < minimalPeriod f x - succ 0 + succ 0\nhM' : n = minimalPeriod f x - succ 0\n\u22a2 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrwa [hM', hM, iterate_minimalPeriod]\n[GOAL]\ncase pos.refine'_1.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n\u271d : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nhr : r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)\nH : \u2200 (m : \u2115), m < minimalPeriod f x - succ 0 \u2192 r (f^[m] x) (f^[succ m] x)\nn : \u2115\nhn : n < minimalPeriod f x - succ 0 + succ 0\nhM' : n < minimalPeriod f x - succ 0\n\u22a2 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nexact H _ hM'\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nH : \u2200 (n : \u2115), n < minimalPeriod f x - succ 0 + succ 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n\u22a2 r (f^[minimalPeriod f x - succ 0] x) (f^[0] x)\n[PROOFSTEP]\nrw [iterate_zero_apply]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nH : \u2200 (n : \u2115), n < minimalPeriod f x - succ 0 + succ 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n\u22a2 r (f^[minimalPeriod f x - succ 0] x) x\n[PROOFSTEP]\nnth_rw 3 [\u2190 @iterate_minimalPeriod \u03b1 f x]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nH : \u2200 (n : \u2115), n < minimalPeriod f x - succ 0 + succ 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n\u22a2 r (f^[minimalPeriod f x - succ 0] x) (f^[minimalPeriod f x] x)\n[PROOFSTEP]\nnth_rw 2 [\u2190 hM]\n[GOAL]\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nhx' : 0 < minimalPeriod f x\nhM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x\nH : \u2200 (n : \u2115), n < minimalPeriod f x - succ 0 + succ 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n\u22a2 r (f^[minimalPeriod f x - succ 0] x) (f^[minimalPeriod f x - succ 0 + succ 0] x)\n[PROOFSTEP]\nexact H _ (Nat.lt_succ_self _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : \u00acx \u2208 periodicPts f\n\u22a2 Cycle.Chain r (periodicOrbit f x) \u2194 \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrw [periodicOrbit_eq_nil_of_not_periodic_pt hx, minimalPeriod_eq_zero_of_nmem_periodicPts hx]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : \u00acx \u2208 periodicPts f\n\u22a2 Cycle.Chain r Cycle.nil \u2194 \u2200 (n : \u2115), n < 0 \u2192 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\n\u22a2 Cycle.Chain r (periodicOrbit f x) \u2194 \u2200 (n : \u2115), r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrw [periodicOrbit_chain r]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\n\u22a2 (\u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)) \u2194 \u2200 (n : \u2115), r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrefine' \u27e8fun H n => _, fun H n _ => H n\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n\u271d : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nH : \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\nn : \u2115\n\u22a2 r (f^[n] x) (f^[n + 1] x)\n[PROOFSTEP]\nrw [iterate_succ_apply, \u2190 iterate_mod_minimalPeriod_eq, \u2190 iterate_mod_minimalPeriod_eq (n := n), \u2190 iterate_succ_apply,\n  minimalPeriod_apply hx]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx\u271d y : \u03b1\nm n\u271d : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nhx : x \u2208 periodicPts f\nH : \u2200 (n : \u2115), n < minimalPeriod f x \u2192 r (f^[n] x) (f^[n + 1] x)\nn : \u2115\n\u22a2 r (f^[n % minimalPeriod f x] x) (f^[succ (n % minimalPeriod f x)] x)\n[PROOFSTEP]\nexact H _ (mod_lt _ (minimalPeriod_pos_of_mem_periodicPts hx))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b1\ng\u271d : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n\u271d : \u2115\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nn : \u2115\n\u22a2 (Prod.map f g)^[n] = Prod.map f^[n] g^[n]\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b1\ng\u271d : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\n\u22a2 (Prod.map f g)^[Nat.zero] = Prod.map f^[Nat.zero] g^[Nat.zero]\n[PROOFSTEP]\nsimp [*, Prod.map_comp_map]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b1\ng\u271d : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nn\u271d : \u2115\nn_ih\u271d : (Prod.map f g)^[n\u271d] = Prod.map f^[n\u271d] g^[n\u271d]\n\u22a2 (Prod.map f g)^[Nat.succ n\u271d] = Prod.map f^[Nat.succ n\u271d] g^[Nat.succ n\u271d]\n[PROOFSTEP]\nsimp [*, Prod.map_comp_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nx : \u03b1 \u00d7 \u03b2\n\u22a2 IsPeriodicPt (Prod.map f g) n x \u2194 IsPeriodicPt f n x.fst \u2227 IsPeriodicPt g n x.snd\n[PROOFSTEP]\nsimp [IsPeriodicPt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b1\ng\u271d : \u03b2 \u2192 \u03b2\nx\u271d : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u2200 (c : \u2115), minimalPeriod (Prod.map f g) x \u2223 c \u2194 Nat.lcm (minimalPeriod f x.fst) (minimalPeriod g x.snd) \u2223 c\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b1\ng\u271d : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2\n\u22a2 \u2200 (c : \u2115),\n    minimalPeriod (Prod.map f g) (fst\u271d, snd\u271d) \u2223 c \u2194\n      Nat.lcm (minimalPeriod f (fst\u271d, snd\u271d).fst) (minimalPeriod g (fst\u271d, snd\u271d).snd) \u2223 c\n[PROOFSTEP]\nsimp [\u2190 isPeriodicPt_iff_minimalPeriod_dvd, Nat.lcm_dvd_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\n\u22a2 minimalPeriod f x.fst \u2223 minimalPeriod (Prod.map f g) x\n[PROOFSTEP]\nrw [minimalPeriod_prod_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\n\u22a2 minimalPeriod f x.fst \u2223 Nat.lcm (minimalPeriod f x.fst) (minimalPeriod g x.snd)\n[PROOFSTEP]\nexact Nat.dvd_lcm_left _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\n\u22a2 minimalPeriod g x.snd \u2223 minimalPeriod (Prod.map f g) x\n[PROOFSTEP]\nrw [minimalPeriod_prod_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\nm n : \u2115\n\u22a2 minimalPeriod g x.snd \u2223 Nat.lcm (minimalPeriod f x.fst) (minimalPeriod g x.snd)\n[PROOFSTEP]\nexact Nat.dvd_lcm_right _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\n\u22a2 a ^ n \u2022 b = b \u2194 minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b \u2223 n\n[PROOFSTEP]\nrw [\u2190 isPeriodicPt_iff_minimalPeriod_dvd, IsPeriodicPt, IsFixedPt, smul_iterate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2124\n\u22a2 a ^ n \u2022 b = b \u2194 \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b) \u2223 n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\na\u271d : \u2115\n\u22a2 a ^ Int.ofNat a\u271d \u2022 b = b \u2194 \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b) \u2223 Int.ofNat a\u271d\n[PROOFSTEP]\nrw [Int.ofNat_eq_coe, zpow_ofNat, Int.coe_nat_dvd, pow_smul_eq_iff_minimalPeriod_dvd]\n[GOAL]\ncase negSucc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\na\u271d : \u2115\n\u22a2 a ^ Int.negSucc a\u271d \u2022 b = b \u2194 \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b) \u2223 Int.negSucc a\u271d\n[PROOFSTEP]\nrw [Int.negSucc_coe, zpow_neg, zpow_ofNat, inv_smul_eq_iff, eq_comm, dvd_neg, Int.coe_nat_dvd,\n  pow_smul_eq_iff_minimalPeriod_dvd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\n\u22a2 a ^ (n % minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b) \u2022 b = a ^ n \u2022 b\n[PROOFSTEP]\nconv_rhs =>\n  rw [\u2190 Nat.mod_add_div n (minimalPeriod ((\u00b7 \u2022 \u00b7) a) b), pow_add, mul_smul,\n    pow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\n| a ^ n \u2022 b\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div n (minimalPeriod ((\u00b7 \u2022 \u00b7) a) b), pow_add, mul_smul,\n    pow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\n| a ^ n \u2022 b\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div n (minimalPeriod ((\u00b7 \u2022 \u00b7) a) b), pow_add, mul_smul,\n    pow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2115\n| a ^ n \u2022 b\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div n (minimalPeriod ((\u00b7 \u2022 \u00b7) a) b), pow_add, mul_smul,\n  pow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2124\n\u22a2 a ^ (n % \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) a) b)) \u2022 b = a ^ n \u2022 b\n[PROOFSTEP]\nconv_rhs =>\n  rw [\u2190 Int.emod_add_ediv n (minimalPeriod ((a \u2022 \u00b7)) b), zpow_add, mul_smul,\n    zpow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2124\n| a ^ n \u2022 b\n[PROOFSTEP]\nrw [\u2190 Int.emod_add_ediv n (minimalPeriod ((a \u2022 \u00b7)) b), zpow_add, mul_smul,\n    zpow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2124\n| a ^ n \u2022 b\n[PROOFSTEP]\nrw [\u2190 Int.emod_add_ediv n (minimalPeriod ((a \u2022 \u00b7)) b), zpow_add, mul_smul,\n    zpow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nn : \u2124\n| a ^ n \u2022 b\n[PROOFSTEP]\nrw [\u2190 Int.emod_add_ediv n (minimalPeriod ((a \u2022 \u00b7)) b), zpow_add, mul_smul,\n  zpow_smul_eq_iff_minimalPeriod_dvd.mpr (dvd_mul_right _ _)]\n", "meta": {"mathlib_filename": "Mathlib.Dynamics.PeriodicPts", "llama_tokens": 20096, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789178257653, "lm_q2_score": 0.7371581684030623, "lm_q1q2_score": 0.5974511545937372}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasShift C \u2124\nT\u2081 T\u2082 T\u2083 A B : Triangle C\niso\u2081 : A.obj\u2081 \u2245 B.obj\u2081\niso\u2082 : A.obj\u2082 \u2245 B.obj\u2082\niso\u2083 : A.obj\u2083 \u2245 B.obj\u2083\ncomm\u2081 : autoParam (A.mor\u2081 \u226b iso\u2082.hom = iso\u2081.hom \u226b B.mor\u2081) _auto\u271d\ncomm\u2082 : autoParam (A.mor\u2082 \u226b iso\u2083.hom = iso\u2082.hom \u226b B.mor\u2082) _auto\u271d\ncomm\u2083 : autoParam (A.mor\u2083 \u226b (shiftFunctor C 1).map iso\u2081.hom = iso\u2083.hom \u226b B.mor\u2083) _auto\u271d\n\u22a2 B.mor\u2081 \u226b iso\u2082.inv = iso\u2081.inv \u226b A.mor\u2081\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono iso\u2082.hom, assoc, Iso.inv_hom_id, comp_id, comm\u2081, Iso.inv_hom_id_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasShift C \u2124\nT\u2081 T\u2082 T\u2083 A B : Triangle C\niso\u2081 : A.obj\u2081 \u2245 B.obj\u2081\niso\u2082 : A.obj\u2082 \u2245 B.obj\u2082\niso\u2083 : A.obj\u2083 \u2245 B.obj\u2083\ncomm\u2081 : autoParam (A.mor\u2081 \u226b iso\u2082.hom = iso\u2081.hom \u226b B.mor\u2081) _auto\u271d\ncomm\u2082 : autoParam (A.mor\u2082 \u226b iso\u2083.hom = iso\u2082.hom \u226b B.mor\u2082) _auto\u271d\ncomm\u2083 : autoParam (A.mor\u2083 \u226b (shiftFunctor C 1).map iso\u2081.hom = iso\u2083.hom \u226b B.mor\u2083) _auto\u271d\n\u22a2 B.mor\u2082 \u226b iso\u2083.inv = iso\u2082.inv \u226b A.mor\u2082\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono iso\u2083.hom, assoc, Iso.inv_hom_id, comp_id, comm\u2082, Iso.inv_hom_id_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasShift C \u2124\nT\u2081 T\u2082 T\u2083 A B : Triangle C\niso\u2081 : A.obj\u2081 \u2245 B.obj\u2081\niso\u2082 : A.obj\u2082 \u2245 B.obj\u2082\niso\u2083 : A.obj\u2083 \u2245 B.obj\u2083\ncomm\u2081 : autoParam (A.mor\u2081 \u226b iso\u2082.hom = iso\u2081.hom \u226b B.mor\u2081) _auto\u271d\ncomm\u2082 : autoParam (A.mor\u2082 \u226b iso\u2083.hom = iso\u2082.hom \u226b B.mor\u2082) _auto\u271d\ncomm\u2083 : autoParam (A.mor\u2083 \u226b (shiftFunctor C 1).map iso\u2081.hom = iso\u2083.hom \u226b B.mor\u2083) _auto\u271d\n\u22a2 B.mor\u2083 \u226b (shiftFunctor C 1).map iso\u2081.inv = iso\u2083.inv \u226b A.mor\u2083\n[PROOFSTEP]\nsimp only [\u2190 cancel_mono (iso\u2081.hom\u27e6(1 : \u2124)\u27e7'), Category.assoc, comm\u2083, Iso.inv_hom_id_assoc, \u2190 Functor.map_comp,\n  Iso.inv_hom_id, Functor.map_id, Category.comp_id]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Triangulated.Basic", "llama_tokens": 915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.7461389817407016, "lm_q1q2_score": 0.5974275151317991}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d : DivisionRing R\nn : \u2115\n\u22a2 \u2191(-\u2191n) = -\u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DivisionRing \u03b1\nm n : \u2124\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 : DivisionRing \u03b1\nn : \u2124\nn_nonzero : \u2191n \u2260 0\nk : \u2124\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 : DivisionRing \u03b1\nn : \u2124\nn_nonzero : \u2191n \u2260 0\nk : \u2124\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DivisionRing \u03b1\nk : \u2124\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 : DivisionRing \u03b1\nn : \u2124\nn_nonzero : \u2191n \u2260 0\nk : \u2124\nthis : n \u2260 0\n\u22a2 \u2191(n * k / n) = \u2191(n * k) / \u2191n\n[PROOFSTEP]\nrw [Int.mul_ediv_cancel_left _ this, mul_comm n k, Int.cast_mul, mul_div_cancel _ n_nonzero]\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Cast.Field", "llama_tokens": 495, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430645886584, "lm_q2_score": 0.7185943805178139, "lm_q1q2_score": 0.5972547356197644}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nx\u271d\u00b2 x\u271d\u00b9 : Opens \u03b1\ncarrier\u271d\u00b9 : Set \u03b1\nis_open'\u271d\u00b9 : IsOpen carrier\u271d\u00b9\ncarrier\u271d : Set \u03b1\nis_open'\u271d : IsOpen carrier\u271d\nx\u271d : { carrier := carrier\u271d\u00b9, is_open' := is_open'\u271d\u00b9 }.carrier = { carrier := carrier\u271d, is_open' := is_open'\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, is_open' := is_open'\u271d\u00b9 } = { carrier := carrier\u271d, is_open' := is_open'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Opens \u03b1\n\u22a2 \u2191(\u2a06 (i : \u03b9), s i) = \u22c3 (i : \u03b9), \u2191(s i)\n[PROOFSTEP]\nsimp [iSup]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Sort u_5\nx : \u03b1\ns : \u03b9 \u2192 Opens \u03b1\n\u22a2 x \u2208 iSup s \u2194 \u2203 i, x \u2208 s i\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Sort u_5\nx : \u03b1\ns : \u03b9 \u2192 Opens \u03b1\n\u22a2 x \u2208 \u2191(iSup s) \u2194 \u2203 i, x \u2208 s i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nUs : Set (Opens \u03b1)\nx : \u03b1\n\u22a2 x \u2208 sSup Us \u2194 \u2203 u, u \u2208 Us \u2227 x \u2208 u\n[PROOFSTEP]\nsimp_rw [sSup_eq_iSup, mem_iSup, exists_prop]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nsrc\u271d : CompleteLattice (Opens \u03b1) := inferInstanceAs (CompleteLattice (Opens \u03b1))\na : Opens \u03b1\ns : Set (Opens \u03b1)\n\u22a2 \u2191(a \u2293 sSup s) = \u2191(\u2a06 (b : Opens \u03b1) (_ : b \u2208 s), a \u2293 b)\n[PROOFSTEP]\nsimp only [coe_inf, coe_iSup, coe_sSup, Set.inter_iUnion\u2082]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nU V : Opens \u03b1\ni : U \u2264 V\n\u22a2 IsOpen (range (inclusion (_ : \u2191U \u2286 \u2191V)))\n[PROOFSTEP]\nrw [Set.range_inclusion i]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nU V : Opens \u03b1\ni : U \u2264 V\n\u22a2 IsOpen {x | \u2191x \u2208 \u2191U}\n[PROOFSTEP]\nexact U.isOpen.preimage continuous_subtype_val\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nU : Opens \u03b1\n\u22a2 \u00acSet.Nonempty \u2191U \u2194 U = \u22a5\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_bot, \u2190 Set.not_nonempty_iff_eq_empty]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nU : Opens \u03b1\n\u22a2 U \u2260 \u22a5 \u2194 Set.Nonempty \u2191U\n[PROOFSTEP]\nrw [Ne.def, \u2190 not_nonempty_iff_eq_bot, not_not]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nt : TopologicalSpace \u03b1\nh : t = \u22a4\nU : Opens \u03b1\n\u22a2 U = \u22a5 \u2228 U = \u22a4\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\n\u22a2 U = \u22a5 \u2228 U = \u22a4\n[PROOFSTEP]\nletI : TopologicalSpace \u03b1 := \u22a4\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\nthis : TopologicalSpace \u03b1 := \u22a4\n\u22a2 U = \u22a5 \u2228 U = \u22a4\n[PROOFSTEP]\nrw [\u2190 coe_eq_empty, \u2190 coe_eq_univ, \u2190 isOpen_top_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b1 : Type u_5\nU : Opens \u03b1\nthis : TopologicalSpace \u03b1 := \u22a4\n\u22a2 IsOpen \u2191U\n[PROOFSTEP]\nexact U.2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\n\u22a2 IsBasis B \u2194 \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\n\u22a2 IsBasis B \u2192 \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\n\u22a2 (\u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U) \u2192 IsBasis B\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\n\u22a2 \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n[PROOFSTEP]\nrintro \u27e8sU, hU\u27e9 x hx\n[GOAL]\ncase mp.mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\nsU : Set \u03b1\nhU : IsOpen sU\nx : \u03b1\nhx : x \u2208 { carrier := sU, is_open' := hU }\n\u22a2 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 { carrier := sU, is_open' := hU }\n[PROOFSTEP]\nrcases h.mem_nhds_iff.mp (IsOpen.mem_nhds hU hx) with \u27e8sV, \u27e8\u27e8V, H\u2081, H\u2082\u27e9, hsV\u27e9\u27e9\n[GOAL]\ncase mp.mk.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\nsU : Set \u03b1\nhU : IsOpen sU\nx : \u03b1\nhx : x \u2208 { carrier := sU, is_open' := hU }\nsV : Set \u03b1\nhsV : x \u2208 sV \u2227 sV \u2286 sU\nV : Opens \u03b1\nH\u2081 : V \u2208 B\nH\u2082 : \u2191V = sV\n\u22a2 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 { carrier := sU, is_open' := hU }\n[PROOFSTEP]\nrefine' \u27e8V, H\u2081, _\u27e9\n[GOAL]\ncase mp.mk.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\nsU : Set \u03b1\nhU : IsOpen sU\nx : \u03b1\nhx : x \u2208 { carrier := sU, is_open' := hU }\nsV : Set \u03b1\nhsV : x \u2208 sV \u2227 sV \u2286 sU\nV : Opens \u03b1\nH\u2081 : V \u2208 B\nH\u2082 : \u2191V = sV\n\u22a2 x \u2208 V \u2227 V \u2264 { carrier := sU, is_open' := hU }\n[PROOFSTEP]\ncases V\n[GOAL]\ncase mp.mk.intro.intro.intro.intro.mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\nsU : Set \u03b1\nhU : IsOpen sU\nx : \u03b1\nhx : x \u2208 { carrier := sU, is_open' := hU }\nsV : Set \u03b1\nhsV : x \u2208 sV \u2227 sV \u2286 sU\ncarrier\u271d : Set \u03b1\nis_open'\u271d : IsOpen carrier\u271d\nH\u2081 : { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2208 B\nH\u2082 : \u2191{ carrier := carrier\u271d, is_open' := is_open'\u271d } = sV\n\u22a2 x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2227\n    { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2264 { carrier := sU, is_open' := hU }\n[PROOFSTEP]\ndsimp at H\u2082 \n[GOAL]\ncase mp.mk.intro.intro.intro.intro.mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\nsU : Set \u03b1\nhU : IsOpen sU\nx : \u03b1\nhx : x \u2208 { carrier := sU, is_open' := hU }\nsV : Set \u03b1\nhsV : x \u2208 sV \u2227 sV \u2286 sU\ncarrier\u271d : Set \u03b1\nis_open'\u271d : IsOpen carrier\u271d\nH\u2081 : { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2208 B\nH\u2082 : carrier\u271d = sV\n\u22a2 x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2227\n    { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2264 { carrier := sU, is_open' := hU }\n[PROOFSTEP]\nsubst H\u2082\n[GOAL]\ncase mp.mk.intro.intro.intro.intro.mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : IsBasis B\nsU : Set \u03b1\nhU : IsOpen sU\nx : \u03b1\nhx : x \u2208 { carrier := sU, is_open' := hU }\ncarrier\u271d : Set \u03b1\nis_open'\u271d : IsOpen carrier\u271d\nH\u2081 : { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2208 B\nhsV : x \u2208 carrier\u271d \u2227 carrier\u271d \u2286 sU\n\u22a2 x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2227\n    { carrier := carrier\u271d, is_open' := is_open'\u271d } \u2264 { carrier := sU, is_open' := hU }\n[PROOFSTEP]\nexact hsV\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n\u22a2 IsBasis B\n[PROOFSTEP]\nrefine' isTopologicalBasis_of_open_of_nhds _ _\n[GOAL]\ncase mpr.refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n\u22a2 \u2200 (u : Set \u03b1), u \u2208 SetLike.coe '' B \u2192 IsOpen u\n[PROOFSTEP]\nrintro sU \u27e8U, -, rfl\u27e9\n[GOAL]\ncase mpr.refine'_1.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\nU : Opens \u03b1\n\u22a2 IsOpen \u2191U\n[PROOFSTEP]\nexact U.2\n[GOAL]\ncase mpr.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n\u22a2 \u2200 (a : \u03b1) (u : Set \u03b1), a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 SetLike.coe '' B \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro x sU hx hsU\n[GOAL]\ncase mpr.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\nx : \u03b1\nsU : Set \u03b1\nhx : x \u2208 sU\nhsU : IsOpen sU\n\u22a2 \u2203 v, v \u2208 SetLike.coe '' B \u2227 x \u2208 v \u2227 v \u2286 sU\n[PROOFSTEP]\nrcases@h \u27e8sU, hsU\u27e9 x hx with \u27e8V, hV, H\u27e9\n[GOAL]\ncase mpr.refine'_2.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\nx : \u03b1\nsU : Set \u03b1\nhx : x \u2208 sU\nhsU : IsOpen sU\nV : Opens \u03b1\nhV : V \u2208 B\nH : x \u2208 V \u2227 V \u2264 { carrier := sU, is_open' := hsU }\n\u22a2 \u2203 v, v \u2208 SetLike.coe '' B \u2227 x \u2208 v \u2227 v \u2286 sU\n[PROOFSTEP]\nexact \u27e8V, \u27e8V, hV, rfl\u27e9, H\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\n\u22a2 IsBasis B \u2194 \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\n\u22a2 IsBasis B \u2192 \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\n[PROOFSTEP]\nintro hB U\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nhB : IsBasis B\nU : Opens \u03b1\n\u22a2 \u2203 Us, Us \u2286 B \u2227 U = sSup Us\n[PROOFSTEP]\nrefine \u27e8{V : Opens \u03b1 | V \u2208 B \u2227 V \u2264 U}, fun U hU => hU.left, ext ?_\u27e9\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nhB : IsBasis B\nU : Opens \u03b1\n\u22a2 \u2191U = \u2191(sSup {V | V \u2208 B \u2227 V \u2264 U})\n[PROOFSTEP]\nrw [coe_sSup, hB.open_eq_sUnion' U.isOpen]\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nhB : IsBasis B\nU : Opens \u03b1\n\u22a2 \u22c3\u2080 {s | s \u2208 SetLike.coe '' B \u2227 s \u2286 \u2191U} = \u22c3 (i : Opens \u03b1) (_ : i \u2208 {V | V \u2208 B \u2227 V \u2264 U}), \u2191i\n[PROOFSTEP]\nsimp_rw [sUnion_eq_biUnion, iUnion, mem_setOf_eq, iSup_and, iSup_image]\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nhB : IsBasis B\nU : Opens \u03b1\n\u22a2 \u2a06 (b : Opens \u03b1) (_ : b \u2208 B) (_ : \u2191b \u2286 \u2191U), \u2191b = \u2a06 (i : Opens \u03b1) (_ : i \u2208 B) (_ : i \u2264 U), \u2191i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\n\u22a2 (\u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us) \u2192 IsBasis B\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\n\u22a2 IsBasis B\n[PROOFSTEP]\nrw [isBasis_iff_nbhd]\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\n\u22a2 \u2200 {U : Opens \u03b1} {x : \u03b1}, x \u2208 U \u2192 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n[PROOFSTEP]\nintro U x hx\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\nU : Opens \u03b1\nx : \u03b1\nhx : x \u2208 U\n\u22a2 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 U\n[PROOFSTEP]\nrcases h U with \u27e8Us, hUs, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\nx : \u03b1\nUs : Set (Opens \u03b1)\nhUs : Us \u2286 B\nhx : x \u2208 sSup Us\n\u22a2 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 sSup Us\n[PROOFSTEP]\nrcases mem_sSup.1 hx with \u27e8U, Us, xU\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nB : Set (Opens \u03b1)\nh : \u2200 (U : Opens \u03b1), \u2203 Us, Us \u2286 B \u2227 U = sSup Us\nx : \u03b1\nUs\u271d : Set (Opens \u03b1)\nhUs : Us\u271d \u2286 B\nhx : x \u2208 sSup Us\u271d\nU : Opens \u03b1\nUs : U \u2208 Us\u271d\nxU : x \u2208 U\n\u22a2 \u2203 U', U' \u2208 B \u2227 x \u2208 U' \u2227 U' \u2264 sSup Us\u271d\n[PROOFSTEP]\nexact \u27e8U, hUs Us, xU, le_sSup Us\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_5\nb : \u03b9 \u2192 Opens \u03b1\nhb : IsBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact \u2191(b i)\nU : Set \u03b1\n\u22a2 IsCompact U \u2227 IsOpen U \u2194 \u2203 s, Set.Finite s \u2227 U = \u22c3 (i : \u03b9) (_ : i \u2208 s), \u2191(b i)\n[PROOFSTEP]\napply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : \u03b9 => (b i).1\n[GOAL]\ncase hb\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_5\nb : \u03b9 \u2192 Opens \u03b1\nhb : IsBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact \u2191(b i)\nU : Set \u03b1\n\u22a2 IsTopologicalBasis (range fun i => (b i).carrier)\n[PROOFSTEP]\nconvert (config := { transparency := .default }) hb\n[GOAL]\ncase h.e'_3\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_5\nb : \u03b9 \u2192 Opens \u03b1\nhb : IsBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact \u2191(b i)\nU : Set \u03b1\n\u22a2 (range fun i => (b i).carrier) = SetLike.coe '' range b\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_5\nb : \u03b9 \u2192 Opens \u03b1\nhb : IsBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact \u2191(b i)\nU x\u271d : Set \u03b1\n\u22a2 (x\u271d \u2208 range fun i => (b i).carrier) \u2194 x\u271d \u2208 SetLike.coe '' range b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hb'\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_5\nb : \u03b9 \u2192 Opens \u03b1\nhb : IsBasis (range b)\nhb' : \u2200 (i : \u03b9), IsCompact \u2191(b i)\nU : Set \u03b1\n\u22a2 \u2200 (i : \u03b9), IsCompact (b i).carrier\n[PROOFSTEP]\nexact hb'\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\n\u22a2 CompleteLattice.IsCompactElement s \u2194 IsCompact \u2191s\n[PROOFSTEP]\nrw [isCompact_iff_finite_subcover, CompleteLattice.isCompactElement_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\n\u22a2 (\u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1) \u2194\n    \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n      (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrefine' \u27e8_, fun H \u03b9 U hU => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\n\u22a2 (\u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1) \u2192\n    \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n      (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nintrov H hU hU'\n[GOAL]\ncase refine'_1\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH : \u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Set \u03b1\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nhU' : \u2191s \u2286 \u22c3 (i : \u03b9), U i\n\u22a2 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := H \u03b9 (fun i => \u27e8U i, hU i\u27e9) (by simpa)\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH : \u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Set \u03b1\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nhU' : \u2191s \u2286 \u22c3 (i : \u03b9), U i\n\u22a2 s \u2264 \u2a06 (i : \u03b9), { carrier := U i, is_open' := (_ : IsOpen (U i)) }\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_1.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH : \u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Set \u03b1\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nhU' : \u2191s \u2286 \u22c3 (i : \u03b9), U i\nt : Finset \u03b9\nht : s \u2264 Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }\n\u22a2 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrefine' \u27e8t, Set.Subset.trans ht _\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH : \u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Set \u03b1\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nhU' : \u2191s \u2286 \u22c3 (i : \u03b9), U i\nt : Finset \u03b9\nht : s \u2264 Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }\n\u22a2 \u2191(Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }) \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrw [coe_finset_sup, Finset.sup_eq_iSup]\n[GOAL]\ncase refine'_1.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH : \u2200 (\u03b9 : Type u_2) (s_1 : \u03b9 \u2192 Opens \u03b1), s \u2264 iSup s_1 \u2192 \u2203 t, s \u2264 Finset.sup t s_1\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Set \u03b1\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nhU' : \u2191s \u2286 \u22c3 (i : \u03b9), U i\nt : Finset \u03b9\nht : s \u2264 Finset.sup t fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }\n\u22a2 \u2a06 (a : \u03b9) (_ : a \u2208 t), (SetLike.coe \u2218 fun i => { carrier := U i, is_open' := (_ : IsOpen (U i)) }) a \u2286\n    \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH :\n  \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n    (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Opens \u03b1\nhU : s \u2264 iSup U\n\u22a2 \u2203 t, s \u2264 Finset.sup t U\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := H (fun i => U i) (fun i => (U i).isOpen) (by simpa using show (s : Set \u03b1) \u2286 \u2191(iSup U) from hU)\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH :\n  \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n    (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Opens \u03b1\nhU : s \u2264 iSup U\n\u22a2 \u2191s \u2286 \u22c3 (i : \u03b9), (fun i => \u2191(U i)) i\n[PROOFSTEP]\nsimpa using show (s : Set \u03b1) \u2286 \u2191(iSup U) from hU\n[GOAL]\ncase refine'_2.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH :\n  \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n    (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Opens \u03b1\nhU : s \u2264 iSup U\nt : Finset \u03b9\nht : \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), \u2191(U i)\n\u22a2 \u2203 t, s \u2264 Finset.sup t U\n[PROOFSTEP]\nrefine' \u27e8t, Set.Subset.trans ht _\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH :\n  \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n    (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Opens \u03b1\nhU : s \u2264 iSup U\nt : Finset \u03b9\nht : \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), \u2191(U i)\n\u22a2 \u22c3 (i : \u03b9) (_ : i \u2208 t), \u2191(U i) \u2286 \u2191(Finset.sup t U)\n[PROOFSTEP]\nsimp only [Set.iUnion_subset_iff]\n[GOAL]\ncase refine'_2.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH :\n  \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n    (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Opens \u03b1\nhU : s \u2264 iSup U\nt : Finset \u03b9\nht : \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), \u2191(U i)\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 \u2191(U i) \u2286 \u2191(Finset.sup t U)\n[PROOFSTEP]\nshow \u2200 i \u2208 t, U i \u2264 t.sup U\n[GOAL]\ncase refine'_2.intro\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Opens \u03b1\nH :\n  \u2200 {\u03b9 : Type u_2} (U : \u03b9 \u2192 Set \u03b1),\n    (\u2200 (i : \u03b9), IsOpen (U i)) \u2192 \u2191s \u2286 \u22c3 (i : \u03b9), U i \u2192 \u2203 t, \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), U i\n\u03b9 : Type u_2\nU : \u03b9 \u2192 Opens \u03b1\nhU : s \u2264 iSup U\nt : Finset \u03b9\nht : \u2191s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), \u2191(U i)\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 U i \u2264 Finset.sup t U\n[PROOFSTEP]\nexact fun i => Finset.le_sup\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : C(\u03b1, \u03b2)\ns : Set (Opens \u03b2)\n\u22a2 \u2191(InfHom.toFun\n        {\n            toInfHom :=\n              { toFun := fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) },\n                map_inf' :=\n                  (_ :\n                    \u2200 (a b : Opens \u03b2),\n                      (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) }) (a \u2293 b) =\n                        (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) }) (a \u2293 b)) },\n            map_top' :=\n              (_ :\n                InfHom.toFun\n                    { toFun := fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) },\n                      map_inf' :=\n                        (_ :\n                          \u2200 (a b : Opens \u03b2),\n                            (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) }) (a \u2293 b) =\n                              (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) })\n                                (a \u2293 b)) }\n                    \u22a4 =\n                  InfHom.toFun\n                    { toFun := fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) },\n                      map_inf' :=\n                        (_ :\n                          \u2200 (a b : Opens \u03b2),\n                            (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) }) (a \u2293 b) =\n                              (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) })\n                                (a \u2293 b)) }\n                    \u22a4) }.toInfHom\n        (sSup s)) =\n    \u2191(sSup\n        ({\n                toInfHom :=\n                  { toFun := fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) },\n                    map_inf' :=\n                      (_ :\n                        \u2200 (a b : Opens \u03b2),\n                          (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) }) (a \u2293 b) =\n                            (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) }) (a \u2293 b)) },\n                map_top' :=\n                  (_ :\n                    InfHom.toFun\n                        { toFun := fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) },\n                          map_inf' :=\n                            (_ :\n                              \u2200 (a b : Opens \u03b2),\n                                (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) })\n                                    (a \u2293 b) =\n                                  (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) })\n                                    (a \u2293 b)) }\n                        \u22a4 =\n                      InfHom.toFun\n                        { toFun := fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) },\n                          map_inf' :=\n                            (_ :\n                              \u2200 (a b : Opens \u03b2),\n                                (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) })\n                                    (a \u2293 b) =\n                                  (fun s => { carrier := \u2191f \u207b\u00b9' \u2191s, is_open' := (_ : IsOpen (\u2191f \u207b\u00b9' s.carrier)) })\n                                    (a \u2293 b)) }\n                        \u22a4) }.toInfHom.toFun ''\n          s))\n[PROOFSTEP]\nsimp only [coe_sSup, preimage_iUnion, biUnion_image, coe_mk]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u2243\u209c \u03b2\n\u22a2 \u2200 {a b : Opens \u03b1},\n    \u2191{ toFun := \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))),\n              invFun := \u2191(comap (Homeomorph.toContinuousMap f)),\n              left_inv :=\n                (_ :\n                  \u2200 (U : Opens \u03b1),\n                    \u2191(comap (Homeomorph.toContinuousMap f))\n                        (\u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))) U) =\n                      U),\n              right_inv :=\n                (_ :\n                  \u2200 (U : Opens \u03b2),\n                    \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f)))\n                        (\u2191(comap (Homeomorph.toContinuousMap f)) U) =\n                      U) }\n          a \u2264\n        \u2191{ toFun := \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))),\n              invFun := \u2191(comap (Homeomorph.toContinuousMap f)),\n              left_inv :=\n                (_ :\n                  \u2200 (U : Opens \u03b1),\n                    \u2191(comap (Homeomorph.toContinuousMap f))\n                        (\u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))) U) =\n                      U),\n              right_inv :=\n                (_ :\n                  \u2200 (U : Opens \u03b2),\n                    \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f)))\n                        (\u2191(comap (Homeomorph.toContinuousMap f)) U) =\n                      U) }\n          b \u2194\n      a \u2264 b\n[PROOFSTEP]\nsimp only [\u2190 SetLike.coe_subset_coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u2243\u209c \u03b2\n\u22a2 \u2200 {a b : Opens \u03b1},\n    \u2191(\u2191{ toFun := \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))),\n                invFun := \u2191(comap (Homeomorph.toContinuousMap f)),\n                left_inv :=\n                  (_ :\n                    \u2200 (U : Opens \u03b1),\n                      \u2191(comap (Homeomorph.toContinuousMap f))\n                          (\u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))) U) =\n                        U),\n                right_inv :=\n                  (_ :\n                    \u2200 (U : Opens \u03b2),\n                      \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f)))\n                          (\u2191(comap (Homeomorph.toContinuousMap f)) U) =\n                        U) }\n            a) \u2286\n        \u2191(\u2191{ toFun := \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))),\n                invFun := \u2191(comap (Homeomorph.toContinuousMap f)),\n                left_inv :=\n                  (_ :\n                    \u2200 (U : Opens \u03b1),\n                      \u2191(comap (Homeomorph.toContinuousMap f))\n                          (\u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f))) U) =\n                        U),\n                right_inv :=\n                  (_ :\n                    \u2200 (U : Opens \u03b2),\n                      \u2191(comap (Homeomorph.toContinuousMap (Homeomorph.symm f)))\n                          (\u2191(comap (Homeomorph.toContinuousMap f)) U) =\n                        U) }\n            b) \u2194\n      \u2191a \u2286 \u2191b\n[PROOFSTEP]\nexact f.symm.surjective.preimage_subset_preimage_iff\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sets.Opens", "llama_tokens": 14218, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.831143031127974, "lm_q2_score": 0.7185943925708561, "lm_q1q2_score": 0.5972547215929067}}
{"text": "[GOAL]\n\u22a2 \u2191LinearMap.det (AlgEquiv.toLinearMap conjAe) = -1\n[PROOFSTEP]\nrw [\u2190 LinearMap.det_toMatrix basisOneI, toMatrix_conjAe, Matrix.det_fin_two_of]\n[GOAL]\n\u22a2 1 * -1 - 0 * 0 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2191LinearEquiv.det (AlgEquiv.toLinearEquiv conjAe) = -1\n[PROOFSTEP]\nrw [\u2190 Units.eq_iff, LinearEquiv.coe_det, AlgEquiv.toLinearEquiv_toLinearMap, det_conjAe, Units.coe_neg_one]\n", "meta": {"mathlib_filename": "Mathlib.Data.Complex.Determinant", "llama_tokens": 183, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8539127529517042, "lm_q2_score": 0.6992544335934766, "lm_q1q2_score": 0.5971022784034903}}
{"text": "[GOAL]\nm n : \u2115\n\u22a2 \u2191m - \u2191n \u2264 \u2191(m - n)\n[PROOFSTEP]\nby_cases h : m \u2265 n\n[GOAL]\ncase pos\nm n : \u2115\nh : m \u2265 n\n\u22a2 \u2191m - \u2191n \u2264 \u2191(m - n)\n[PROOFSTEP]\nexact le_of_eq (Int.ofNat_sub h).symm\n[GOAL]\ncase neg\nm n : \u2115\nh : \u00acm \u2265 n\n\u22a2 \u2191m - \u2191n \u2264 \u2191(m - n)\n[PROOFSTEP]\nsimp [le_of_not_ge h, ofNat_le]\n[GOAL]\nn : \u2115\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 a ^ 2 = b ^ 2\n[PROOFSTEP]\nrw [sq, sq]\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]\nexact natAbs_eq_iff_mul_self_eq\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a < natAbs b \u2194 a ^ 2 < b ^ 2\n[PROOFSTEP]\nrw [sq, sq]\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]\nexact natAbs_lt_iff_mul_self_lt\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a \u2264 natAbs b \u2194 a ^ 2 \u2264 b ^ 2\n[PROOFSTEP]\nrw [sq, sq]\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]\nexact natAbs_le_iff_mul_self_le\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 natAbs a = natAbs b \u2194 a = b\n[PROOFSTEP]\nrw [\u2190 sq_eq_sq ha hb, \u2190 natAbs_eq_iff_sq_eq]\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : a \u2264 0\nhb : b \u2264 0\n\u22a2 natAbs a = natAbs b \u2194 a = b\n[PROOFSTEP]\nsimpa only [Int.natAbs_neg, neg_inj] using\n  natAbs_inj_of_nonneg_of_nonneg (neg_nonneg_of_nonpos ha) (neg_nonneg_of_nonpos hb)\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : 0 \u2264 a\nhb : b \u2264 0\n\u22a2 natAbs a = natAbs b \u2194 a = -b\n[PROOFSTEP]\nsimpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg ha (neg_nonneg_of_nonpos hb)\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : a \u2264 0\nhb : 0 \u2264 b\n\u22a2 natAbs a = natAbs b \u2194 -a = b\n[PROOFSTEP]\nsimpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg (neg_nonneg_of_nonpos ha) hb\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na : \u2124\nx\u271d : a \u2208 Iic 0\nb : \u2124\nhb : b \u2208 Iic 0\nhab : a < b\n\u22a2 natAbs b < natAbs a\n[PROOFSTEP]\nsimpa [Int.natAbs_neg] using natAbs_lt_natAbs_of_nonneg_of_lt (Right.nonneg_neg_iff.mpr hb) (neg_lt_neg_iff.mpr hab)\n[GOAL]\na b : \u2124\nn\u271d n : \u2115\nh : \u2191(n + 1) \u2264 0\n\u22a2 0 < \u2191(n + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\na b\u271d : \u2124\nn\u271d : \u2115\nb : Bool\nn : \u2124\n\u22a2 div2 (bit b n) = n\n[PROOFSTEP]\nrw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : \u2124) = 0), zero_add]\n[GOAL]\na b\u271d : \u2124\nn\u271d : \u2115\nb : Bool\nn : \u2124\n\u22a2 (bif b then 1 else 0) / 2 = 0\ncase H a b\u271d : \u2124 n\u271d : \u2115 b : Bool n : \u2124 \u22a2 2 \u2260 0\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 (bif false then 1 else 0) / 2 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 (bif true then 1 else 0) / 2 = 0\n[PROOFSTEP]\nshow ofNat _ = _\n[GOAL]\ncase true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 ofNat (1 / 2) = 0\n[PROOFSTEP]\nrw [Nat.div_eq_zero]\n[GOAL]\ncase true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 ofNat 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\na b : \u2124\nn\u271d : \u2115\nn : \u2124\n\u22a2 1 < 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H\na b\u271d : \u2124\nn\u271d : \u2115\nb : Bool\nn : \u2124\n\u22a2 2 \u2260 0\n[PROOFSTEP]\ndecide\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Lemmas", "llama_tokens": 1778, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.853912760387131, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.5971022675474117}}
{"text": "[GOAL]\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 IsMinimal M \u03b1 \u2194 \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 IsMinimal M \u03b1 \u2192 \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\n[PROOFSTEP]\nintro _ _\n[GOAL]\ncase mp\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\na\u271d : IsMinimal M \u03b1\ns\u271d : Set \u03b1\n\u22a2 IsClosed s\u271d \u2192 (\u2200 (c : M), c \u2022 s\u271d \u2286 s\u271d) \u2192 s\u271d = \u2205 \u2228 s\u271d = univ\n[PROOFSTEP]\nexact eq_empty_or_univ_of_smul_invariant_closed M\n[GOAL]\ncase mpr\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\n\u22a2 (\u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ) \u2192 IsMinimal M \u03b1\n[PROOFSTEP]\nrefine' fun H \u21a6 \u27e8fun _ \u21a6 dense_iff_closure_eq.2 <| (H _ _ _).resolve_left _\u27e9\n[GOAL]\ncase mpr.refine'_1\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 IsClosed (closure (orbit M x\u271d))\ncase mpr.refine'_2\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 \u2200 (c : M), c \u2022 closure (orbit M x\u271d) \u2286 closure (orbit M x\u271d)\ncase mpr.refine'_3\nM : Type u_1\nG : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MulAction M \u03b1\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : ContinuousConstSMul M \u03b1\nH : \u2200 (s : Set \u03b1), IsClosed s \u2192 (\u2200 (c : M), c \u2022 s \u2286 s) \u2192 s = \u2205 \u2228 s = univ\nx\u271d : \u03b1\n\u22a2 \u00acclosure (orbit M x\u271d) = \u2205\n[PROOFSTEP]\nexacts [isClosed_closure, fun _ \u21a6 smul_closure_orbit_subset _ _, (orbit_nonempty _).closure.ne_empty]\n", "meta": {"mathlib_filename": "Mathlib.Dynamics.Minimal", "llama_tokens": 1232, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127492339909, "lm_q2_score": 0.6992544210587585, "lm_q1q2_score": 0.5971022651003071}}
{"text": "[GOAL]\nq : \u211a\n\u22a2 sqrt (q * q) = |q|\n[PROOFSTEP]\nrw [sqrt, mul_self_num, mul_self_den, Int.sqrt_eq, Nat.sqrt_eq, abs_def, divInt_ofNat]\n[GOAL]\nx : \u211a\nx\u271d : \u2203 q, q * q = x\nn : \u211a\nhn : n * n = x\n\u22a2 sqrt x * sqrt x = x\n[PROOFSTEP]\nrw [\u2190 hn, sqrt_eq, abs_mul_abs_self]\n", "meta": {"mathlib_filename": "Mathlib.Data.Rat.Sqrt", "llama_tokens": 146, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8918110511888303, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.5965148140074176}}
{"text": "[GOAL]\nm : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\n\u22a2 det (mvPolynomialX m m R) \u2260 0\n[PROOFSTEP]\nintro h_det\n[GOAL]\nm : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nh_det : det (mvPolynomialX m m R) = 0\n\u22a2 False\n[PROOFSTEP]\nhave := congr_arg Matrix.det (mvPolynomialX_mapMatrix_eval (1 : Matrix m m R))\n[GOAL]\nm : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nh_det : det (mvPolynomialX m m R) = 0\nthis : det (\u2191(RingHom.mapMatrix (MvPolynomial.eval fun p => OfNat.ofNat 1 p.fst p.snd)) (mvPolynomialX m m R)) = det 1\n\u22a2 False\n[PROOFSTEP]\nrw [det_one, \u2190 RingHom.map_det, h_det, RingHom.map_zero] at this \n[GOAL]\nm : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\nh_det : det (mvPolynomialX m m R) = 0\nthis : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one this\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "llama_tokens": 549, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677430095496, "lm_q2_score": 0.7025300636233416, "lm_q1q2_score": 0.5957228324470403}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : GroupWithZero \u03b2\nsrc\u271d\u00b9 : DivInvMonoid (Germ (\u2191\u03c6) \u03b2) := divInvMonoid\nsrc\u271d : MonoidWithZero (Germ (\u2191\u03c6) \u03b2) := monoidWithZero\n\u22a2 ((fun x x_1 => x \u2218 x_1) Inv.inv fun x => 0) =\u1da0[\u2191\u03c6] fun x => 0\n[PROOFSTEP]\nsimp only [Function.comp, inv_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : GroupWithZero \u03b2\nsrc\u271d\u00b9 : DivInvMonoid (Germ (\u2191\u03c6) \u03b2) := divInvMonoid\nsrc\u271d : MonoidWithZero (Germ (\u2191\u03c6) \u03b2) := monoidWithZero\n\u22a2 (fun x => 0) =\u1da0[\u2191\u03c6] fun x => 0\n[PROOFSTEP]\nexact EventuallyEq.refl _ fun _ => 0\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\n\u22a2 \u2191f < \u2191g \u2194 \u2200* (x : \u03b1), f x < g x\n[PROOFSTEP]\nsimp only [lt_iff_le_not_le, eventually_and, coe_le, eventually_not, EventuallyLE]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : Preorder \u03b2\n\u22a2 (fun x x_1 => x < x_1) = LiftRel fun x x_1 => x < x_1\n[PROOFSTEP]\next \u27e8f\u27e9 \u27e8g\u27e9\n[GOAL]\ncase h.mk.h.mk.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : Preorder \u03b2\nx\u271d\u00b9 : Germ (\u2191\u03c6) \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d : Germ (\u2191\u03c6) \u03b2\ng : \u03b1 \u2192 \u03b2\n\u22a2 Quot.mk Setoid.r f < Quot.mk Setoid.r g \u2194 LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)\n[PROOFSTEP]\nexact coe_lt\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\n\u22a2 max \u2191a \u2191b = map\u2082 max \u2191a \u2191b\n[PROOFSTEP]\ncases' le_total (a : \u03b2*) b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191a \u2264 \u2191b\n\u22a2 max \u2191a \u2191b = map\u2082 max \u2191a \u2191b\n[PROOFSTEP]\nrw [max_eq_right h, map\u2082_coe, coe_eq]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191a \u2264 \u2191b\n\u22a2 b =\u1da0[\u2191\u03c6] fun x => max (a x) (b x)\n[PROOFSTEP]\nexact h.mono fun i hi => (max_eq_right hi).symm\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191b \u2264 \u2191a\n\u22a2 max \u2191a \u2191b = map\u2082 max \u2191a \u2191b\n[PROOFSTEP]\nrw [max_eq_left h, map\u2082_coe, coe_eq]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191b \u2264 \u2191a\n\u22a2 a =\u1da0[\u2191\u03c6] fun x => max (a x) (b x)\n[PROOFSTEP]\nexact h.mono fun i hi => (max_eq_left hi).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\nK : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\n\u22a2 min \u2191a \u2191b = map\u2082 min \u2191a \u2191b\n[PROOFSTEP]\ncases' le_total (a : \u03b2*) b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\nK : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191a \u2264 \u2191b\n\u22a2 min \u2191a \u2191b = map\u2082 min \u2191a \u2191b\n[PROOFSTEP]\nrw [min_eq_left h, map\u2082_coe, coe_eq]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\nK : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191a \u2264 \u2191b\n\u22a2 a =\u1da0[\u2191\u03c6] fun x => min (a x) (b x)\n[PROOFSTEP]\nexact h.mono fun i hi => (min_eq_left hi).symm\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\nK : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191b \u2264 \u2191a\n\u22a2 min \u2191a \u2191b = map\u2082 min \u2191a \u2191b\n[PROOFSTEP]\nrw [min_eq_right h, map\u2082_coe, coe_eq]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\nK : LinearOrder \u03b2\nx y : Germ (\u2191\u03c6) \u03b2\na b : \u03b1 \u2192 \u03b2\nh : \u2191b \u2264 \u2191a\n\u22a2 b =\u1da0[\u2191\u03c6] fun x => min (a x) (b x)\n[PROOFSTEP]\nexact h.mono fun i hi => (min_eq_right hi).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : \u03b2\n\u22a2 \u2191(max x y) = max \u2191x \u2191y\n[PROOFSTEP]\nrw [max_def, map\u2082_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrder \u03b2\nx y : \u03b2\n\u22a2 \u2191(min x y) = min \u2191x \u2191y\n[PROOFSTEP]\nrw [min_def, map\u2082_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c6 : Ultrafilter \u03b1\ninst\u271d : LinearOrderedAddCommGroup \u03b2\nx : \u03b2\n\u22a2 \u2191|x| = |\u2191x|\n[PROOFSTEP]\nrw [abs_def, map_const]\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.FilterProduct", "llama_tokens": 1923, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388167733099, "lm_q2_score": 0.7185943985973773, "lm_q1q2_score": 0.5956707905132381}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\n\u22a2 sqrt (1 + \u2016x\u2016 ^ 2) \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\nrw [sqrt_le_left (by positivity)]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\n\u22a2 0 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\n\u22a2 1 + \u2016x\u2016 ^ 2 \u2264 (1 + \u2016x\u2016) ^ 2\n[PROOFSTEP]\nsimp [add_sq]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\n\u22a2 1 + \u2016x\u2016 \u2264 sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\nrw [\u2190 sqrt_mul zero_le_two]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\n\u22a2 1 + \u2016x\u2016 \u2264 sqrt (2 * (1 + \u2016x\u2016 ^ 2))\n[PROOFSTEP]\nhave := sq_nonneg (\u2016x\u2016 - 1)\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\nthis : 0 \u2264 (\u2016x\u2016 - 1) ^ 2\n\u22a2 1 + \u2016x\u2016 \u2264 sqrt (2 * (1 + \u2016x\u2016 ^ 2))\n[PROOFSTEP]\napply le_sqrt_of_sq_le\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nx : E\nthis : 0 \u2264 (\u2016x\u2016 - 1) ^ 2\n\u22a2 (1 + \u2016x\u2016) ^ 2 \u2264 2 * (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 (1 + \u2016x\u2016 ^ 2) ^ (-r / 2) = 2 ^ (r / 2) * ((sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2)) ^ r)\u207b\u00b9\n[PROOFSTEP]\nrw [rpow_div_two_eq_sqrt, rpow_div_two_eq_sqrt, mul_rpow, mul_inv, rpow_neg, mul_inv_cancel_left\u2080]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 sqrt 2 ^ r \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase hx\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 0 \u2264 sqrt (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 0 \u2264 sqrt 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h\u2081\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 0 \u2264 sqrt (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase hx\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 0 \u2264 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase hx\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 0 \u2264 1 + \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 2 ^ (r / 2) * ((sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2)) ^ r)\u207b\u00b9 \u2264 2 ^ (r / 2) * ((1 + \u2016x\u2016) ^ r)\u207b\u00b9\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.h.h\u2081\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 1 + \u2016x\u2016 \u2264 sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\napply one_add_norm_le_sqrt_two_mul_sqrt\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 2 ^ (r / 2) * ((1 + \u2016x\u2016) ^ r)\u207b\u00b9 = 2 ^ (r / 2) * (1 + \u2016x\u2016) ^ (-r)\n[PROOFSTEP]\nrw [rpow_neg]\n[GOAL]\ncase hx\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr : \u211d\nx : E\nhr : 0 < r\n\u22a2 0 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr t : \u211d\nhr : 0 < r\nht : 0 < t\nx : E\n\u22a2 t \u2264 (1 + \u2016x\u2016) ^ (-r) \u2194 \u2016x\u2016 \u2264 t ^ (-r\u207b\u00b9) - 1\n[PROOFSTEP]\nrw [le_sub_iff_add_le', neg_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr t : \u211d\nhr : 0 < r\nht : 0 < t\nx : E\n\u22a2 t \u2264 (1 + \u2016x\u2016) ^ (-r) \u2194 1 + \u2016x\u2016 \u2264 t ^ (-r)\u207b\u00b9\n[PROOFSTEP]\nexact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr t : \u211d\nhr : 0 < r\nht : 0 < t\nx : E\n\u22a2 0 < 1 + \u2016x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr t : \u211d\nhr : 0 < r\nht : 1 < t\n\u22a2 Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1) = \u2205\n[PROOFSTEP]\nrw [Metric.closedBall_eq_empty, sub_neg]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr t : \u211d\nhr : 0 < r\nht : 1 < t\n\u22a2 t ^ (-r\u207b\u00b9) < 1\n[PROOFSTEP]\nexact Real.rpow_lt_one_of_one_lt_of_neg ht (by simp only [hr, Right.neg_neg_iff, inv_pos])\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nr t : \u211d\nhr : 0 < r\nht : 1 < t\n\u22a2 -r\u207b\u00b9 < 0\n[PROOFSTEP]\nsimp only [hr, Right.neg_neg_iff, inv_pos]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\n\u22a2 \u222b\u207b (x : \u211d) in Ioc 0 1, ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) < \u22a4\n[PROOFSTEP]\nhave hr : 0 < r := lt_of_le_of_lt n.cast_nonneg hnr\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\n\u22a2 \u222b\u207b (x : \u211d) in Ioc 0 1, ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) < \u22a4\n[PROOFSTEP]\nhave h_int : \u2200 x : \u211d, x \u2208 Ioc (0 : \u211d) 1 \u2192 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * n))) :=\n  fun x hx \u21a6 by\n  apply ENNReal.ofReal_le_ofReal\n  rw [\u2190 neg_mul, rpow_mul hx.1.le, rpow_nat_cast]\n  refine' pow_le_pow_of_le_left _ (by simp only [sub_le_self_iff, zero_le_one]) n\n  rw [le_sub_iff_add_le', add_zero]\n  refine' Real.one_le_rpow_of_pos_of_le_one_of_nonpos hx.1 hx.2 _\n  rw [Right.neg_nonpos_iff, inv_nonneg]\n  exact hr.le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n)))\n[PROOFSTEP]\napply ENNReal.ofReal_le_ofReal\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 (x ^ (-r\u207b\u00b9) - 1) ^ n \u2264 x ^ (-(r\u207b\u00b9 * \u2191n))\n[PROOFSTEP]\nrw [\u2190 neg_mul, rpow_mul hx.1.le, rpow_nat_cast]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 (x ^ (-r\u207b\u00b9) - 1) ^ n \u2264 (x ^ (-r\u207b\u00b9)) ^ n\n[PROOFSTEP]\nrefine' pow_le_pow_of_le_left _ (by simp only [sub_le_self_iff, zero_le_one]) n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 x ^ (-r\u207b\u00b9) - 1 \u2264 x ^ (-r\u207b\u00b9)\n[PROOFSTEP]\nsimp only [sub_le_self_iff, zero_le_one]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 0 \u2264 x ^ (-r\u207b\u00b9) - 1\n[PROOFSTEP]\nrw [le_sub_iff_add_le', add_zero]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 1 \u2264 x ^ (-r\u207b\u00b9)\n[PROOFSTEP]\nrefine' Real.one_le_rpow_of_pos_of_le_one_of_nonpos hx.1 hx.2 _\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 -r\u207b\u00b9 \u2264 0\n[PROOFSTEP]\nrw [Right.neg_nonpos_iff, inv_nonneg]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 0 \u2264 r\n[PROOFSTEP]\nexact hr.le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nh_int : \u2200 (x : \u211d), x \u2208 Ioc 0 1 \u2192 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n)))\n\u22a2 \u222b\u207b (x : \u211d) in Ioc 0 1, ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) < \u22a4\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (set_lintegral_mono' measurableSet_Ioc h_int) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nh_int : \u2200 (x : \u211d), x \u2208 Ioc 0 1 \u2192 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n)))\n\u22a2 \u222b\u207b (x : \u211d) in Ioc 0 1, ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n))) < \u22a4\n[PROOFSTEP]\nrefine' IntegrableOn.set_lintegral_lt_top _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nh_int : \u2200 (x : \u211d), x \u2208 Ioc 0 1 \u2192 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n)))\n\u22a2 IntegrableOn (fun x => x ^ (-(r\u207b\u00b9 * \u2191n))) (Ioc 0 1)\n[PROOFSTEP]\nrw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le zero_le_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nh_int : \u2200 (x : \u211d), x \u2208 Ioc 0 1 \u2192 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n)))\n\u22a2 IntervalIntegrable (fun x => x ^ (-(r\u207b\u00b9 * \u2191n))) volume 0 1\n[PROOFSTEP]\napply intervalIntegral.intervalIntegrable_rpow'\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nr : \u211d\nn : \u2115\nhnr : \u2191n < r\nhr : 0 < r\nh_int : \u2200 (x : \u211d), x \u2208 Ioc 0 1 \u2192 ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * \u2191n)))\n\u22a2 -1 < -(r\u207b\u00b9 * \u2191n)\n[PROOFSTEP]\nrwa [neg_lt_neg_iff, inv_mul_lt_iff' hr, one_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\n\u22a2 \u222b\u207b (x : E), ENNReal.ofReal ((1 + \u2016x\u2016) ^ (-r)) < \u22a4\n[PROOFSTEP]\nhave hr : 0 < r := lt_of_le_of_lt (finrank \u211d E).cast_nonneg hnr\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\n\u22a2 \u222b\u207b (x : E), ENNReal.ofReal ((1 + \u2016x\u2016) ^ (-r)) < \u22a4\n[PROOFSTEP]\nhave h_meas : Measurable fun \u03c9 : E => (1 + \u2016\u03c9\u2016) ^ (-r) :=\n  -- porting note: was `by measurability`(measurable_norm.const_add _).pow_const _\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\n\u22a2 \u222b\u207b (x : E), ENNReal.ofReal ((1 + \u2016x\u2016) ^ (-r)) < \u22a4\n[PROOFSTEP]\nhave h_pos : \u2200 x : E, 0 \u2264 (1 + \u2016x\u2016) ^ (-r) := fun x \u21a6 by positivity\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nx : E\n\u22a2 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\n\u22a2 \u222b\u207b (x : E), ENNReal.ofReal ((1 + \u2016x\u2016) ^ (-r)) < \u22a4\n[PROOFSTEP]\nrw [lintegral_eq_lintegral_meas_le volume h_pos h_meas]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} < \u22a4\n[PROOFSTEP]\nhave h_int : \u2200 t, 0 < t \u2192 volume {a : E | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = volume (Metric.closedBall (0 : E) (t ^ (-r\u207b\u00b9) - 1)) :=\n  fun t ht \u21a6 by\n  congr 1\n  ext x\n  simp only [mem_setOf_eq, mem_closedBall_zero_iff]\n  exact le_rpow_one_add_norm_iff_norm_le hr (mem_Ioi.mp ht) x\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nt : \u211d\nht : 0 < t\n\u22a2 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nt : \u211d\nht : 0 < t\n\u22a2 {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1)\n[PROOFSTEP]\next x\n[GOAL]\ncase e_a.h\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nt : \u211d\nht : 0 < t\nx : E\n\u22a2 x \u2208 {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} \u2194 x \u2208 Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1)\n[PROOFSTEP]\nsimp only [mem_setOf_eq, mem_closedBall_zero_iff]\n[GOAL]\ncase e_a.h\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nt : \u211d\nht : 0 < t\nx : E\n\u22a2 t \u2264 (1 + \u2016x\u2016) ^ (-r) \u2194 \u2016x\u2016 \u2264 t ^ (-r\u207b\u00b9) - 1\n[PROOFSTEP]\nexact le_rpow_one_add_norm_iff_norm_le hr (mem_Ioi.mp ht) x\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} < \u22a4\n[PROOFSTEP]\nrw [set_lintegral_congr_fun measurableSet_Ioi (eventually_of_forall h_int)]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\n\u22a2 \u222b\u207b (x : \u211d) in Ioi 0, \u2191\u2191volume (Metric.closedBall 0 (x ^ (-r\u207b\u00b9) - 1)) < \u22a4\n[PROOFSTEP]\nset f := fun t : \u211d \u21a6 volume (Metric.closedBall (0 : E) (t ^ (-r\u207b\u00b9) - 1))\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\n\u22a2 lintegral (Measure.restrict volume (Ioi 0)) f < \u22a4\n[PROOFSTEP]\nset mB :=\n  volume\n    (Metric.ball (0 : E) 1)\n      -- the next two inequalities are in fact equalities but we don't need that\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\n\u22a2 lintegral (Measure.restrict volume (Ioi 0)) f < \u22a4\n[PROOFSTEP]\ncalc\n  \u222b\u207b t in Ioi 0, f t \u2264 \u222b\u207b t in Ioc 0 1 \u222a Ioi 1, f t := lintegral_mono_set Ioi_subset_Ioc_union_Ioi\n  _ \u2264 (\u222b\u207b t in Ioc 0 1, f t) + \u222b\u207b t in Ioi 1, f t := (lintegral_union_le _ _ _)\n  _ < \u221e := ENNReal.add_lt_top.2 \u27e8?_, ?_\u27e9\n[GOAL]\ncase calc_1\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\n\u22a2 \u222b\u207b (t : \u211d) in Ioc 0 1, f t < \u22a4\n[PROOFSTEP]\nhave h_int' : \u2200 t \u2208 Ioc (0 : \u211d) 1, f t = ENNReal.ofReal ((t ^ (-r\u207b\u00b9) - 1) ^ finrank \u211d E) * mB := fun t ht \u21a6\n  by\n  refine' volume.addHaar_closedBall (0 : E) _\n  rw [sub_nonneg]\n  exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 f t = ENNReal.ofReal ((t ^ (-r\u207b\u00b9) - 1) ^ finrank \u211d E) * mB\n[PROOFSTEP]\nrefine' volume.addHaar_closedBall (0 : E) _\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 0 \u2264 t ^ (-r\u207b\u00b9) - 1\n[PROOFSTEP]\nrw [sub_nonneg]\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 1 \u2264 t ^ (-r\u207b\u00b9)\n[PROOFSTEP]\nexact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 -r\u207b\u00b9 \u2264 0\n[PROOFSTEP]\nsimp [hr.le]\n[GOAL]\ncase calc_1\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nh_int' : \u2200 (t : \u211d), t \u2208 Ioc 0 1 \u2192 f t = ENNReal.ofReal ((t ^ (-r\u207b\u00b9) - 1) ^ finrank \u211d E) * mB\n\u22a2 \u222b\u207b (t : \u211d) in Ioc 0 1, f t < \u22a4\n[PROOFSTEP]\nrw [set_lintegral_congr_fun measurableSet_Ioc (ae_of_all _ h_int'), lintegral_mul_const' _ _ measure_ball_lt_top.ne]\n[GOAL]\ncase calc_1\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nh_int' : \u2200 (t : \u211d), t \u2208 Ioc 0 1 \u2192 f t = ENNReal.ofReal ((t ^ (-r\u207b\u00b9) - 1) ^ finrank \u211d E) * mB\n\u22a2 (\u222b\u207b (a : \u211d) in Ioc 0 1, ENNReal.ofReal ((a ^ (-r\u207b\u00b9) - 1) ^ finrank \u211d E)) * \u2191\u2191volume (Metric.ball 0 1) < \u22a4\n[PROOFSTEP]\nexact ENNReal.mul_lt_top (finite_integral_rpow_sub_one_pow_aux (finrank \u211d E) hnr).ne measure_ball_lt_top.ne\n[GOAL]\ncase calc_2\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 1, f t < \u22a4\n[PROOFSTEP]\nhave h_int'' : \u2200 t \u2208 Ioi (1 : \u211d), f t = 0 := fun t ht => by\n  simp only [closedBall_rpow_sub_one_eq_empty_aux E hr ht, measure_empty]\n    -- The integral over the constant zero function is finite:\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nt : \u211d\nht : t \u2208 Ioi 1\n\u22a2 f t = 0\n[PROOFSTEP]\nsimp only [closedBall_rpow_sub_one_eq_empty_aux E hr ht, measure_empty]\n  -- The integral over the constant zero function is finite:\n[GOAL]\ncase calc_2\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nh_int'' : \u2200 (t : \u211d), t \u2208 Ioi 1 \u2192 f t = 0\n\u22a2 \u222b\u207b (t : \u211d) in Ioi 1, f t < \u22a4\n[PROOFSTEP]\nrw [set_lintegral_congr_fun measurableSet_Ioi (ae_of_all volume <| h_int''), lintegral_const 0, zero_mul]\n[GOAL]\ncase calc_2\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nh_meas : Measurable fun \u03c9 => (1 + \u2016\u03c9\u2016) ^ (-r)\nh_pos : \u2200 (x : E), 0 \u2264 (1 + \u2016x\u2016) ^ (-r)\nh_int : \u2200 (t : \u211d), 0 < t \u2192 \u2191\u2191volume {a | t \u2264 (1 + \u2016a\u2016) ^ (-r)} = \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nf : \u211d \u2192 \u211d\u22650\u221e := fun t => \u2191\u2191volume (Metric.closedBall 0 (t ^ (-r\u207b\u00b9) - 1))\nmB : \u211d\u22650\u221e := \u2191\u2191volume (Metric.ball 0 1)\nh_int'' : \u2200 (t : \u211d), t \u2208 Ioi 1 \u2192 f t = 0\n\u22a2 0 < \u22a4\n[PROOFSTEP]\nexact WithTop.zero_lt_top\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\n\u22a2 Integrable fun x => (1 + \u2016x\u2016) ^ (-r)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\n\u22a2 AEStronglyMeasurable (fun x => (1 + \u2016x\u2016) ^ (-r)) volume\n[PROOFSTEP]\nexact ((measurable_norm.const_add _).pow_const _).aestronglyMeasurable\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\n\u22a2 HasFiniteIntegral fun x => (1 + \u2016x\u2016) ^ (-r)\n[PROOFSTEP]\nhave : (\u222b\u207b a : E, \u2016(1 + \u2016a\u2016) ^ (-r)\u2016\u208a) = \u222b\u207b a : E, ENNReal.ofReal ((1 + \u2016a\u2016) ^ (-r)) :=\n  lintegral_nnnorm_eq_of_nonneg fun _ => rpow_nonneg_of_nonneg (by positivity) _\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nx\u271d : E\n\u22a2 0 \u2264 1 + \u2016x\u271d\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nthis : \u222b\u207b (a : E), \u2191\u2016(1 + \u2016a\u2016) ^ (-r)\u2016\u208a = \u222b\u207b (a : E), ENNReal.ofReal ((1 + \u2016a\u2016) ^ (-r))\n\u22a2 HasFiniteIntegral fun x => (1 + \u2016x\u2016) ^ (-r)\n[PROOFSTEP]\nrw [HasFiniteIntegral, this]\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nthis : \u222b\u207b (a : E), \u2191\u2016(1 + \u2016a\u2016) ^ (-r)\u2016\u208a = \u222b\u207b (a : E), ENNReal.ofReal ((1 + \u2016a\u2016) ^ (-r))\n\u22a2 \u222b\u207b (a : E), ENNReal.ofReal ((1 + \u2016a\u2016) ^ (-r)) < \u22a4\n[PROOFSTEP]\nexact finite_integral_one_add_norm hnr\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\n\u22a2 Integrable fun x => (1 + \u2016x\u2016 ^ 2) ^ (-r / 2)\n[PROOFSTEP]\nhave hr : 0 < r := lt_of_le_of_lt (finrank \u211d E).cast_nonneg hnr\n[GOAL]\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\n\u22a2 Integrable fun x => (1 + \u2016x\u2016 ^ 2) ^ (-r / 2)\n[PROOFSTEP]\nrefine ((integrable_one_add_norm hnr).const_mul <| (2 : \u211d) ^ (r / 2)).mono' ?_ (eventually_of_forall fun x => ?_)\n[GOAL]\ncase refine_1\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\n\u22a2 AEStronglyMeasurable (fun x => (1 + \u2016x\u2016 ^ 2) ^ (-r / 2)) volume\n[PROOFSTEP]\nexact (((measurable_id.norm.pow_const _).const_add _).pow_const _).aestronglyMeasurable\n[GOAL]\ncase refine_2\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nx : E\n\u22a2 \u2016(1 + \u2016x\u2016 ^ 2) ^ (-r / 2)\u2016 \u2264 2 ^ (r / 2) * (1 + \u2016x\u2016) ^ (-r)\n[PROOFSTEP]\nrefine (abs_of_pos ?_).trans_le (rpow_neg_one_add_norm_sq_le x hr)\n[GOAL]\ncase refine_2\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : MeasureSpace E\ninst\u271d\u00b9 : BorelSpace E\ninst\u271d : Measure.IsAddHaarMeasure volume\nr : \u211d\nhnr : \u2191(finrank \u211d E) < r\nhr : 0 < r\nx : E\n\u22a2 0 < (1 + \u2016x\u2016 ^ 2) ^ (-r / 2)\n[PROOFSTEP]\npositivity\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.JapaneseBracket", "llama_tokens": 15060, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950947024555, "lm_q2_score": 0.7057850216484838, "lm_q1q2_score": 0.5956085176836218}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\n\u22a2 length ((a :: as) {0 \u21a6 a\u271d}) = max (length (a :: as)) (0 + 1)\n[PROOFSTEP]\nrw [max_eq_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\n\u22a2 length ((a :: as) {0 \u21a6 a\u271d}) = length (a :: as)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\n\u22a2 0 + 1 \u2264 length (a :: as)\n[PROOFSTEP]\nsimp [Nat.le_add_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\n\u22a2 1 \u2264 Nat.succ (length as)\n[PROOFSTEP]\nexact Nat.succ_le_succ (Nat.zero_le _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nm : \u2115\n\u22a2 length ([] {m + 1 \u21a6 a}) = max (length []) (m + 1 + 1)\n[PROOFSTEP]\nhave := @length_set m []\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nm : \u2115\nthis : length ([] {m \u21a6 a}) = max (length []) (m + 1)\n\u22a2 length ([] {m + 1 \u21a6 a}) = max (length []) (m + 1 + 1)\n[PROOFSTEP]\nsimp [set, length, @length_set m, Nat.zero_max]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nm : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\n\u22a2 length ((head\u271d :: as) {m + 1 \u21a6 a}) = max (length (head\u271d :: as)) (m + 1 + 1)\n[PROOFSTEP]\nsimp [set, length, @length_set m, Nat.max_succ_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nk : \u2115\n\u22a2 get k [] = default\n[PROOFSTEP]\ncases k\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\n\u22a2 get Nat.zero [] = default\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn\u271d : \u2115\n\u22a2 get (Nat.succ n\u271d) [] = default\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nh1 : length (a :: as) \u2264 0\n\u22a2 get 0 (a :: as) = default\n[PROOFSTEP]\ncases h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nk : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : length (head\u271d :: as) \u2264 k + 1\n\u22a2 get (k + 1) (head\u271d :: as) = default\n[PROOFSTEP]\napply get_eq_default_of_le k\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nk : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : length (head\u271d :: as) \u2264 k + 1\n\u22a2 length as \u2264 k\n[PROOFSTEP]\nrw [\u2190 Nat.succ_le_succ_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nk : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : length (head\u271d :: as) \u2264 k + 1\n\u22a2 Nat.succ (length as) \u2264 Nat.succ k\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\n\u22a2 get 0 (as {0 \u21a6 a}) = a\n[PROOFSTEP]\ncases as\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\n\u22a2 get 0 ([] {0 \u21a6 a}) = a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na head\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 get 0 ((head\u271d :: tail\u271d) {0 \u21a6 a}) = a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nas : List \u03b1\n\u22a2 get (k + 1) (as {k + 1 \u21a6 a}) = a\n[PROOFSTEP]\ncases as\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\n\u22a2 get (k + 1) ([] {k + 1 \u21a6 a}) = a\n[PROOFSTEP]\nsimp [get_set]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 get (k + 1) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = a\n[PROOFSTEP]\nsimp [get_set]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nh : a \u2208 []\n\u22a2 \u2203 n, a = get n []\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a \u2208 b :: as\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\nrw [mem_cons] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a = b \u2228 a \u2208 as\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\ncases h with\n| inl h => exact \u27e80, h\u27e9\n| inr h =>\n  rcases eq_get_of_mem h with \u27e8n, h\u27e9\n  exact \u27e8n + 1, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a = b \u2228 a \u2208 as\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\ncases h with\n| inl h => exact \u27e80, h\u27e9\n| inr h =>\n  rcases eq_get_of_mem h with \u27e8n, h\u27e9\n  exact \u27e8n + 1, h\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a = b\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\n\n| inl h => exact \u27e80, h\u27e9\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a = b\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\nexact \u27e80, h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a \u2208 as\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\n\n| inr h =>\n  rcases eq_get_of_mem h with \u27e8n, h\u27e9\n  exact \u27e8n + 1, h\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh : a \u2208 as\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\nrcases eq_get_of_mem h with \u27e8n, h\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na b : \u03b1\nas : List \u03b1\nh\u271d : a \u2208 as\nn : \u2115\nh : a = get n as\n\u22a2 \u2203 n, a = get n (b :: as)\n[PROOFSTEP]\nexact \u27e8n + 1, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nx\u271d : \u2115\nh1 : x\u271d < length []\n\u22a2 get x\u271d [] \u2208 []\n[PROOFSTEP]\ncases h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nx\u271d : 0 < length (a :: as)\n\u22a2 get 0 (a :: as) \u2208 a :: as\n[PROOFSTEP]\nrw [mem_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nx\u271d : 0 < length (a :: as)\n\u22a2 get 0 (a :: as) = a \u2228 get 0 (a :: as) \u2208 as\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (a :: as)\n\u22a2 get (n + 1) (a :: as) \u2208 a :: as\n[PROOFSTEP]\nrw [mem_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (a :: as)\n\u22a2 get (n + 1) (a :: as) = a \u2228 get (n + 1) (a :: as) \u2208 as\n[PROOFSTEP]\napply Or.inr\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (a :: as)\n\u22a2 get (n + 1) (a :: as) \u2208 as\n[PROOFSTEP]\nunfold get\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (a :: as)\n\u22a2 get (Nat.add n 0) as \u2208 as\n[PROOFSTEP]\napply mem_get_of_le\n[GOAL]\ncase h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (a :: as)\n\u22a2 Nat.add n 0 < length as\n[PROOFSTEP]\napply Nat.lt_of_succ_lt_succ h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nx\u271d : \u2115\nh1 : get x\u271d [] \u2260 default\n\u22a2 get x\u271d [] \u2208 []\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nx\u271d : \u2115\nh1 : get x\u271d [] \u2260 default\n\u22a2 False\n[PROOFSTEP]\napply h1\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nx\u271d : \u2115\nh1 : get x\u271d [] \u2260 default\n\u22a2 get x\u271d [] = default\n[PROOFSTEP]\nrw [get_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nx\u271d : get 0 (a :: as) \u2260 default\n\u22a2 get 0 (a :: as) \u2208 a :: as\n[PROOFSTEP]\nrw [mem_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nx\u271d : get 0 (a :: as) \u2260 default\n\u22a2 get 0 (a :: as) = a \u2228 get 0 (a :: as) \u2208 as\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : get (n + 1) (a :: as) \u2260 default\n\u22a2 get (n + 1) (a :: as) \u2208 a :: as\n[PROOFSTEP]\nrw [mem_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : get (n + 1) (a :: as) \u2260 default\n\u22a2 get (n + 1) (a :: as) = a \u2228 get (n + 1) (a :: as) \u2208 as\n[PROOFSTEP]\nunfold get\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : get (n + 1) (a :: as) \u2260 default\n\u22a2 get (Nat.add n 0) as = a \u2228 get (Nat.add n 0) as \u2208 as\n[PROOFSTEP]\napply Or.inr (mem_get_of_ne_zero _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nn : \u2115\na : \u03b1\nas : List \u03b1\nh1 : get (n + 1) (a :: as) \u2260 default\n\u22a2 get (Nat.add n 0) as \u2260 default\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nm : \u2115\nh1 : m \u2260 0\n\u22a2 get m (as {0 \u21a6 a}) = get m as\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nh1 : Nat.zero \u2260 0\n\u22a2 get Nat.zero (as {0 \u21a6 a}) = get Nat.zero as\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 0\n\u22a2 get (Nat.succ n\u271d) (as {0 \u21a6 a}) = get (Nat.succ n\u271d) as\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 0\n\u22a2 get (Nat.succ n\u271d) (as {0 \u21a6 a}) = get (Nat.succ n\u271d) as\n[PROOFSTEP]\ncases as\n[GOAL]\ncase succ.nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 0\n\u22a2 get (Nat.succ n\u271d) ([] {0 \u21a6 a}) = get (Nat.succ n\u271d) []\n[PROOFSTEP]\nsimp only [set, get, get_nil]\n[GOAL]\ncase succ.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 0\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 get (Nat.succ n\u271d) ((head\u271d :: tail\u271d) {0 \u21a6 a}) = get (Nat.succ n\u271d) (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [set, get, get_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nas : List \u03b1\nk m : \u2115\nh1 : m \u2260 k + 1\n\u22a2 get m (as {k + 1 \u21a6 a}) = get m as\n[PROOFSTEP]\ncases as\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : m \u2260 k + 1\n\u22a2 get m ([] {k + 1 \u21a6 a}) = get m []\n[PROOFSTEP]\ncases m\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : m \u2260 k + 1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 get m ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get m (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncases m\n[GOAL]\ncase nil.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ([] {k + 1 \u21a6 a}) = get Nat.zero []\ncase nil.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk n\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 k + 1\n\u22a2 get (Nat.succ n\u271d) ([] {k + 1 \u21a6 a}) = get (Nat.succ n\u271d) []\ncase cons.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get Nat.zero (head\u271d :: tail\u271d)\ncase cons.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 k + 1\n\u22a2 get (Nat.succ n\u271d) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get (Nat.succ n\u271d) (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase nil => simp only [set, get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ([] {k + 1 \u21a6 a}) = get Nat.zero []\n[PROOFSTEP]\ncase nil => simp only [set, get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ([] {k + 1 \u21a6 a}) = get Nat.zero []\n[PROOFSTEP]\nsimp only [set, get]\n[GOAL]\ncase nil.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk n\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 k + 1\n\u22a2 get (Nat.succ n\u271d) ([] {k + 1 \u21a6 a}) = get (Nat.succ n\u271d) []\ncase cons.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get Nat.zero (head\u271d :: tail\u271d)\ncase cons.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 k + 1\n\u22a2 get (Nat.succ n\u271d) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get (Nat.succ n\u271d) (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase nil\n  m =>\n  have h3 : get m (nil {k \u21a6 a}) = default :=\n    by\n    rw [get_set_eq_of_ne k m, get_nil]\n    intro hc\n    apply h1\n    simp [hc]\n  apply h3\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 get (Nat.succ m) ([] {k + 1 \u21a6 a}) = get (Nat.succ m) []\n[PROOFSTEP]\ncase nil\n  m =>\n  have h3 : get m (nil {k \u21a6 a}) = default :=\n    by\n    rw [get_set_eq_of_ne k m, get_nil]\n    intro hc\n    apply h1\n    simp [hc]\n  apply h3\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 get (Nat.succ m) ([] {k + 1 \u21a6 a}) = get (Nat.succ m) []\n[PROOFSTEP]\nhave h3 : get m (nil {k \u21a6 a}) = default :=\n  by\n  rw [get_set_eq_of_ne k m, get_nil]\n  intro hc\n  apply h1\n  simp [hc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 get m ([] {k \u21a6 a}) = default\n[PROOFSTEP]\nrw [get_set_eq_of_ne k m, get_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 m \u2260 k\n[PROOFSTEP]\nintro hc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\nhc : m = k\n\u22a2 False\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\nhc : m = k\n\u22a2 Nat.succ m = k + 1\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk m : \u2115\nh1 : Nat.succ m \u2260 k + 1\nh3 : get m ([] {k \u21a6 a}) = default\n\u22a2 get (Nat.succ m) ([] {k + 1 \u21a6 a}) = get (Nat.succ m) []\n[PROOFSTEP]\napply h3\n[GOAL]\ncase cons.zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get Nat.zero (head\u271d :: tail\u271d)\ncase cons.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 k + 1\n\u22a2 get (Nat.succ n\u271d) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get (Nat.succ n\u271d) (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase zero => simp only [set, get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get Nat.zero (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase zero => simp only [set, get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : Nat.zero \u2260 k + 1\n\u22a2 get Nat.zero ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get Nat.zero (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [set, get]\n[GOAL]\ncase cons.succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nn\u271d : \u2115\nh1 : Nat.succ n\u271d \u2260 k + 1\n\u22a2 get (Nat.succ n\u271d) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get (Nat.succ n\u271d) (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase _ _ m =>\n  apply get_set_eq_of_ne k m\n  intro hc\n  apply h1\n  simp [hc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nm : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 get (Nat.succ m) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get (Nat.succ m) (head\u271d :: tail\u271d)\n[PROOFSTEP]\ncase _ _ m =>\n  apply get_set_eq_of_ne k m\n  intro hc\n  apply h1\n  simp [hc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nm : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 get (Nat.succ m) ((head\u271d :: tail\u271d) {k + 1 \u21a6 a}) = get (Nat.succ m) (head\u271d :: tail\u271d)\n[PROOFSTEP]\napply get_set_eq_of_ne k m\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nm : \u2115\nh1 : Nat.succ m \u2260 k + 1\n\u22a2 m \u2260 k\n[PROOFSTEP]\nintro hc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nm : \u2115\nh1 : Nat.succ m \u2260 k + 1\nhc : m = k\n\u22a2 False\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na\u271d : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na : \u03b1\nk : \u2115\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nm : \u2115\nh1 : Nat.succ m \u2260 k + 1\nhc : m = k\n\u22a2 Nat.succ m = k + 1\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d : \u2115\nh : x\u271d < length []\n\u22a2 get x\u271d (map f []) = f (get x\u271d [])\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (head\u271d :: as)\n\u22a2 get (n + 1) (map f (head\u271d :: as)) = f (get (n + 1) (head\u271d :: as))\n[PROOFSTEP]\nhave h2 : n < length as := by\n  rw [\u2190 Nat.succ_le_iff, \u2190 Nat.lt_succ_iff]\n  apply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (head\u271d :: as)\n\u22a2 n < length as\n[PROOFSTEP]\nrw [\u2190 Nat.succ_le_iff, \u2190 Nat.lt_succ_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (head\u271d :: as)\n\u22a2 Nat.succ n < Nat.succ (length as)\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nhead\u271d : \u03b1\nas : List \u03b1\nh1 : n + 1 < length (head\u271d :: as)\nh2 : n < length as\n\u22a2 get (n + 1) (map f (head\u271d :: as)) = f (get (n + 1) (head\u271d :: as))\n[PROOFSTEP]\napply get_map h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\n\u22a2 f default = default \u2192 get n (map f as) = f (get n as)\n[PROOFSTEP]\nintro h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\n\u22a2 get n (map f as) = f (get n as)\n[PROOFSTEP]\nby_cases h2 : n < as.length\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : n < length as\n\u22a2 get n (map f as) = f (get n as)\n[PROOFSTEP]\napply get_map h2\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : \u00acn < length as\n\u22a2 get n (map f as) = f (get n as)\n[PROOFSTEP]\nrw [not_lt] at h2 \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 get n (map f as) = f (get n as)\n[PROOFSTEP]\nrw [get_eq_default_of_le _ h2, get_eq_default_of_le, h1]\n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 length (map f as) \u2264 n\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 length as \u2264 n\n[PROOFSTEP]\napply h2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nas : List \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 p default \u2192 (\u2200 (x : \u03b1), x \u2208 as \u2192 p x) \u2192 \u2200 (n : \u2115), p (get n as)\n[PROOFSTEP]\nintro h1 h2 n\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nas : List \u03b1\np : \u03b1 \u2192 Prop\nh1 : p default\nh2 : \u2200 (x : \u03b1), x \u2208 as \u2192 p x\nn : \u2115\n\u22a2 p (get n as)\n[PROOFSTEP]\nby_cases h3 : n < as.length\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nas : List \u03b1\np : \u03b1 \u2192 Prop\nh1 : p default\nh2 : \u2200 (x : \u03b1), x \u2208 as \u2192 p x\nn : \u2115\nh3 : n < length as\n\u22a2 p (get n as)\n[PROOFSTEP]\napply h2 _ (mem_get_of_le h3)\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nas : List \u03b1\np : \u03b1 \u2192 Prop\nh1 : p default\nh2 : \u2200 (x : \u03b1), x \u2208 as \u2192 p x\nn : \u2115\nh3 : \u00acn < length as\n\u22a2 p (get n as)\n[PROOFSTEP]\nrw [not_lt] at h3 \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nas : List \u03b1\np : \u03b1 \u2192 Prop\nh1 : p default\nh2 : \u2200 (x : \u03b1), x \u2208 as \u2192 p x\nn : \u2115\nh3 : length as \u2264 n\n\u22a2 p (get n as)\n[PROOFSTEP]\nrw [get_eq_default_of_le _ h3]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nas : List \u03b1\np : \u03b1 \u2192 Prop\nh1 : p default\nh2 : \u2200 (x : \u03b1), x \u2208 as \u2192 p x\nn : \u2115\nh3 : length as \u2264 n\n\u22a2 p default\n[PROOFSTEP]\napply h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\n\u22a2 as1 = as2 \u2192 Equiv as1 as2\n[PROOFSTEP]\nintro h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nh1 : as1 = as2\n\u22a2 Equiv as1 as2\n[PROOFSTEP]\nrw [h1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nh1 : as1 = as2\n\u22a2 Equiv as2 as2\n[PROOFSTEP]\napply equiv_refl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : length (head\u271d :: tail\u271d) = length []\nx\u271d : Equiv (head\u271d :: tail\u271d) []\n\u22a2 head\u271d :: tail\u271d = []\n[PROOFSTEP]\ncases h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh1 : length [] = length (head\u271d :: tail\u271d)\nx\u271d : Equiv [] (head\u271d :: tail\u271d)\n\u22a2 [] = head\u271d :: tail\u271d\n[PROOFSTEP]\ncases h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\n\u22a2 a1 :: as1 = a2 :: as2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_head\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\n\u22a2 a1 = a2\n[PROOFSTEP]\napply h2 0\n[GOAL]\ncase e_tail\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\n\u22a2 as1 = as2\n[PROOFSTEP]\nhave h3 : as1.length = as2.length := by simpa [add_left_inj, add_comm, length] using h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\n\u22a2 length as1 = length as2\n[PROOFSTEP]\nsimpa [add_left_inj, add_comm, length] using h1\n[GOAL]\ncase e_tail\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\nh3 : length as1 = length as2\n\u22a2 as1 = as2\n[PROOFSTEP]\napply eq_of_equiv h3\n[GOAL]\ncase e_tail\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\nh3 : length as1 = length as2\n\u22a2 Equiv as1 as2\n[PROOFSTEP]\nintro m\n[GOAL]\ncase e_tail\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas as1\u271d as2\u271d as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\na1 : \u03b1\nas1 : List \u03b1\na2 : \u03b1\nas2 : List \u03b1\nh1 : length (a1 :: as1) = length (a2 :: as2)\nh2 : Equiv (a1 :: as1) (a2 :: as2)\nh3 : length as1 = length as2\nm : \u2115\n\u22a2 get m as1 = get m as2\n[PROOFSTEP]\napply h2 (m + 1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d : AddGroup \u03b1\nk : \u2115\nas : List \u03b1\n\u22a2 get k (neg as) = -get k as\n[PROOFSTEP]\nunfold neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d : AddGroup \u03b1\nk : \u2115\nas : List \u03b1\n\u22a2 get k (map (fun a => -a) as) = -get k as\n[PROOFSTEP]\nrw [@get_map' \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9]\n  -- porting note: had to add a `\u27e80\u27e9` b/c of instance troubles\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d : AddGroup \u03b1\nk : \u2115\nas : List \u03b1\n\u22a2 -default = default\n[PROOFSTEP]\napply neg_zero\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d : Neg \u03b1\nas : List \u03b1\n\u22a2 length (neg as) = length as\n[PROOFSTEP]\nsimp only [neg, length_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nb : \u03b2\nbs : List \u03b2\n\u22a2 pointwise f [] (b :: bs) = map (f default) (b :: bs)\n[PROOFSTEP]\nsimp only [nil_pointwise bs, pointwise, eq_self_iff_true, and_self_iff, map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : \u03b1\nas : List \u03b1\n\u22a2 pointwise f (a :: as) [] = map (fun a => f a default) (a :: as)\n[PROOFSTEP]\nsimp only [pointwise_nil as, pointwise, eq_self_iff_true, and_self_iff, List.map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nk : \u2115\n\u22a2 get k (pointwise f [] []) = f (get k []) (get k [])\n[PROOFSTEP]\nsimp only [h1, get_nil, pointwise, get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nb : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get 0 (pointwise f [] (b :: tail\u271d)) = f (get 0 []) (get 0 (b :: tail\u271d))\n[PROOFSTEP]\nsimp only [get_pointwise, get_nil, pointwise, get, Nat.zero_eq, map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nk : \u2115\nb : \u03b2\nbs : List \u03b2\n\u22a2 get (k + 1) (pointwise f [] (b :: bs)) = f (get (k + 1) []) (get (k + 1) (b :: bs))\n[PROOFSTEP]\nhave : get k (map (f default) bs) = f default (get k bs) := by\n  simpa [nil_pointwise, get_nil] using get_pointwise h1 k [] bs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nk : \u2115\nb : \u03b2\nbs : List \u03b2\n\u22a2 get k (map (f default) bs) = f default (get k bs)\n[PROOFSTEP]\nsimpa [nil_pointwise, get_nil] using get_pointwise h1 k [] bs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nk : \u2115\nb : \u03b2\nbs : List \u03b2\nthis : get k (map (f default) bs) = f default (get k bs)\n\u22a2 get (k + 1) (pointwise f [] (b :: bs)) = f (get (k + 1) []) (get (k + 1) (b :: bs))\n[PROOFSTEP]\nsimpa [get, get_nil, pointwise, map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\na : \u03b1\ntail\u271d : List \u03b1\n\u22a2 get 0 (pointwise f (a :: tail\u271d) []) = f (get 0 (a :: tail\u271d)) (get 0 [])\n[PROOFSTEP]\nsimp only [get_pointwise, get_nil, pointwise, get, Nat.zero_eq, map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nk : \u2115\na : \u03b1\nas : List \u03b1\n\u22a2 get (k + 1) (pointwise f (a :: as) []) = f (get (k + 1) (a :: as)) (get (k + 1) [])\n[PROOFSTEP]\nsimpa [get, get_nil, pointwise, map, pointwise_nil, get_nil] using get_pointwise h1 k as []\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\na : \u03b1\ntail\u271d\u00b9 : List \u03b1\nb : \u03b2\ntail\u271d : List \u03b2\n\u22a2 get 0 (pointwise f (a :: tail\u271d\u00b9) (b :: tail\u271d)) = f (get 0 (a :: tail\u271d\u00b9)) (get 0 (b :: tail\u271d))\n[PROOFSTEP]\nsimp only [pointwise, get]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Inhabited \u03b1\ninst\u271d\u00b9 : Inhabited \u03b2\ninst\u271d : Inhabited \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nh1 : f default default = default\nk : \u2115\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\n\u22a2 get (k + 1) (pointwise f (head\u271d\u00b9 :: as) (head\u271d :: bs)) = f (get (k + 1) (head\u271d\u00b9 :: as)) (get (k + 1) (head\u271d :: bs))\n[PROOFSTEP]\nsimp only [get, Nat.add_eq, add_zero, get_pointwise h1 k as bs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhead\u271d : \u03b2\nbs : List \u03b2\n\u22a2 length (pointwise f [] (head\u271d :: bs)) = max (length []) (length (head\u271d :: bs))\n[PROOFSTEP]\nsimp only [pointwise, length, length_map, max_eq_right (Nat.zero_le (length bs + 1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhead\u271d : \u03b1\nas : List \u03b1\n\u22a2 length (pointwise f (head\u271d :: as) []) = max (length (head\u271d :: as)) (length [])\n[PROOFSTEP]\nsimp only [pointwise, length, length_map, max_eq_left (Nat.zero_le (length as + 1))]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\n\u22a2 length (pointwise f (head\u271d\u00b9 :: as) (head\u271d :: bs)) = max (length (head\u271d\u00b9 :: as)) (length (head\u271d :: bs))\n[PROOFSTEP]\nsimp only [pointwise, length, Nat.max_succ_succ, @length_pointwise _ as bs]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nk : \u2115\nxs ys : List \u03b1\n\u22a2 get k (add xs ys) = get k xs + get k ys\n[PROOFSTEP]\napply @get_pointwise _ _ _ \u27e80\u27e9 \u27e80\u27e9 \u27e80\u27e9\n[GOAL]\ncase h1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nk : \u2115\nxs ys : List \u03b1\n\u22a2 default + default = default\n[PROOFSTEP]\napply zero_add\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\n\u22a2 add [] as = as\n[PROOFSTEP]\nrw [add, @nil_pointwise \u03b1 \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\n\u22a2 map (fun x => default + x) as = as\n[PROOFSTEP]\napply Eq.trans _ (map_id as)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\n\u22a2 map (fun x => default + x) as = map id as\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_f.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\nx : \u03b1\n\u22a2 default + x = id x\n[PROOFSTEP]\nexact zero_add x\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\n\u22a2 add as [] = as\n[PROOFSTEP]\nrw [add, @pointwise_nil \u03b1 \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\n\u22a2 map (fun a => a + default) as = as\n[PROOFSTEP]\napply Eq.trans _ (map_id as)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\n\u22a2 map (fun a => a + default) as = map id as\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_f.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nas : List \u03b1\nx : \u03b1\n\u22a2 x + default = id x\n[PROOFSTEP]\nexact add_zero x\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\n\u22a2 add (map f as) (map g as) = map (fun x => f x + g x) as\n[PROOFSTEP]\napply @eq_of_equiv _ (\u27e80\u27e9 : Inhabited \u03b1)\n[GOAL]\ncase a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\n\u22a2 length (add (map f as) (map g as)) = length (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [length_map, length_add, max_eq_left, length_map]\n[GOAL]\ncase a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\n\u22a2 length (map g as) \u2264 length (map f as)\n[PROOFSTEP]\napply le_of_eq\n[GOAL]\ncase a.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\n\u22a2 length (map g as) = length (map f as)\n[PROOFSTEP]\nrw [length_map, length_map]\n[GOAL]\ncase a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\n\u22a2 Equiv (add (map f as) (map g as)) (map (fun x => f x + g x) as)\n[PROOFSTEP]\nintro m\n[GOAL]\ncase a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\n\u22a2 get m (add (map f as) (map g as)) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [get_add]\n[GOAL]\ncase a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\n\u22a2 get m (map f as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nby_cases h : m < length as\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : m < length as\n\u22a2 get m (map f as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrepeat' rw [@get_map \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9 _ _ _ h]\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : m < length as\n\u22a2 get m (map f as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [@get_map \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9 _ _ _ h]\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : m < length as\n\u22a2 f (get m as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [@get_map \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9 _ _ _ h]\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : m < length as\n\u22a2 f (get m as) + g (get m as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [@get_map \u03b1 \u03b1 \u27e80\u27e9 \u27e80\u27e9 _ _ _ h]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : \u00acm < length as\n\u22a2 get m (map f as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [not_lt] at h \n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 get m (map f as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrepeat' rw [@get_eq_default_of_le _ \u27e80\u27e9 m] <;> try rw [length_map]; apply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 get m (map f as) + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [@get_eq_default_of_le _ \u27e80\u27e9 m]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\ntry rw [length_map]; apply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length (map f as) \u2264 m\n[PROOFSTEP]\ntry rw [length_map]; apply h\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length (map f as) \u2264 m\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length as \u2264 m\n[PROOFSTEP]\napply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + get m (map g as) = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [@get_eq_default_of_le _ \u27e80\u27e9 m]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\ntry rw [length_map]; apply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length (map g as) \u2264 m\n[PROOFSTEP]\ntry rw [length_map]; apply h\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length (map g as) \u2264 m\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length as \u2264 m\n[PROOFSTEP]\napply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = get m (map (fun x => f x + g x) as)\n[PROOFSTEP]\nrw [@get_eq_default_of_le _ \u27e80\u27e9 m]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = default\n[PROOFSTEP]\ntry rw [length_map]; apply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = default\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length (map (fun x => f x + g x) as) \u2264 m\n[PROOFSTEP]\ntry rw [length_map]; apply h\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length (map (fun x => f x + g x) as) \u2264 m\n[PROOFSTEP]\nrw [length_map]\n[GOAL]\ncase neg.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 length as \u2264 m\n[PROOFSTEP]\napply h\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = default\n[PROOFSTEP]\nrw [@get_eq_default_of_le _ \u27e80\u27e9 m]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddMonoid \u03b1\nf g : \u03b1 \u2192 \u03b1\nas : List \u03b1\nm : \u2115\nh : length as \u2264 m\n\u22a2 default + default = default\n[PROOFSTEP]\nexact zero_add _\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddGroup \u03b1\nk : \u2115\nxs ys : List \u03b1\n\u22a2 get k (sub xs ys) = get k xs - get k ys\n[PROOFSTEP]\napply @get_pointwise _ _ _ \u27e80\u27e9 \u27e80\u27e9 \u27e80\u27e9\n[GOAL]\ncase h1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u\ninst\u271d : AddGroup \u03b1\nk : \u2115\nxs ys : List \u03b1\n\u22a2 Sub.sub default default = default\n[PROOFSTEP]\napply sub_zero\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\n\u22a2 sub [] as = neg as\n[PROOFSTEP]\nrw [sub, @nil_pointwise _ _ _ \u27e80\u27e9 \u27e80\u27e9]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\n\u22a2 map (Sub.sub default) as = neg as\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_f.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\nx : \u03b1\n\u22a2 Sub.sub default x = -x\n[PROOFSTEP]\nexact zero_sub x\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\n\u22a2 sub as [] = as\n[PROOFSTEP]\nrw [sub, @pointwise_nil _ _ _ \u27e80\u27e9 \u27e80\u27e9]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\n\u22a2 map (fun a => Sub.sub a default) as = as\n[PROOFSTEP]\napply Eq.trans _ (map_id as)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\n\u22a2 map (fun a => Sub.sub a default) as = map id as\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_f.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\ninst\u271d : AddGroup \u03b1\nas : List \u03b1\nx : \u03b1\n\u22a2 Sub.sub x default = id x\n[PROOFSTEP]\nexact sub_zero x\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Func", "llama_tokens": 22237, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637505099167, "lm_q2_score": 0.6926419704455588, "lm_q1q2_score": 0.595439194073808}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedAddCommGroup W\ninst\u271d : NormedSpace \ud835\udd5c W\nn : \u2115\u221e\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 ContDiff \ud835\udd5c n \u2191f\n[PROOFSTEP]\nrw [f.decomp]\n[GOAL]\n\ud835\udd5c : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedAddCommGroup W\ninst\u271d : NormedSpace \ud835\udd5c W\nn : \u2115\u221e\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 ContDiff \ud835\udd5c n (\u2191(contLinear f) + Function.const V (\u2191f 0))\n[PROOFSTEP]\napply f.contLinear.contDiff.add\n[GOAL]\n\ud835\udd5c : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedAddCommGroup W\ninst\u271d : NormedSpace \ud835\udd5c W\nn : \u2115\u221e\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 ContDiff \ud835\udd5c n fun x => Function.const V (\u2191f 0) x\n[PROOFSTEP]\nsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedAddCommGroup W\ninst\u271d : NormedSpace \ud835\udd5c W\nn : \u2115\u221e\nf : V \u2192A[\ud835\udd5c] W\n\u22a2 ContDiff \ud835\udd5c n fun x => Function.const V (\u2191f 0) x\n[PROOFSTEP]\nexact contDiff_const\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.AffineMap", "llama_tokens": 574, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.7090191214879991, "lm_q1q2_score": 0.5953874400426029}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 min a b = a \u2227 a \u2264 b \u2228 min a b = b \u2227 b < a\n[PROOFSTEP]\nby_cases h : a \u2264 b\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : a \u2264 b\n\u22a2 min a b = a \u2227 a \u2264 b \u2228 min a b = b \u2227 b < a\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : a \u2264 b\n\u22a2 min a b = a \u2227 a \u2264 b\n[PROOFSTEP]\nexact \u27e8min_eq_left h, h\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 min a b = a \u2227 a \u2264 b \u2228 min a b = b \u2227 b < a\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 min a b = b \u2227 b < a\n[PROOFSTEP]\nexact \u27e8min_eq_right (le_of_lt (not_le.mp h)), not_le.mp h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 min a b = c \u2194 a = c \u2227 a \u2264 b \u2228 b = c \u2227 b \u2264 a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 min a b = c \u2192 a = c \u2227 a \u2264 b \u2228 b = c \u2227 b \u2264 a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nh : min a b = c\n\u22a2 a = c \u2227 a \u2264 b \u2228 b = c \u2227 b \u2264 a\n[PROOFSTEP]\nrefine' Or.imp (fun h' => _) (fun h' => _) (le_total a b)\n[GOAL]\ncase mp.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nh : min a b = c\nh' : a \u2264 b\n\u22a2 a = c \u2227 a \u2264 b\n[PROOFSTEP]\nexact \u27e8by simpa [h'] using h, h'\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nh : min a b = c\nh' : a \u2264 b\n\u22a2 a = c\n[PROOFSTEP]\nsimpa [h'] using h\n[GOAL]\ncase mp.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nh : min a b = c\nh' : b \u2264 a\n\u22a2 b = c \u2227 b \u2264 a\n[PROOFSTEP]\nexact \u27e8by simpa [h'] using h, h'\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nh : min a b = c\nh' : b \u2264 a\n\u22a2 b = c\n[PROOFSTEP]\nsimpa [h'] using h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 a = c \u2227 a \u2264 b \u2228 b = c \u2227 b \u2264 a \u2192 min a b = c\n[PROOFSTEP]\nrintro (\u27e8rfl, h\u27e9 | \u27e8rfl, h\u27e9)\n[GOAL]\ncase mpr.inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b d : \u03b1\nh : a \u2264 b\n\u22a2 min a b = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mpr.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b d : \u03b1\nh : b \u2264 a\n\u22a2 min a b = b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 min a c < min b c \u2194 a < b \u2227 a < c\n[PROOFSTEP]\nsimp_rw [lt_min_iff, min_lt_iff, or_iff_left (lt_irrefl _)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 (a < b \u2228 c < b) \u2227 a < c \u2194 a < b \u2227 a < c\n[PROOFSTEP]\nexact and_congr_left fun h => or_iff_left_of_imp h.trans\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 min a b < min a c \u2194 b < c \u2227 b < a\n[PROOFSTEP]\nsimp_rw [min_comm a, min_lt_min_left_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 \u2200 (a : \u03b1), max a a = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\n\u22a2 \u2200 (a : \u03b1), min a a = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : MonotoneOn f s\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 f (max a b) = max (f a) (f b)\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : MonotoneOn f s\nha : a \u2208 s\nhb : b \u2208 s\nh : a \u2264 b\n\u22a2 f (max a b) = max (f a) (f b)\n[PROOFSTEP]\nsimp only [max_eq_right, max_eq_left, hf ha hb, hf hb ha, h]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : MonotoneOn f s\nha : a \u2208 s\nhb : b \u2208 s\nh : b \u2264 a\n\u22a2 f (max a b) = max (f a) (f b)\n[PROOFSTEP]\nsimp only [max_eq_right, max_eq_left, hf ha hb, hf hb ha, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : Monotone f\n\u22a2 f (max a b) = max (f a) (f b)\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : Monotone f\nh : a \u2264 b\n\u22a2 f (max a b) = max (f a) (f b)\n[PROOFSTEP]\nsimp [h, hf h]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : Monotone f\nh : b \u2264 a\n\u22a2 f (max a b) = max (f a) (f b)\n[PROOFSTEP]\nsimp [h, hf h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : Antitone f\n\u22a2 f (max a b) = min (f a) (f b)\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : Antitone f\nh : a \u2264 b\n\u22a2 f (max a b) = min (f a) (f b)\n[PROOFSTEP]\nsimp [h, hf h]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b c d : \u03b1\nhf : Antitone f\nh : b \u2264 a\n\u22a2 f (max a b) = min (f a) (f b)\n[PROOFSTEP]\nsimp [h, hf h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 min a b = a \u2228 min a b = b\n[PROOFSTEP]\ncases le_total a b\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\nh\u271d : a \u2264 b\n\u22a2 min a b = a \u2228 min a b = b\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b\u271d c d a b : \u03b1\nh\u271d : b \u2264 a\n\u22a2 min a b = a \u2228 min a b = b\n[PROOFSTEP]\nsimp [*]\n", "meta": {"mathlib_filename": "Mathlib.Order.MinMax", "llama_tokens": 3352, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.7577943767446202, "lm_q1q2_score": 0.5951024266060407}}
{"text": "[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2074 : DivisionRing K\u271d\ninst\u271d\u00b3 : TopologicalSpace K\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Field \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalDivisionRing \u03b1\nK : Subfield \u03b1\nsrc\u271d : Subring \u03b1 := Subring.topologicalClosure K.toSubring\nx : \u03b1\nhx :\n  x \u2208\n    {\n              toSubsemiring :=\n                {\n                  toSubmonoid :=\n                    {\n                      toSubsemigroup :=\n                        { carrier := _root_.closure \u2191K,\n                          mul_mem' := (_ : \u2200 {a b : \u03b1}, 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) },\n                  add_mem' := (_ : \u2200 {a b : \u03b1}, 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) },\n              neg_mem' :=\n                (_ : \u2200 {x : \u03b1}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n\u22a2 x\u207b\u00b9 \u2208\n    {\n              toSubsemiring :=\n                {\n                  toSubmonoid :=\n                    {\n                      toSubsemigroup :=\n                        { carrier := _root_.closure \u2191K,\n                          mul_mem' := (_ : \u2200 {a b : \u03b1}, 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) },\n                  add_mem' := (_ : \u2200 {a b : \u03b1}, 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) },\n              neg_mem' :=\n                (_ : \u2200 {x : \u03b1}, x \u2208 src\u271d.carrier \u2192 -x \u2208 src\u271d.carrier) }.toSubsemiring.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\ndsimp only at hx \u22a2\n[GOAL]\nK\u271d : Type u_1\ninst\u271d\u2074 : DivisionRing K\u271d\ninst\u271d\u00b3 : TopologicalSpace K\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Field \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalDivisionRing \u03b1\nK : Subfield \u03b1\nsrc\u271d : Subring \u03b1 := Subring.topologicalClosure K.toSubring\nx : \u03b1\nhx : x \u2208 _root_.closure \u2191K\n\u22a2 x\u207b\u00b9 \u2208 _root_.closure \u2191K\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | h)\n[GOAL]\ncase inl\nK\u271d : Type u_1\ninst\u271d\u2074 : DivisionRing K\u271d\ninst\u271d\u00b3 : TopologicalSpace K\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Field \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalDivisionRing \u03b1\nK : Subfield \u03b1\nsrc\u271d : Subring \u03b1 := Subring.topologicalClosure K.toSubring\nhx : 0 \u2208 _root_.closure \u2191K\n\u22a2 0\u207b\u00b9 \u2208 _root_.closure \u2191K\n[PROOFSTEP]\nrwa [inv_zero]\n[GOAL]\ncase inr\nK\u271d : Type u_1\ninst\u271d\u2074 : DivisionRing K\u271d\ninst\u271d\u00b3 : TopologicalSpace K\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Field \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalDivisionRing \u03b1\nK : Subfield \u03b1\nsrc\u271d : Subring \u03b1 := Subring.topologicalClosure K.toSubring\nx : \u03b1\nhx : x \u2208 _root_.closure \u2191K\nh : x \u2260 0\n\u22a2 x\u207b\u00b9 \u2208 _root_.closure \u2191K\n[PROOFSTEP]\nrw [\u2190 @inv_coe_set \u03b1 (Subfield \u03b1) _ _ SubfieldClass.toInvMemClass K, \u2190 Set.image_inv]\n[GOAL]\ncase inr\nK\u271d : Type u_1\ninst\u271d\u2074 : DivisionRing K\u271d\ninst\u271d\u00b3 : TopologicalSpace K\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Field \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalDivisionRing \u03b1\nK : Subfield \u03b1\nsrc\u271d : Subring \u03b1 := Subring.topologicalClosure K.toSubring\nx : \u03b1\nhx : x \u2208 _root_.closure \u2191K\nh : x \u2260 0\n\u22a2 x\u207b\u00b9 \u2208 _root_.closure (Inv.inv '' \u2191K)\n[PROOFSTEP]\nexact mem_closure_image (continuousAt_inv\u2080 h) hx\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : TopologicalSpace K\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a \u2260 0\nx : \ud835\udd5c\n\u22a2 (fun y => (y - b) / a) ((fun x => a * x + b) x) = x\n[PROOFSTEP]\nsimp only [add_sub_cancel]\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : TopologicalSpace K\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a \u2260 0\nx : \ud835\udd5c\n\u22a2 a * x / a = x\n[PROOFSTEP]\nexact mul_div_cancel_left x h\n[GOAL]\nK : Type u_1\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : TopologicalSpace K\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a \u2260 0\ny : \ud835\udd5c\n\u22a2 (fun x => a * x + b) ((fun y => (y - b) / a) y) = y\n[PROOFSTEP]\nsimp [mul_div_cancel' _ h]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : TopologicalSpace K\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LinearOrderedSemifield \u03b2\na\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh1 : IsLocalMin f a\nh2 : \u2200\u1da0 (z : \u03b1) in \ud835\udcdd a, 0 < f z\n\u22a2 IsLocalMax f\u207b\u00b9 a\n[PROOFSTEP]\nfilter_upwards [h1, h2] with z h3 h4 using (inv_le_inv h4 h2.self_of_nhds).mpr h3\n[GOAL]\nK : Type u_1\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b2 : T1Space \ud835\udd5c\ninst\u271d\u00b9 : Ring \ud835\udd5c\ninst\u271d : NoZeroDivisors \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhsq : EqOn (f ^ 2) 1 S\n\u22a2 EqOn f 1 S \u2228 EqOn f (-1) S\n[PROOFSTEP]\nhave : DiscreteTopology ({1, -1} : Set \ud835\udd5c) := discrete_of_t1_of_finite\n[GOAL]\nK : Type u_1\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b2 : T1Space \ud835\udd5c\ninst\u271d\u00b9 : Ring \ud835\udd5c\ninst\u271d : NoZeroDivisors \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhsq : EqOn (f ^ 2) 1 S\nthis : DiscreteTopology \u2191{1, -1}\n\u22a2 EqOn f 1 S \u2228 EqOn f (-1) S\n[PROOFSTEP]\nhave hmaps : MapsTo f S {1, -1}\n[GOAL]\ncase hmaps\nK : Type u_1\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b2 : T1Space \ud835\udd5c\ninst\u271d\u00b9 : Ring \ud835\udd5c\ninst\u271d : NoZeroDivisors \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhsq : EqOn (f ^ 2) 1 S\nthis : DiscreteTopology \u2191{1, -1}\n\u22a2 MapsTo f S {1, -1}\n[PROOFSTEP]\nsimpa only [EqOn, Pi.one_apply, Pi.pow_apply, sq_eq_one_iff] using hsq\n[GOAL]\nK : Type u_1\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b2 : T1Space \ud835\udd5c\ninst\u271d\u00b9 : Ring \ud835\udd5c\ninst\u271d : NoZeroDivisors \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhsq : EqOn (f ^ 2) 1 S\nthis : DiscreteTopology \u2191{1, -1}\nhmaps : MapsTo f S {1, -1}\n\u22a2 EqOn f 1 S \u2228 EqOn f (-1) S\n[PROOFSTEP]\nsimpa using hS.eqOn_const_of_mapsTo hf hmaps\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\n\u22a2 EqOn f g S \u2228 EqOn f (-g) S\n[PROOFSTEP]\nhave hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\nx : \u03b1\nhx : x \u2208 S\n\u22a2 ((f / g) ^ 2) x = OfNat.ofNat 1 x\n[PROOFSTEP]\nsimpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq\u271d : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\nhsq : EqOn ((f / g) ^ 2) 1 S\n\u22a2 EqOn f g S \u2228 EqOn f (-g) S\n[PROOFSTEP]\nsimpa (config := { contextual := true }) [EqOn, div_eq_iff (hg_ne _)] using\n  hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : f y = g y\nx : \u03b1\nhx : x \u2208 S\n\u22a2 f x = g x\n[PROOFSTEP]\nrcases hS.eq_or_eq_neg_of_sq_eq hf hg @hsq @hg_ne with (h | h)\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : f y = g y\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f g S\n\u22a2 f x = g x\n[PROOFSTEP]\nexact h hx\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : f y = g y\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f (-g) S\n\u22a2 f x = g x\n[PROOFSTEP]\nrw [h _, Pi.neg_apply, neg_eq_iff_add_eq_zero, \u2190 two_mul, mul_eq_zero, iff_false_iff.2 (hg_ne _)] at hy' \u22a2\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : 2 = 0 \u2228 False\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f (-g) S\n\u22a2 2 = 0 \u2228 False\n[PROOFSTEP]\nassumption\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : 2 = 0 \u2228 False\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f (-g) S\n\u22a2 x \u2208 S\n[PROOFSTEP]\nassumption\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : 2 = 0 \u2228 g y = 0\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f (-g) S\n\u22a2 y \u2208 S\n[PROOFSTEP]\nassumption\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : (-g) y = g y\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f (-g) S\n\u22a2 x \u2208 S\n[PROOFSTEP]\nassumption\n[GOAL]\nK : Type u_1\ninst\u271d\u2077 : DivisionRing K\ninst\u271d\u2076 : TopologicalSpace K\n\u03b1 : Type u_2\n\ud835\udd5c : Type u_3\nf g : \u03b1 \u2192 \ud835\udd5c\nS : Set \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : T1Space \ud835\udd5c\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : HasContinuousInv\u2080 \ud835\udd5c\ninst\u271d : ContinuousMul \ud835\udd5c\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : \u2200 {x : \u03b1}, x \u2208 S \u2192 g x \u2260 0\ny : \u03b1\nhy : y \u2208 S\nhy' : f y = g y\nx : \u03b1\nhx : x \u2208 S\nh : EqOn f (-g) S\n\u22a2 y \u2208 S\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Field", "llama_tokens": 5935, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428947, "lm_q2_score": 0.7549149868676283, "lm_q1q2_score": 0.5948210682113505}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\na : \u03b1\ninst\u271d : Invertible a\n\u22a2 0 < a * \u215fa\n[PROOFSTEP]\nsimp only [mul_invOf_self, zero_lt_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\na : \u03b1\ninst\u271d : Invertible a\n\u22a2 \u215fa \u2264 0 \u2194 a \u2264 0\n[PROOFSTEP]\nsimp only [\u2190 not_lt, invOf_pos]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\na : \u03b1\ninst\u271d : Invertible a\n\u22a2 0 < a * \u215fa\n[PROOFSTEP]\nsimp only [mul_invOf_self, zero_lt_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\na : \u03b1\ninst\u271d : Invertible a\n\u22a2 \u215fa < 0 \u2194 a < 0\n[PROOFSTEP]\nsimp only [\u2190 not_le, invOf_nonneg]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Invertible", "llama_tokens": 306, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8757870046160257, "lm_q2_score": 0.6791786991753929, "lm_q1q2_score": 0.5948158785498261}}
{"text": "[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible (det A)\n\u22a2 \u215f(det A) \u2022 adjugate A * A = 1\n[PROOFSTEP]\nrw [smul_mul_assoc, adjugate_mul, smul_smul, invOf_mul_self, one_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible (det A)\n\u22a2 A * \u215f(det A) \u2022 adjugate A = 1\n[PROOFSTEP]\nrw [mul_smul_comm, mul_adjugate, smul_smul, invOf_mul_self, one_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible (det A)\ninst\u271d : Invertible A\n\u22a2 \u215fA = \u215f(det A) \u2022 adjugate A\n[PROOFSTEP]\nletI := invertibleOfDetInvertible A\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible (det A)\ninst\u271d : Invertible A\nthis : Invertible A := invertibleOfDetInvertible A\n\u22a2 \u215fA = \u215f(det A) \u2022 adjugate A\n[PROOFSTEP]\nconvert (rfl : \u215fA = _)\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : B * A = 1\n\u22a2 det B * det A = 1\n[PROOFSTEP]\nrw [\u2190 det_mul, h, det_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : B * A = 1\n\u22a2 det A * det B = 1\n[PROOFSTEP]\nrw [mul_comm, \u2190 det_mul, h, det_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : A * B = 1\n\u22a2 det B * det A = 1\n[PROOFSTEP]\nrw [mul_comm, \u2190 det_mul, h, det_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : A * B = 1\n\u22a2 det A * det B = 1\n[PROOFSTEP]\nrw [\u2190 det_mul, h, det_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (det A)\n\u22a2 det \u215fA = \u215f(det A)\n[PROOFSTEP]\nletI := detInvertibleOfInvertible A\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (det A)\nthis : Invertible (det A) := detInvertibleOfInvertible A\n\u22a2 det \u215fA = \u215f(det A)\n[PROOFSTEP]\nconvert (rfl : _ = \u215fA.det)\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\nh : A * B = 1\n\u22a2 B * A = 1\n[PROOFSTEP]\nletI : Invertible B.det := detInvertibleOfLeftInverse _ _ h\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\nh : A * B = 1\nthis : Invertible (det B) := detInvertibleOfLeftInverse B A h\n\u22a2 B * A = 1\n[PROOFSTEP]\nletI : Invertible B := invertibleOfDetInvertible B\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\nh : A * B = 1\nthis\u271d : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n\u22a2 B * A = 1\n[PROOFSTEP]\ncalc\n  B * A = B * A * (B * \u215fB) := by rw [mul_invOf_self, Matrix.mul_one]\n  _ = B * (A * B * \u215fB) := by simp only [Matrix.mul_assoc]\n  _ = B * \u215fB := by rw [h, Matrix.one_mul]\n  _ = 1 := mul_invOf_self B\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\nh : A * B = 1\nthis\u271d : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n\u22a2 B * A = B * A * (B * \u215fB)\n[PROOFSTEP]\nrw [mul_invOf_self, Matrix.mul_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\nh : A * B = 1\nthis\u271d : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n\u22a2 B * A * (B * \u215fB) = B * (A * B * \u215fB)\n[PROOFSTEP]\nsimp only [Matrix.mul_assoc]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\nh : A * B = 1\nthis\u271d : Invertible (det B) := detInvertibleOfLeftInverse B A h\nthis : Invertible B := invertibleOfDetInvertible B\n\u22a2 B * (A * B * \u215fB) = B * \u215fB\n[PROOFSTEP]\nrw [h, Matrix.one_mul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 Invertible (det A\u1d40)\n[PROOFSTEP]\nsimpa using detInvertibleOfInvertible A\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\u1d40\n\u22a2 Invertible A\n[PROOFSTEP]\nrw [\u2190 transpose_transpose A]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\u1d40\n\u22a2 Invertible A\u1d40\u1d40\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\ninst\u271d : Invertible A\u1d34\n\u22a2 Invertible A\n[PROOFSTEP]\nrw [\u2190 conjTranspose_conjTranspose A, \u2190 star_eq_conjTranspose]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\ninst\u271d : Invertible A\u1d34\n\u22a2 Invertible (star A\u1d34)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 IsUnit A \u2194 IsUnit (det A)\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, (invertibleEquivDetInvertible A).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 IsUnit (det A\u1d40)\n[PROOFSTEP]\nrw [det_transpose]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 IsUnit (det A)\n[PROOFSTEP]\nexact h\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : \u00acIsUnit (det A)\n\u22a2 A\u207b\u00b9 = 0\n[PROOFSTEP]\nrw [inv_def, Ring.inverse_non_unit _ h, zero_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A\u207b\u00b9 = \u2191(IsUnit.unit h)\u207b\u00b9 \u2022 adjugate A\n[PROOFSTEP]\nrw [inv_def, \u2190 Ring.inverse_unit h.unit, IsUnit.unit_spec]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 \u215fA = A\u207b\u00b9\n[PROOFSTEP]\nletI := detInvertibleOfInvertible A\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\nthis : Invertible (det A) := detInvertibleOfInvertible A\n\u22a2 \u215fA = A\u207b\u00b9\n[PROOFSTEP]\nrw [inv_def, Ring.inverse_invertible, invOf_eq]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\nA : (Matrix n n \u03b1)\u02e3\n\u22a2 \u2191A\u207b\u00b9 = (\u2191A)\u207b\u00b9\n[PROOFSTEP]\nletI := A.invertible\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\nA : (Matrix n n \u03b1)\u02e3\nthis : Invertible \u2191A := Units.invertible A\n\u22a2 \u2191A\u207b\u00b9 = (\u2191A)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 invOf_eq_nonsing_inv, invOf_units]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 A\u207b\u00b9 = Ring.inverse A\n[PROOFSTEP]\nby_cases h_det : IsUnit A.det\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh_det : IsUnit (det A)\n\u22a2 A\u207b\u00b9 = Ring.inverse A\n[PROOFSTEP]\ncases (A.isUnit_iff_isUnit_det.mpr h_det).nonempty_invertible\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh_det : IsUnit (det A)\nval\u271d : Invertible A\n\u22a2 A\u207b\u00b9 = Ring.inverse A\n[PROOFSTEP]\nrw [\u2190 invOf_eq_nonsing_inv, Ring.inverse_invertible]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh_det : \u00acIsUnit (det A)\n\u22a2 A\u207b\u00b9 = Ring.inverse A\n[PROOFSTEP]\nhave h := mt A.isUnit_iff_isUnit_det.mp h_det\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh_det : \u00acIsUnit (det A)\nh : \u00acIsUnit A\n\u22a2 A\u207b\u00b9 = Ring.inverse A\n[PROOFSTEP]\nrw [Ring.inverse_non_unit _ h, nonsing_inv_apply_not_isUnit A h_det]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 A\u207b\u00b9\u1d40 = A\u1d40\u207b\u00b9\n[PROOFSTEP]\nrw [inv_def, inv_def, transpose_smul, det_transpose, adjugate_transpose]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : StarRing \u03b1\n\u22a2 A\u207b\u00b9\u1d34 = A\u1d34\u207b\u00b9\n[PROOFSTEP]\nrw [inv_def, inv_def, conjTranspose_smul, det_conjTranspose, adjugate_conjTranspose, Ring.inverse_star]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A * A\u207b\u00b9 = 1\n[PROOFSTEP]\ncases (A.isUnit_iff_isUnit_det.mpr h).nonempty_invertible\n[GOAL]\ncase intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\nval\u271d : Invertible A\n\u22a2 A * A\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 invOf_eq_nonsing_inv, mul_invOf_self]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A\u207b\u00b9 * A = 1\n[PROOFSTEP]\ncases (A.isUnit_iff_isUnit_det.mpr h).nonempty_invertible\n[GOAL]\ncase intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\nval\u271d : Invertible A\n\u22a2 A\u207b\u00b9 * A = 1\n[PROOFSTEP]\nrw [\u2190 invOf_eq_nonsing_inv, invOf_mul_self]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 Invertible A\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 invOf_eq_nonsing_inv]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 Invertible \u215fA\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 A\u207b\u00b9\u207b\u00b9 = A\n[PROOFSTEP]\nsimp only [\u2190 invOf_eq_nonsing_inv, invOf_invOf]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B\u271d : Matrix n n \u03b1\nB : Matrix m n \u03b1\nh : IsUnit (det A)\n\u22a2 B * A * A\u207b\u00b9 = B\n[PROOFSTEP]\nsimp [Matrix.mul_assoc, mul_nonsing_inv A h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B\u271d : Matrix n n \u03b1\nB : Matrix n m \u03b1\nh : IsUnit (det A)\n\u22a2 A * (A\u207b\u00b9 * B) = B\n[PROOFSTEP]\nsimp [\u2190 Matrix.mul_assoc, mul_nonsing_inv A h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B\u271d : Matrix n n \u03b1\nB : Matrix m n \u03b1\nh : IsUnit (det A)\n\u22a2 B * A\u207b\u00b9 * A = B\n[PROOFSTEP]\nsimp [Matrix.mul_assoc, nonsing_inv_mul A h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B\u271d : Matrix n n \u03b1\nB : Matrix n m \u03b1\nh : IsUnit (det A)\n\u22a2 A\u207b\u00b9 * (A * B) = B\n[PROOFSTEP]\nsimp [\u2190 Matrix.mul_assoc, nonsing_inv_mul A h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA\u271d B\u271d A B C : Matrix n n \u03b1\ninst\u271d : Invertible A\nh : A\u207b\u00b9 * B = C\n\u22a2 B = A * C\n[PROOFSTEP]\nrw [\u2190 h, mul_inv_cancel_left_of_invertible]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA\u271d B\u271d A B C : Matrix n n \u03b1\ninst\u271d : Invertible A\nh : B = A * C\n\u22a2 A\u207b\u00b9 * B = C\n[PROOFSTEP]\nrw [h, inv_mul_cancel_left_of_invertible]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA\u271d B\u271d A B C : Matrix n n \u03b1\ninst\u271d : Invertible A\nh : B * A\u207b\u00b9 = C\n\u22a2 B = C * A\n[PROOFSTEP]\nrw [\u2190 h, inv_mul_cancel_right_of_invertible]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA\u271d B\u271d A B C : Matrix n n \u03b1\ninst\u271d : Invertible A\nh : B = C * A\n\u22a2 B * A\u207b\u00b9 = C\n[PROOFSTEP]\nrw [h, mul_inv_cancel_right_of_invertible]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\nx\u271d\u00b9 x\u271d : Matrix n m \u03b1\nh : (fun x => A * x) x\u271d\u00b9 = (fun x => A * x) x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d\n[PROOFSTEP]\nsimpa only [inv_mul_cancel_left_of_invertible] using congr_arg (A\u207b\u00b9 * \u00b7) h\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\na x : Matrix m n \u03b1\nhax : (fun x => x * A) a = (fun x => x * A) x\n\u22a2 a = x\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right_of_invertible] using congr_arg (\u00b7 * A\u207b\u00b9) hax\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 A\u207b\u00b9 * A = 1 \u2227 A * A\u207b\u00b9 = 1 \u2228 A\u207b\u00b9 = 0\n[PROOFSTEP]\nby_cases h : IsUnit A.det\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A\u207b\u00b9 * A = 1 \u2227 A * A\u207b\u00b9 = 1 \u2228 A\u207b\u00b9 = 0\n[PROOFSTEP]\nexact Or.inl \u27e8nonsing_inv_mul _ h, mul_nonsing_inv _ h\u27e9\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : \u00acIsUnit (det A)\n\u22a2 A\u207b\u00b9 * A = 1 \u2227 A * A\u207b\u00b9 = 1 \u2228 A\u207b\u00b9 = 0\n[PROOFSTEP]\nexact Or.inr (nonsing_inv_apply_not_isUnit _ h)\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 det A\u207b\u00b9 * det A = 1\n[PROOFSTEP]\nrw [\u2190 det_mul, A.nonsing_inv_mul h, det_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\nby_cases h : IsUnit A.det\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\ncases h.nonempty_invertible\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\nval\u271d : Invertible (det A)\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\nletI := invertibleOfDetInvertible A\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\nval\u271d : Invertible (det A)\nthis : Invertible A := invertibleOfDetInvertible A\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\nrw [Ring.inverse_invertible, \u2190 invOf_eq_nonsing_inv, det_invOf]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : \u00acIsUnit (det A)\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\ncases isEmpty_or_nonempty n\n[GOAL]\ncase neg.inl\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : \u00acIsUnit (det A)\nh\u271d : IsEmpty n\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\nrw [det_isEmpty, det_isEmpty, Ring.inverse_one]\n[GOAL]\ncase neg.inr\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : \u00acIsUnit (det A)\nh\u271d : Nonempty n\n\u22a2 det A\u207b\u00b9 = Ring.inverse (det A)\n[PROOFSTEP]\nrw [Ring.inverse_non_unit _ h, nonsing_inv_apply_not_isUnit _ h, det_zero \u2039_\u203a]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A\u207b\u00b9\u207b\u00b9 = 1 * A\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nrw [Matrix.one_mul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 1 * A\u207b\u00b9\u207b\u00b9 = A * A\u207b\u00b9 * A\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nrw [A.mul_nonsing_inv h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A * A\u207b\u00b9 * A\u207b\u00b9\u207b\u00b9 = A\n[PROOFSTEP]\nrw [Matrix.mul_assoc, A\u207b\u00b9.mul_nonsing_inv (A.isUnit_nonsing_inv_det h), Matrix.mul_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\n\u22a2 IsUnit (det A\u207b\u00b9) \u2194 IsUnit (det A)\n[PROOFSTEP]\nrw [Matrix.det_nonsing_inv, isUnit_ring_inverse]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible (det A)\n\u22a2 unitOfDetInvertible A = nonsingInvUnit A (_ : IsUnit (det A))\n[PROOFSTEP]\next\n[GOAL]\ncase a.a.h\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible (det A)\ni\u271d x\u271d : n\n\u22a2 \u2191(unitOfDetInvertible A) i\u271d x\u271d = \u2191(nonsingInvUnit A (_ : IsUnit (det A))) i\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : B * A = 1\ng : C * A = 1\n\u22a2 B = C\n[PROOFSTEP]\nrw [\u2190 inv_eq_left_inv h, \u2190 inv_eq_left_inv g]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : A * B = 1\ng : A * C = 1\n\u22a2 B = C\n[PROOFSTEP]\nrw [\u2190 inv_eq_right_inv h, \u2190 inv_eq_right_inv g]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : A * B = 1\ng : C * A = 1\n\u22a2 B = C\n[PROOFSTEP]\nrw [\u2190 inv_eq_right_inv h, \u2190 inv_eq_left_inv g]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : A\u207b\u00b9 = B\u207b\u00b9\nh' : IsUnit (det A)\n\u22a2 A = B\n[PROOFSTEP]\nrefine' left_inv_eq_left_inv (mul_nonsing_inv _ h') _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : A\u207b\u00b9 = B\u207b\u00b9\nh' : IsUnit (det A)\n\u22a2 B * A\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : A\u207b\u00b9 = B\u207b\u00b9\nh' : IsUnit (det A)\n\u22a2 B * B\u207b\u00b9 = 1\n[PROOFSTEP]\nrefine' mul_nonsing_inv _ _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B C : Matrix n n \u03b1\nh : A\u207b\u00b9 = B\u207b\u00b9\nh' : IsUnit (det A)\n\u22a2 IsUnit (det B)\n[PROOFSTEP]\nrwa [\u2190 isUnit_nonsing_inv_det_iff, \u2190 h, isUnit_nonsing_inv_det_iff]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\ncases' subsingleton_or_nontrivial \u03b1 with ht ht\n[GOAL]\ncase inl\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Subsingleton \u03b1\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\ncases' (Fintype.card n).zero_le.eq_or_lt with hc hc\n[GOAL]\ncase inr.inl\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc : 0 = Fintype.card n\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nrw [eq_comm, Fintype.card_eq_zero_iff] at hc \n[GOAL]\ncase inr.inl\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc : IsEmpty n\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nhaveI := hc\n[GOAL]\ncase inr.inl\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc this : IsEmpty n\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase inr.inl.a.h\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc this : IsEmpty n\ni x\u271d : n\n\u22a2 0\u207b\u00b9 i x\u271d = OfNat.ofNat 0 i x\u271d\n[PROOFSTEP]\nexact (IsEmpty.false i).elim\n[GOAL]\ncase inr.inr\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc : 0 < Fintype.card n\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nhave hn : Nonempty n := Fintype.card_pos_iff.mp hc\n[GOAL]\ncase inr.inr\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc : 0 < Fintype.card n\nhn : Nonempty n\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nrefine' nonsing_inv_apply_not_isUnit _ _\n[GOAL]\ncase inr.inr\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nht : Nontrivial \u03b1\nhc : 0 < Fintype.card n\nhn : Nonempty n\n\u22a2 \u00acIsUnit (det 0)\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nsrc\u271d\u00b9 : One (Matrix n n \u03b1) := one\nsrc\u271d : Inv (Matrix n n \u03b1) := inv\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nk : \u03b1\ninst\u271d : Invertible k\nh : IsUnit (det A)\n\u22a2 \u215fk \u2022 A\u207b\u00b9 * k \u2022 A = 1\n[PROOFSTEP]\nsimp [h, smul_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nk : \u03b1\u02e3\nh : IsUnit (det A)\n\u22a2 k\u207b\u00b9 \u2022 A\u207b\u00b9 * k \u2022 A = 1\n[PROOFSTEP]\nsimp [h, smul_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 (adjugate A)\u207b\u00b9 = (IsUnit.unit h)\u207b\u00b9 \u2022 A\n[PROOFSTEP]\nrefine' inv_eq_left_inv _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 (IsUnit.unit h)\u207b\u00b9 \u2022 A * adjugate A = 1\n[PROOFSTEP]\nrw [smul_mul, mul_adjugate, Units.smul_def, smul_smul, h.val_inv_mul, one_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1\u271d : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\u271d\nA B : Matrix n n \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Semiring \u03b1\nv : n \u2192 \u03b1\ninst\u271d\u00b9 : Invertible v\ninst\u271d : Invertible (diagonal v)\n\u22a2 \u215f(diagonal v) = diagonal \u215fv\n[PROOFSTEP]\nletI := diagonalInvertible v\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1\u271d : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\u271d\nA B : Matrix n n \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Semiring \u03b1\nv : n \u2192 \u03b1\ninst\u271d\u00b9 : Invertible v\ninst\u271d : Invertible (diagonal v)\nthis : Invertible (diagonal v) := diagonalInvertible v\n\u22a2 \u215f(diagonal v) = diagonal \u215fv\n[PROOFSTEP]\nconvert (rfl : \u215f(diagonal v) = _)\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\n\u22a2 (diag \u215f(diagonal v) * v) i = OfNat.ofNat 1 i\n[PROOFSTEP]\nletI : Invertible (diagonal v).det := detInvertibleOfInvertible _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 (diag \u215f(diagonal v) * v) i = OfNat.ofNat 1 i\n[PROOFSTEP]\nrw [invOf_eq, diag_smul, adjugate_diagonal, diag_diagonal]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 ((\u215f(det (diagonal v)) \u2022 fun i => \u220f j in erase univ i, v j) * v) i = OfNat.ofNat 1 i\n[PROOFSTEP]\ndsimp\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 (\u215f(det (diagonal v)) * \u220f j in erase univ i, v j) * v i = 1\n[PROOFSTEP]\nrw [mul_assoc, prod_erase_mul _ _ (Finset.mem_univ _), \u2190 det_diagonal]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 \u215f(det (diagonal v)) * det (diagonal fun x => v x) = 1\n[PROOFSTEP]\nexact mul_invOf_self _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\n\u22a2 (v * diag \u215f(diagonal v)) i = OfNat.ofNat 1 i\n[PROOFSTEP]\nletI : Invertible (diagonal v).det := detInvertibleOfInvertible _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 (v * diag \u215f(diagonal v)) i = OfNat.ofNat 1 i\n[PROOFSTEP]\nrw [invOf_eq, diag_smul, adjugate_diagonal, diag_diagonal]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 (v * \u215f(det (diagonal v)) \u2022 fun i => \u220f j in erase univ i, v j) i = OfNat.ofNat 1 i\n[PROOFSTEP]\ndsimp\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 v i * (\u215f(det (diagonal v)) * \u220f j in erase univ i, v j) = 1\n[PROOFSTEP]\nrw [mul_left_comm, mul_prod_erase _ _ (Finset.mem_univ _), \u2190 det_diagonal]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\ninst\u271d : Invertible (diagonal v)\ni : n\nthis : Invertible (det (diagonal v)) := detInvertibleOfInvertible (diagonal v)\n\u22a2 \u215f(det (diagonal v)) * det (diagonal fun x => v x) = 1\n[PROOFSTEP]\nexact mul_invOf_self _\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\n\u22a2 IsUnit (diagonal v) \u2194 IsUnit v\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, (diagonalInvertibleEquivInvertible v).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\n\u22a2 (diagonal v)\u207b\u00b9 = diagonal (Ring.inverse v)\n[PROOFSTEP]\nrw [nonsing_inv_eq_ring_inverse]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\nby_cases h : IsUnit v\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\nh : IsUnit v\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\nhave := isUnit_diagonal.mpr h\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\nh : IsUnit v\nthis : IsUnit (diagonal v)\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\ncases this.nonempty_invertible\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\nh : IsUnit v\nthis : IsUnit (diagonal v)\nval\u271d : Invertible (diagonal v)\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\ncases h.nonempty_invertible\n[GOAL]\ncase pos.intro.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\nh : IsUnit v\nthis : IsUnit (diagonal v)\nval\u271d\u00b9 : Invertible (diagonal v)\nval\u271d : Invertible v\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\nrw [Ring.inverse_invertible, Ring.inverse_invertible, invOf_diagonal_eq]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\nh : \u00acIsUnit v\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\nhave := isUnit_diagonal.not.mpr h\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\nv : n \u2192 \u03b1\nh : \u00acIsUnit v\nthis : \u00acIsUnit (diagonal v)\n\u22a2 Ring.inverse (diagonal v) = diagonal (Ring.inverse v)\n[PROOFSTEP]\nrw [Ring.inverse_non_unit _ h, Pi.zero_def, diagonal_zero, Ring.inverse_non_unit _ this]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\n\u22a2 A\u207b\u00b9\u207b\u00b9\u207b\u00b9 = A\u207b\u00b9\n[PROOFSTEP]\nby_cases h : IsUnit A.det\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nh : IsUnit (det A)\n\u22a2 A\u207b\u00b9\u207b\u00b9\u207b\u00b9 = A\u207b\u00b9\n[PROOFSTEP]\nrw [nonsing_inv_nonsing_inv _ h]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nh : \u00acIsUnit (det A)\n\u22a2 A\u207b\u00b9\u207b\u00b9\u207b\u00b9 = A\u207b\u00b9\n[PROOFSTEP]\nsimp [nonsing_inv_apply_not_isUnit _ h]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\n\u22a2 (A * B)\u207b\u00b9 = B\u207b\u00b9 * A\u207b\u00b9\n[PROOFSTEP]\nsimp only [inv_def]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B\u271d A B : Matrix n n \u03b1\n\u22a2 Ring.inverse (det (A * B)) \u2022 adjugate (A * B) = Ring.inverse (det B) \u2022 adjugate B * Ring.inverse (det A) \u2022 adjugate A\n[PROOFSTEP]\nrw [Matrix.smul_mul, Matrix.mul_smul, smul_smul, det_mul, adjugate_mul_distrib, Ring.mul_inverse_rev]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 (List.prod [])\u207b\u00b9 = List.prod (List.map Inv.inv (List.reverse []))\n[PROOFSTEP]\nrw [List.reverse_nil, List.map_nil, List.prod_nil, inv_one]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nXs : List (Matrix n n \u03b1)\n\u22a2 (List.prod (A :: Xs))\u207b\u00b9 = List.prod (List.map Inv.inv (List.reverse (A :: Xs)))\n[PROOFSTEP]\nrw [List.reverse_cons', List.map_concat, List.prod_concat, List.prod_cons, mul_inv_rev, list_prod_inv_reverse Xs]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nb : n \u2192 \u03b1\nh : IsUnit (det A)\n\u22a2 det A \u2022 mulVec A\u207b\u00b9 b = \u2191(cramer A) b\n[PROOFSTEP]\nrw [cramer_eq_adjugate_mulVec, A.nonsing_inv_apply h, \u2190 smul_mulVec_assoc, smul_smul, h.mul_val_inv, one_smul]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA\u271d B A : Matrix n n \u03b1\nb : n \u2192 \u03b1\nh : IsUnit (det A)\n\u22a2 det A \u2022 vecMul b A\u207b\u00b9 = \u2191(cramer A\u1d40) b\n[PROOFSTEP]\nrw [\u2190 A\u207b\u00b9.transpose_transpose, vecMul_transpose, transpose_nonsing_inv, \u2190 det_transpose,\n  A\u1d40.det_smul_inv_mulVec_eq_cramer _ (isUnit_det_transpose A h)]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d : Invertible A\n\u22a2 submatrix A \u2191e\u2081 \u2191e\u2082 * submatrix \u215fA \u2191e\u2082 \u2191e\u2081 = 1\n[PROOFSTEP]\nrw [Matrix.submatrix_mul_equiv, mul_invOf_self, submatrix_one_equiv]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082)\n\u22a2 A * submatrix \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2082.symm \u2191e\u2081.symm = 1\n[PROOFSTEP]\nhave : A = (A.submatrix e\u2081 e\u2082).submatrix e\u2081.symm e\u2082.symm := by\n  simp\n    -- Porting note: was\n        -- conv in _ * _ =>\n        --   congr\n        --   rw [this]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082)\n\u22a2 A = submatrix (submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2081.symm \u2191e\u2082.symm\n[PROOFSTEP]\nsimp\n  -- Porting note: was\n      -- conv in _ * _ =>\n      --   congr\n      --   rw [this]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082)\nthis : A = submatrix (submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2081.symm \u2191e\u2082.symm\n\u22a2 A * submatrix \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2082.symm \u2191e\u2081.symm = 1\n[PROOFSTEP]\nrw [congr_arg\u2082 (\u00b7 * \u00b7) this rfl]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082)\nthis : A = submatrix (submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2081.symm \u2191e\u2082.symm\n\u22a2 submatrix (submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2081.symm \u2191e\u2082.symm * submatrix \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) \u2191e\u2082.symm \u2191e\u2081.symm = 1\n[PROOFSTEP]\nrw [Matrix.submatrix_mul_equiv, mul_invOf_self, submatrix_one_equiv]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082)\n\u22a2 \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) = submatrix \u215fA \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nletI := submatrixEquivInvertible A e\u2081 e\u2082\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082)\nthis : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082) := submatrixEquivInvertible A e\u2081 e\u2082\n\u22a2 \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) = submatrix \u215fA \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nconvert (rfl : \u215f(A.submatrix e\u2081 e\u2082) = _)\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\n\u22a2 IsUnit (submatrix A \u2191e\u2081 \u2191e\u2082) \u2194 IsUnit A\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, (submatrixEquivInvertibleEquivInvertible A _ _).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\n\u22a2 (submatrix A \u2191e\u2081 \u2191e\u2082)\u207b\u00b9 = submatrix A\u207b\u00b9 \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nby_cases h : IsUnit A\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\nh : IsUnit A\n\u22a2 (submatrix A \u2191e\u2081 \u2191e\u2082)\u207b\u00b9 = submatrix A\u207b\u00b9 \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\ncases h.nonempty_invertible\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\nh : IsUnit A\nval\u271d : Invertible A\n\u22a2 (submatrix A \u2191e\u2081 \u2191e\u2082)\u207b\u00b9 = submatrix A\u207b\u00b9 \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nletI := submatrixEquivInvertible A e\u2081 e\u2082\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\nh : IsUnit A\nval\u271d : Invertible A\nthis : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082) := submatrixEquivInvertible A e\u2081 e\u2082\n\u22a2 (submatrix A \u2191e\u2081 \u2191e\u2082)\u207b\u00b9 = submatrix A\u207b\u00b9 \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nrw [\u2190 invOf_eq_nonsing_inv, \u2190 invOf_eq_nonsing_inv, invOf_submatrix_equiv_eq A]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\nh : \u00acIsUnit A\n\u22a2 (submatrix A \u2191e\u2081 \u2191e\u2082)\u207b\u00b9 = submatrix A\u207b\u00b9 \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nhave := (isUnit_submatrix_equiv e\u2081 e\u2082).not.mpr h\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA\u271d B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\ne\u2081 e\u2082 : n \u2243 m\nh : \u00acIsUnit A\nthis : \u00acIsUnit (submatrix A \u2191e\u2081 \u2191e\u2082)\n\u22a2 (submatrix A \u2191e\u2081 \u2191e\u2082)\u207b\u00b9 = submatrix A\u207b\u00b9 \u2191e\u2082 \u2191e\u2081\n[PROOFSTEP]\nsimp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ h, Ring.inverse_non_unit _ this, submatrix_zero,\n  Pi.zero_apply]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m m \u03b1\nh : IsUnit M\nN : Matrix m m \u03b1\n\u22a2 det (M * N * M\u207b\u00b9) = det N\n[PROOFSTEP]\nrw [\u2190 h.unit_spec, \u2190 coe_units_inv, det_units_conj]\n[GOAL]\nl : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m m \u03b1\nh : IsUnit M\nN : Matrix m m \u03b1\n\u22a2 det (M\u207b\u00b9 * N * M) = det N\n[PROOFSTEP]\nrw [\u2190 h.unit_spec, \u2190 coe_units_inv, det_units_conj']\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "llama_tokens": 19973, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430562234878, "lm_q2_score": 0.7154239957834733, "lm_q1q2_score": 0.5946196863510956}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f)\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.lt_succ_iff] at hf \n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k)\n\u22a2 Measurable (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) f)\n[PROOFSTEP]\nrefine' Finset.measurable_sup' _ _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k < Nat.succ n \u2192 Measurable (f k)\n\u22a2 \u2200 (n_1 : \u2115), n_1 \u2208 range (n + 1) \u2192 Measurable (f n_1)\n[PROOFSTEP]\nsimpa [Finset.mem_range]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 Measurable fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x\n[PROOFSTEP]\nconvert Finset.measurable_range_sup' hf using 1\n[GOAL]\ncase h.e'_5\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\n\u22a2 (fun x => sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =\n    sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_5.h\nM : Type u_1\ninst\u271d\u00b3 : MeasurableSpace M\n\u03b1 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 M\n\u03b4 : Type u_3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : MeasurableSup\u2082 \u03b1\nf : \u2115 \u2192 \u03b4 \u2192 \u03b1\nn : \u2115\nhf : \u2200 (k : \u2115), k \u2264 n \u2192 Measurable (f k)\nx : \u03b4\n\u22a2 (sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) fun k => f k x) =\n    sup' (range (n + 1)) (_ : Finset.Nonempty (range (n + 1))) (fun k => f k) x\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Lattice", "llama_tokens": 1258, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933315126791, "lm_q2_score": 0.724870282120402, "lm_q1q2_score": 0.594316310522232}}
{"text": "[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nha : a \u2264 a'\nhr : r \u2264 1\n\u22a2 \u2191(lineMap a b) r \u2264 \u2191(lineMap a' b) r\n[PROOFSTEP]\nsimp only [lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nha : a \u2264 a'\nhr : r \u2264 1\n\u22a2 (1 - r) \u2022 a + r \u2022 b \u2264 (1 - r) \u2022 a' + r \u2022 b\n[PROOFSTEP]\nexact add_le_add_right (smul_le_smul_of_nonneg ha (sub_nonneg.2 hr)) _\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nha : a < a'\nhr : r < 1\n\u22a2 \u2191(lineMap a b) r < \u2191(lineMap a' b) r\n[PROOFSTEP]\nsimp only [lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nha : a < a'\nhr : r < 1\n\u22a2 (1 - r) \u2022 a + r \u2022 b < (1 - r) \u2022 a' + r \u2022 b\n[PROOFSTEP]\nexact add_lt_add_right (smul_lt_smul_of_pos ha (sub_pos.2 hr)) _\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b \u2264 b'\nhr : 0 \u2264 r\n\u22a2 \u2191(lineMap a b) r \u2264 \u2191(lineMap a b') r\n[PROOFSTEP]\nsimp only [lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b \u2264 b'\nhr : 0 \u2264 r\n\u22a2 (1 - r) \u2022 a + r \u2022 b \u2264 (1 - r) \u2022 a + r \u2022 b'\n[PROOFSTEP]\nexact add_le_add_left (smul_le_smul_of_nonneg hb hr) _\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b < b'\nhr : 0 < r\n\u22a2 \u2191(lineMap a b) r < \u2191(lineMap a b') r\n[PROOFSTEP]\nsimp only [lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b < b'\nhr : 0 < r\n\u22a2 (1 - r) \u2022 a + r \u2022 b < (1 - r) \u2022 a + r \u2022 b'\n[PROOFSTEP]\nexact add_lt_add_left (smul_lt_smul_of_pos hb hr) _\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nha : a < a'\nhb : b < b'\nh\u2080 : 0 \u2264 r\nh\u2081 : r \u2264 1\n\u22a2 \u2191(lineMap a b) r < \u2191(lineMap a' b') r\n[PROOFSTEP]\nrcases h\u2080.eq_or_lt with (rfl | h\u2080)\n[GOAL]\ncase inl\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr' : k\nha : a < a'\nhb : b < b'\nh\u2080 : 0 \u2264 0\nh\u2081 : 0 \u2264 1\n\u22a2 \u2191(lineMap a b) 0 < \u2191(lineMap a' b') 0\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nha : a < a'\nhb : b < b'\nh\u2080\u271d : 0 \u2264 r\nh\u2081 : r \u2264 1\nh\u2080 : 0 < r\n\u22a2 \u2191(lineMap a b) r < \u2191(lineMap a' b') r\n[PROOFSTEP]\nexact (lineMap_mono_left ha.le h\u2081).trans_lt (lineMap_strict_mono_right hb h\u2080)\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nh : r < r'\n\u22a2 \u2191(lineMap a b) r < \u2191(lineMap a b) r' \u2194 a < b\n[PROOFSTEP]\nsimp only [lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nh : r < r'\n\u22a2 (1 - r) \u2022 a + r \u2022 b < (1 - r') \u2022 a + r' \u2022 b \u2194 a < b\n[PROOFSTEP]\nrw [\u2190 lt_sub_iff_add_lt, add_sub_assoc, \u2190 sub_lt_iff_lt_add', \u2190 sub_smul, \u2190 sub_smul, sub_sub_sub_cancel_left,\n  smul_lt_smul_iff_of_pos (sub_pos.2 h)]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nh : 0 < r\n\u22a2 a < \u2191(lineMap a b) r \u2194 \u2191(lineMap a b) 0 < \u2191(lineMap a b) r\n[PROOFSTEP]\nrw [lineMap_apply_zero]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nh : r < 1\n\u22a2 \u2191(lineMap a b) r < b \u2194 \u2191(lineMap a b) r < \u2191(lineMap a b) 1\n[PROOFSTEP]\nrw [lineMap_apply_one]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na b : E\nr r' : k\nh : r < r'\n\u22a2 \u2191(lineMap a b) r \u2264 \u2191(lineMap a b) r' \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only [lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na b : E\nr r' : k\nh : r < r'\n\u22a2 (1 - r) \u2022 a + r \u2022 b \u2264 (1 - r') \u2022 a + r' \u2022 b \u2194 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 le_sub_iff_add_le, add_sub_assoc, \u2190 sub_le_iff_le_add', \u2190 sub_smul, \u2190 sub_smul, sub_sub_sub_cancel_left,\n  smul_le_smul_iff_of_pos (sub_pos.2 h)]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na b : E\nr r' : k\nh : 0 < r\n\u22a2 a \u2264 \u2191(lineMap a b) r \u2194 \u2191(lineMap a b) 0 \u2264 \u2191(lineMap a b) r\n[PROOFSTEP]\nrw [lineMap_apply_zero]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na b : E\nr r' : k\nh : r < 1\n\u22a2 \u2191(lineMap a b) r \u2264 b \u2194 \u2191(lineMap a b) r \u2264 \u2191(lineMap a b) 1\n[PROOFSTEP]\nrw [lineMap_apply_one]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r : k\nh : 0 < r * (b - a)\n\u22a2 f (\u2191(lineMap a b) r) \u2264 \u2191(lineMap (f a) (f b)) r \u2194 slope f a (\u2191(lineMap a b) r) \u2264 slope f a b\n[PROOFSTEP]\nrw [lineMap_apply, lineMap_apply, slope, slope, vsub_eq_sub, vsub_eq_sub, vsub_eq_sub, vadd_eq_add, vadd_eq_add,\n  smul_eq_mul, add_sub_cancel, smul_sub, smul_sub, smul_sub, sub_le_iff_le_add, mul_inv_rev, mul_smul, mul_smul, \u2190\n  smul_sub, \u2190 smul_sub, \u2190 smul_add, smul_smul, \u2190 mul_inv_rev, inv_smul_le_iff h, smul_smul,\n  mul_inv_cancel_right\u2080 (right_ne_zero_of_mul h.ne'), smul_add, smul_inv_smul\u2080 (left_ne_zero_of_mul h.ne')]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r : k\nh : 0 < (1 - r) * (b - a)\n\u22a2 f (\u2191(lineMap a b) r) \u2264 \u2191(lineMap (f a) (f b)) r \u2194 slope f a b \u2264 slope f (\u2191(lineMap a b) r) b\n[PROOFSTEP]\nrw [\u2190 lineMap_apply_one_sub, \u2190 lineMap_apply_one_sub _ _ r]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r : k\nh : 0 < (1 - r) * (b - a)\n\u22a2 f (\u2191(lineMap b a) (1 - r)) \u2264 \u2191(lineMap (f b) (f a)) (1 - r) \u2194 slope f a b \u2264 slope f (\u2191(lineMap b a) (1 - r)) b\n[PROOFSTEP]\nrevert h\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r : k\n\u22a2 0 < (1 - r) * (b - a) \u2192\n    (f (\u2191(lineMap b a) (1 - r)) \u2264 \u2191(lineMap (f b) (f a)) (1 - r) \u2194 slope f a b \u2264 slope f (\u2191(lineMap b a) (1 - r)) b)\n[PROOFSTEP]\ngeneralize 1 - r = r'\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r r' : k\n\u22a2 0 < r' * (b - a) \u2192 (f (\u2191(lineMap b a) r') \u2264 \u2191(lineMap (f b) (f a)) r' \u2194 slope f a b \u2264 slope f (\u2191(lineMap b a) r') b)\n[PROOFSTEP]\nclear! r\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r' : k\n\u22a2 0 < r' * (b - a) \u2192 (f (\u2191(lineMap b a) r') \u2264 \u2191(lineMap (f b) (f a)) r' \u2194 slope f a b \u2264 slope f (\u2191(lineMap b a) r') b)\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r' : k\nh : 0 < r' * (b - a)\n\u22a2 f (\u2191(lineMap b a) r') \u2264 \u2191(lineMap (f b) (f a)) r' \u2194 slope f a b \u2264 slope f (\u2191(lineMap b a) r') b\n[PROOFSTEP]\nsimp_rw [lineMap_apply, slope, vsub_eq_sub, vadd_eq_add, smul_eq_mul]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r' : k\nh : 0 < r' * (b - a)\n\u22a2 f (r' * (a - b) + b) \u2264 r' \u2022 (f a - f b) + f b \u2194\n    (b - a)\u207b\u00b9 \u2022 (f b - f a) \u2264 (b - (r' * (a - b) + b))\u207b\u00b9 \u2022 (f b - f (r' * (a - b) + b))\n[PROOFSTEP]\nrw [sub_add_eq_sub_sub_swap, sub_self, zero_sub, neg_mul_eq_mul_neg, neg_sub, le_inv_smul_iff h, smul_smul,\n  mul_inv_cancel_right\u2080, le_sub_comm, \u2190 neg_sub (f b), smul_neg, neg_add_eq_sub]\n[GOAL]\ncase h\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r' : k\nh : 0 < r' * (b - a)\n\u22a2 b - a \u2260 0\n[PROOFSTEP]\nexact right_ne_zero_of_mul h.ne'\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\nf : k \u2192 E\na b r : k\nhab : a < b\nh\u2080 : 0 < r\nh\u2081 : r < 1\n\u22a2 f (\u2191(lineMap a b) r) \u2264 \u2191(lineMap (f a) (f b)) r \u2194 slope f a (\u2191(lineMap a b) r) \u2264 slope f (\u2191(lineMap a b) r) b\n[PROOFSTEP]\nrw [map_le_lineMap_iff_slope_le_slope_left (mul_pos h\u2080 (sub_pos.2 hab)), \u2190 lineMap_slope_lineMap_slope_lineMap f a b r,\n  right_le_lineMap_iff_le h\u2081]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Ordered", "llama_tokens": 5152, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933447152498, "lm_q2_score": 0.7248702702332475, "lm_q1q2_score": 0.5943163103461843}}
{"text": "[GOAL]\nR : Type u_1\nS\u271d : Type u_2\nM\u271d : Type u_3\nA : Type u_4\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : AddCommMonoid M\u271d\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Algebra R S\u271d\ninst\u271d : Module S\u271d M\u271d\nr : R\nS : S\u271d\nM : RestrictScalars R S\u271d M\u271d\n\u22a2 (r \u2022 S) \u2022 M = r \u2022 S \u2022 M\n[PROOFSTEP]\nrw [Algebra.smul_def, mul_smul]\n[GOAL]\nR : Type u_1\nS\u271d : Type u_2\nM\u271d : Type u_3\nA : Type u_4\ninst\u271d\u2074 : Semiring S\u271d\ninst\u271d\u00b3 : AddCommMonoid M\u271d\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Algebra R S\u271d\ninst\u271d : Module S\u271d M\u271d\nr : R\nS : S\u271d\nM : RestrictScalars R S\u271d M\u271d\n\u22a2 \u2191(algebraMap R S\u271d) r \u2022 S \u2022 M = r \u2022 S \u2022 M\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Module S M\nr : R\ns : S\nx : M\n\u22a2 \u2191(AddEquiv.symm (addEquiv R S M)) ((r \u2022 s) \u2022 x) = r \u2022 \u2191(AddEquiv.symm (addEquiv R S M)) (s \u2022 x)\n[PROOFSTEP]\nrw [Algebra.smul_def, mul_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\nA : Type u_4\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Module S M\nr : R\ns : S\nx : M\n\u22a2 \u2191(AddEquiv.symm (addEquiv R S M)) (\u2191(algebraMap R S) r \u2022 s \u2022 x) = r \u2022 \u2191(AddEquiv.symm (addEquiv R S M)) (s \u2022 x)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Algebra.RestrictScalars", "llama_tokens": 683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424450764199, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.5942999944925776}}
{"text": "[GOAL]\nX\u271d Y\u271d : CompHaus\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : (compHausToTop \u22d9 topToLocale).map a\u2081\u271d = (compHausToTop \u22d9 topToLocale).map a\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\ndsimp at h \n[GOAL]\nX\u271d Y\u271d : CompHaus\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : (Opens.comap a\u2081\u271d).op = (Opens.comap a\u2082\u271d).op\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\nexact Opens.comap_injective (Quiver.Hom.op_inj h)\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.Locale", "llama_tokens": 226, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8615382165412808, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.5938631317372207}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nh : a = b\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c\u271d c : \u03b1\nh : c * a = b\n\u22a2 a * ?m.8269 c h = b\n[PROOFSTEP]\nrw [mul_comm] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c\u271d c : \u03b1\nh\u271d : c * a = b\nh : a * c = b\n\u22a2 a * ?m.8269 c h\u271d = b\n[PROOFSTEP]\napply h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c : \u03b1\n\u22a2 (\u2203 c, b = c * a) \u2192 a \u2223 b\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na c\u271d c : \u03b1\n\u22a2 a \u2223 c * a\n[PROOFSTEP]\nexact \u27e8c, mul_comm _ _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c\u271d : \u03b1\nh : a \u2223 b\nc : \u03b1\n\u22a2 a \u2223 c * b\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c\u271d : \u03b1\nh : a \u2223 b\nc : \u03b1\n\u22a2 a \u2223 b * c\n[PROOFSTEP]\nexact h.mul_right _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na\u271d b c\u271d a c e f : \u03b1\n\u22a2 a * e * (c * f) = a * c * (e * f)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c : \u03b1\nh : a * b \u2223 c\nd : \u03b1\nceq : c = a * b * d\n\u22a2 b * (a * d) = c\n[PROOFSTEP]\nsimp [ceq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na\u271d b\u271d a b : \u03b1\nh : a \u2223 b\n\u22a2 a ^ 0 \u2223 b ^ 0\n[PROOFSTEP]\nrw [pow_zero, pow_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na\u271d b\u271d a b : \u03b1\nh : a \u2223 b\nn : \u2115\n\u22a2 a ^ (n + 1) \u2223 b ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na\u271d b\u271d a b : \u03b1\nh : a \u2223 b\nn : \u2115\n\u22a2 a * a ^ n \u2223 b * b ^ n\n[PROOFSTEP]\nexact mul_dvd_mul h (pow_dvd_pow_of_dvd h n)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Divisibility.Basic", "llama_tokens": 862, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.7690802423634961, "lm_q1q2_score": 0.5936095843847641}}
{"text": "[GOAL]\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 Associated a b\n[PROOFSTEP]\nrefine' Int.natAbs_eq_natAbs_iff.trans _\n[GOAL]\na b : \u2124\n\u22a2 a = b \u2228 a = -b \u2194 Associated a b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\na b : \u2124\n\u22a2 a = b \u2228 a = -b \u2192 Associated a b\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase mp.inl\na : \u2124\n\u22a2 Associated a a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.inr\nb : \u2124\n\u22a2 Associated (-b) b\n[PROOFSTEP]\nexact \u27e8-1, by simp\u27e9\n[GOAL]\nb : \u2124\n\u22a2 -b * \u2191(-1) = b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\na b : \u2124\n\u22a2 Associated a b \u2192 a = b \u2228 a = -b\n[PROOFSTEP]\nrintro \u27e8u, rfl\u27e9\n[GOAL]\ncase mpr.intro\na : \u2124\nu : \u2124\u02e3\n\u22a2 a = a * \u2191u \u2228 a = -(a * \u2191u)\n[PROOFSTEP]\nobtain rfl | rfl := Int.units_eq_one_or u\n[GOAL]\ncase mpr.intro.inl\na : \u2124\n\u22a2 a = a * \u21911 \u2228 a = -(a * \u21911)\n[PROOFSTEP]\nexact Or.inl (by simp)\n[GOAL]\na : \u2124\n\u22a2 a = a * \u21911\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.intro.inr\na : \u2124\n\u22a2 a = a * \u2191(-1) \u2228 a = -(a * \u2191(-1))\n[PROOFSTEP]\nexact Or.inr (by simp)\n[GOAL]\na : \u2124\n\u22a2 a = -(a * \u2191(-1))\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Associated", "llama_tokens": 538, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802264851919, "lm_q2_score": 0.7718434978390747, "lm_q1q2_score": 0.5936095721291983}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf fab : \u03b1 \u2192 \u03b2\nfbc : \u03b2 \u2192 \u03b3\nga ga' : \u03b1 \u2192 \u03b1\ngb gb' : \u03b2 \u2192 \u03b2\ngc gc' : \u03b3 \u2192 \u03b3\nh : Semiconj f ga gb\nh' : Semiconj f ga' gb'\nx : \u03b1\n\u22a2 f ((ga \u2218 ga') x) = (gb \u2218 gb') (f x)\n[PROOFSTEP]\nsimp only [comp_apply, h.eq, h'.eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf fab : \u03b1 \u2192 \u03b2\nfbc : \u03b2 \u2192 \u03b3\nga ga' : \u03b1 \u2192 \u03b1\ngb gb' : \u03b2 \u2192 \u03b2\ngc gc' : \u03b3 \u2192 \u03b3\nhab : Semiconj fab ga gb\nhbc : Semiconj fbc gb gc\nx : \u03b1\n\u22a2 (fbc \u2218 fab) (ga x) = gc ((fbc \u2218 fab) x)\n[PROOFSTEP]\nsimp only [comp_apply, hab.eq, hbc.eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf fab : \u03b1 \u2192 \u03b2\nfbc : \u03b2 \u2192 \u03b3\nga ga' : \u03b1 \u2192 \u03b1\ngb gb' : \u03b2 \u2192 \u03b2\ngc gc' : \u03b3 \u2192 \u03b3\nh : Semiconj f ga gb\nha : RightInverse ga' ga\nhb : LeftInverse gb' gb\nx : \u03b1\n\u22a2 f (ga' x) = gb' (f x)\n[PROOFSTEP]\nrw [\u2190 hb (f (ga' x)), \u2190 h.eq, ha x]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nga : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngb : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nf' : \u03b2 \u2192 \u03b3\ngc : \u03b3 \u2192 \u03b3 \u2192 \u03b3\nhf' : Semiconj\u2082 f' gb gc\nhf : Semiconj\u2082 f ga gb\nx y : \u03b1\n\u22a2 (f' \u2218 f) (ga x y) = gc ((f' \u2218 f) x) ((f' \u2218 f) y)\n[PROOFSTEP]\nsimp only [hf'.eq, hf.eq, comp_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nga : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngb : \u03b2 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : IsAssociative \u03b1 ga\nh : Semiconj\u2082 f ga gb\nh_surj : Surjective f\nx\u2081 x\u2082 x\u2083 : \u03b1\n\u22a2 gb (gb (f x\u2081) (f x\u2082)) (f x\u2083) = gb (f x\u2081) (gb (f x\u2082) (f x\u2083))\n[PROOFSTEP]\nsimp only [\u2190 h.eq, @IsAssociative.assoc _ ga]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nga : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngb : \u03b2 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : IsAssociative \u03b2 gb\nh : Semiconj\u2082 f ga gb\nh_inj : Injective f\nx\u2081 x\u2082 x\u2083 : \u03b1\n\u22a2 f (ga (ga x\u2081 x\u2082) x\u2083) = f (ga x\u2081 (ga x\u2082 x\u2083))\n[PROOFSTEP]\nsimp only [h.eq, @IsAssociative.assoc _ gb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nga : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngb : \u03b2 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : IsIdempotent \u03b1 ga\nh : Semiconj\u2082 f ga gb\nh_surj : Surjective f\nx : \u03b1\n\u22a2 gb (f x) (f x) = f x\n[PROOFSTEP]\nsimp only [\u2190 h.eq, @IsIdempotent.idempotent _ ga]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nga : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ngb : \u03b2 \u2192 \u03b2 \u2192 \u03b2\ninst\u271d : IsIdempotent \u03b2 gb\nh : Semiconj\u2082 f ga gb\nh_inj : Injective f\nx : \u03b1\n\u22a2 f (ga x x) = f x\n[PROOFSTEP]\nrw [h.eq, @IsIdempotent.idempotent _ gb]\n", "meta": {"mathlib_filename": "Mathlib.Logic.Function.Conjugate", "llama_tokens": 1186, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324938410784, "lm_q2_score": 0.7279754430043072, "lm_q1q2_score": 0.5934692358554652}}
{"text": "[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Nonempty \u03c3\ninst\u271d : Nontrivial R\n\u22a2 max (lift #(\u03c3 \u2192\u2080 \u2115)) (lift #R) = max (max (lift #R) (lift #\u03c3)) \u2135\u2080\n[PROOFSTEP]\nrw [mk_finsupp_nat, max_assoc, lift_max, lift_aleph0, max_comm]\n[GOAL]\n\u03c3\u271d : Type u\nR\u271d : Type v\ninst\u271d\u00b9 : CommSemiring R\u271d\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommSemiring R\n\u22a2 #(MvPolynomial \u03c3 R) \u2264 max (max (lift #R) (lift #\u03c3)) \u2135\u2080\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\n\u03c3\u271d : Type u\nR\u271d : Type v\ninst\u271d\u00b9 : CommSemiring R\u271d\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommSemiring R\nh\u271d : Subsingleton R\n\u22a2 #(MvPolynomial \u03c3 R) \u2264 max (max (lift #R) (lift #\u03c3)) \u2135\u2080\n[PROOFSTEP]\nexact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph0)\n[GOAL]\ncase inr\n\u03c3\u271d : Type u\nR\u271d : Type v\ninst\u271d\u00b9 : CommSemiring R\u271d\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 #(MvPolynomial \u03c3 R) \u2264 max (max (lift #R) (lift #\u03c3)) \u2135\u2080\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03c3\n[GOAL]\ncase inr.inl\n\u03c3\u271d : Type u\nR\u271d : Type v\ninst\u271d\u00b9 : CommSemiring R\u271d\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommSemiring R\nh\u271d\u00b9 : Nontrivial R\nh\u271d : IsEmpty \u03c3\n\u22a2 #(MvPolynomial \u03c3 R) \u2264 max (max (lift #R) (lift #\u03c3)) \u2135\u2080\n[PROOFSTEP]\nexact cardinal_mk_eq_lift.trans_le (le_max_of_le_left <| le_max_left _ _)\n[GOAL]\ncase inr.inr\n\u03c3\u271d : Type u\nR\u271d : Type v\ninst\u271d\u00b9 : CommSemiring R\u271d\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommSemiring R\nh\u271d\u00b9 : Nontrivial R\nh\u271d : Nonempty \u03c3\n\u22a2 #(MvPolynomial \u03c3 R) \u2264 max (max (lift #R) (lift #\u03c3)) \u2135\u2080\n[PROOFSTEP]\nexact cardinal_mk_eq_max_lift.le\n[GOAL]\n\u03c3 R : Type u\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Nonempty \u03c3\ninst\u271d : Nontrivial R\n\u22a2 #(MvPolynomial \u03c3 R) = max (max #R #\u03c3) \u2135\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03c3 R : Type u\ninst\u271d : CommSemiring R\n\u22a2 max (max (lift #R) (lift #\u03c3)) \u2135\u2080 \u2264 max (max #R #\u03c3) \u2135\u2080\n[PROOFSTEP]\nrw [lift_id, lift_id]\n", "meta": {"mathlib_filename": "Mathlib.Data.MvPolynomial.Cardinal", "llama_tokens": 940, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677622198946, "lm_q2_score": 0.6992544147913994, "lm_q1q2_score": 0.5929452013330448}}
{"text": "[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na : (\u2a02[R]^i) M\nb : (\u2a02[R]^j) M\n\u22a2 \u2191toTensorAlgebra (GradedMonoid.GMul.mul a b) = \u2191toTensorAlgebra a * \u2191toTensorAlgebra b\n[PROOFSTEP]\nrw [TensorPower.gMul_eq_coe_linearMap, \u2190 LinearMap.compr\u2082_apply, \u2190 @LinearMap.mul_apply' R, \u2190 LinearMap.compl\u2082_apply, \u2190\n  LinearMap.comp_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na : (\u2a02[R]^i) M\nb : (\u2a02[R]^j) M\n\u22a2 \u2191(\u2191(LinearMap.compr\u2082 (LinearMap.compr\u2082 (TensorProduct.mk R ((\u2a02[R]^i) M) ((\u2a02[R]^j) M)) \u2191mulEquiv) toTensorAlgebra) a)\n      b =\n    \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R ((fun x => TensorAlgebra R M) a)) toTensorAlgebra)\n              toTensorAlgebra)\n          a)\n      b\n[PROOFSTEP]\nrefine' LinearMap.congr_fun (LinearMap.congr_fun _ a) b\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na : (\u2a02[R]^i) M\nb : (\u2a02[R]^j) M\n\u22a2 LinearMap.compr\u2082 (LinearMap.compr\u2082 (TensorProduct.mk R ((\u2a02[R]^i) M) ((\u2a02[R]^j) M)) \u2191mulEquiv) toTensorAlgebra =\n    LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R ((fun x => TensorAlgebra R M) a)) toTensorAlgebra) toTensorAlgebra\n[PROOFSTEP]\nclear! a b\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na : (\u2a02[R]^i) M\n\u22a2 LinearMap.compr\u2082 (LinearMap.compr\u2082 (TensorProduct.mk R ((\u2a02[R]^i) M) ((\u2a02[R]^j) M)) \u2191mulEquiv) toTensorAlgebra =\n    LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R ((fun x => TensorAlgebra R M) a)) toTensorAlgebra) toTensorAlgebra\n[PROOFSTEP]\next\n  (a b)\n    -- Porting note: pulled the next two lines out of the long `simp only` below.\n[GOAL]\ncase H.H.H.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na\u271d : (\u2a02[R]^i) M\na : Fin i \u2192 M\nb : Fin j \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (\u2191(LinearMap.compMultilinearMap\n                (LinearMap.compr\u2082 (LinearMap.compr\u2082 (TensorProduct.mk R ((\u2a02[R]^i) M) ((\u2a02[R]^j) M)) \u2191mulEquiv)\n                  toTensorAlgebra)\n                (PiTensorProduct.tprod R))\n            a)\n          (PiTensorProduct.tprod R))\n      b =\n    \u2191(LinearMap.compMultilinearMap\n          (\u2191(LinearMap.compMultilinearMap\n                (LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R ((fun x => TensorAlgebra R M) a\u271d)) toTensorAlgebra)\n                  toTensorAlgebra)\n                (PiTensorProduct.tprod R))\n            a)\n          (PiTensorProduct.tprod R))\n      b\n[PROOFSTEP]\nsimp only [LinearMap.compMultilinearMap_apply]\n[GOAL]\ncase H.H.H.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na\u271d : (\u2a02[R]^i) M\na : Fin i \u2192 M\nb : Fin j \u2192 M\n\u22a2 \u2191(\u2191(LinearMap.compr\u2082 (LinearMap.compr\u2082 (TensorProduct.mk R ((\u2a02[R]^i) M) ((\u2a02[R]^j) M)) \u2191mulEquiv) toTensorAlgebra)\n          (\u2191(PiTensorProduct.tprod R) a))\n      (\u2191(PiTensorProduct.tprod R) b) =\n    \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R ((fun x => TensorAlgebra R M) a\u271d)) toTensorAlgebra)\n              toTensorAlgebra)\n          (\u2191(PiTensorProduct.tprod R) a))\n      (\u2191(PiTensorProduct.tprod R) b)\n[PROOFSTEP]\nrw [LinearMap.compr\u2082_apply, \u2190 gMul_eq_coe_linearMap]\n[GOAL]\ncase H.H.H.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na\u271d : (\u2a02[R]^i) M\na : Fin i \u2192 M\nb : Fin j \u2192 M\n\u22a2 \u2191toTensorAlgebra (GradedMonoid.GMul.mul (\u2191(PiTensorProduct.tprod R) a) (\u2191(PiTensorProduct.tprod R) b)) =\n    \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R ((fun x => TensorAlgebra R M) a\u271d)) toTensorAlgebra)\n              toTensorAlgebra)\n          (\u2191(PiTensorProduct.tprod R) a))\n      (\u2191(PiTensorProduct.tprod R) b)\n[PROOFSTEP]\nsimp only [LinearMap.compr\u2082_apply, LinearMap.mul_apply', LinearMap.compl\u2082_apply, LinearMap.comp_apply,\n  LinearMap.compMultilinearMap_apply, PiTensorProduct.lift.tprod, TensorPower.tprod_mul_tprod,\n  TensorPower.toTensorAlgebra_tprod, TensorAlgebra.tprod_apply, \u2190 gMul_eq_coe_linearMap]\n[GOAL]\ncase H.H.H.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na\u271d : (\u2a02[R]^i) M\na : Fin i \u2192 M\nb : Fin j \u2192 M\n\u22a2 List.prod (List.ofFn fun i_1 => \u2191(TensorAlgebra.\u03b9 R) (Fin.append a b i_1)) =\n    List.prod (List.ofFn fun i => \u2191(TensorAlgebra.\u03b9 R) (a i)) *\n      List.prod (List.ofFn fun i => \u2191(TensorAlgebra.\u03b9 R) (b i))\n[PROOFSTEP]\nrefine' Eq.trans _ List.prod_append\n[GOAL]\ncase H.H.H.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na\u271d : (\u2a02[R]^i) M\na : Fin i \u2192 M\nb : Fin j \u2192 M\n\u22a2 List.prod (List.ofFn fun i_1 => \u2191(TensorAlgebra.\u03b9 R) (Fin.append a b i_1)) =\n    List.prod ((List.ofFn fun i => \u2191(TensorAlgebra.\u03b9 R) (a i)) ++ List.ofFn fun i => \u2191(TensorAlgebra.\u03b9 R) (b i))\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase H.H.H.H.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni j : \u2115\na\u271d : (\u2a02[R]^i) M\na : Fin i \u2192 M\nb : Fin j \u2192 M\n\u22a2 (List.ofFn fun i_1 => \u2191(TensorAlgebra.\u03b9 R) (Fin.append a b i_1)) =\n    (List.ofFn fun i => \u2191(TensorAlgebra.\u03b9 R) (a i)) ++ List.ofFn fun i => \u2191(TensorAlgebra.\u03b9 R) (b i)\n[PROOFSTEP]\nerw [\u2190 List.map_ofFn _ (TensorAlgebra.\u03b9 R), \u2190 List.map_ofFn _ (TensorAlgebra.\u03b9 R), \u2190\n  List.map_ofFn _ (TensorAlgebra.\u03b9 R), \u2190 List.map_append, List.ofFn_fin_append]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\n\u22a2 \u2191toTensorAlgebra (\u2191DirectSum.GAlgebra.toFun r) = \u2191(algebraMap R (TensorAlgebra R M)) r\n[PROOFSTEP]\nrw [TensorPower.galgebra_toFun_def, TensorPower.algebraMap\u2080_eq_smul_one, LinearMap.map_smul,\n  TensorPower.toTensorAlgebra_gOne, Algebra.algebraMap_eq_smul_one]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 AlgHom.comp ofDirectSum toDirectSum = AlgHom.id R (TensorAlgebra R M)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d : M\n\u22a2 \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.comp ofDirectSum toDirectSum)) (\u03b9 R)) x\u271d =\n    \u2191(LinearMap.comp (AlgHom.toLinearMap (AlgHom.id R (TensorAlgebra R M))) (\u03b9 R)) x\u271d\n[PROOFSTEP]\nsimp [DirectSum.lof_eq_of, tprod_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn m : \u2115\nh : n = m\nx : (\u2a02[R]^n) M\n\u22a2 GradedMonoid.mk m (\u2191(PiTensorProduct.reindex R M (Fin.castIso h).toEquiv) x) = GradedMonoid.mk n x\n[PROOFSTEP]\nrw [Fin.castIso_to_equiv, mk_reindex_cast h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\nrefine' Fin.consInduction _ _ x\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x : Fin 1), Fin.elim0 a)) (List.finRange 0)) =\n    GradedMonoid.mk 0 (\u2191(PiTensorProduct.tprod R) Fin.elim0)\n[PROOFSTEP]\nclear x\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 \u2200 {n : \u2115} (x\u2080 : M) (x : Fin n \u2192 M),\n    List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n        GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x) \u2192\n      List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), Fin.cons x\u2080 x a)) (List.finRange (n + 1))) =\n        GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))\n[PROOFSTEP]\nclear x\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\n\u22a2 List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x : Fin 1), Fin.elim0 a)) (List.finRange 0)) =\n    GradedMonoid.mk 0 (\u2191(PiTensorProduct.tprod R) Fin.elim0)\n[PROOFSTEP]\nrw [List.finRange_zero, List.map_nil, List.prod_nil]\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\n\u22a2 1 = GradedMonoid.mk 0 (\u2191(PiTensorProduct.tprod R) Fin.elim0)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\n\u22a2 \u2200 {n : \u2115} (x\u2080 : M) (x : Fin n \u2192 M),\n    List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n        GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x) \u2192\n      List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), Fin.cons x\u2080 x a)) (List.finRange (n + 1))) =\n        GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))\n[PROOFSTEP]\nintro n x\u2080 x ih\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), Fin.cons x\u2080 x a)) (List.finRange (n + 1))) =\n    GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))\n[PROOFSTEP]\nrw [List.finRange_succ_eq_map, List.map_cons, List.prod_cons, List.map_map]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), Fin.cons x\u2080 x 0) *\n      List.prod\n        (List.map ((fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), Fin.cons x\u2080 x a)) \u2218 Fin.succ) (List.finRange n)) =\n    GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))\n[PROOFSTEP]\nsimp_rw [Function.comp, Fin.cons_zero, Fin.cons_succ]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 GradedMonoid.mk 1 (\u2a02\u209c[R] (x : Fin 1), x\u2080) *\n      List.prod (List.map (fun x_1 => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_2 : Fin 1), x x_1)) (List.finRange n)) =\n    GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))\n[PROOFSTEP]\nrw [ih, GradedMonoid.mk_mul_mk, TensorPower.tprod_mul_tprod]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 GradedMonoid.mk (1 + n) (\u2191(PiTensorProduct.tprod R) (Fin.append (fun x => x\u2080) x)) =\n    GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))\n[PROOFSTEP]\nrefine' TensorPower.gradedMonoid_eq_of_cast (add_comm _ _) _\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 \u2191(TensorPower.cast R M (_ : 1 + n = n + 1))\n      (GradedMonoid.mk (1 + n) (\u2191(PiTensorProduct.tprod R) (Fin.append (fun x => x\u2080) x))).snd =\n    (GradedMonoid.mk (n + 1) (\u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x))).snd\n[PROOFSTEP]\ndsimp only [GradedMonoid.mk]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 \u2191(TensorPower.cast R M (_ : 1 + n = n + 1)) (\u2191(PiTensorProduct.tprod R) (Fin.append (fun x => x\u2080) x)) =\n    \u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x)\n[PROOFSTEP]\nrw [TensorPower.cast_tprod]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 \u2191(PiTensorProduct.tprod R) (Fin.append (fun x => x\u2080) x \u2218 \u2191(Fin.castIso (_ : n + 1 = 1 + n))) =\n    \u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x)\n[PROOFSTEP]\nsimp_rw [Fin.append_left_eq_cons, Function.comp]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn\u271d n : \u2115\nx\u2080 : M\nx : Fin n \u2192 M\nih :\n  List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n\u22a2 (\u2a02\u209c[R] (x_1 : Fin (n + 1)),\n      Fin.cons x\u2080 x (\u2191(Fin.castIso (_ : 1 + n = n + 1)) (\u2191(Fin.castIso (_ : n + 1 = 1 + n)) x_1))) =\n    \u2191(PiTensorProduct.tprod R) (Fin.cons x\u2080 x)\n[PROOFSTEP]\ncongr 1 with i\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 \u2191toDirectSum (\u2191(tprod R M n) x) = \u2191(DirectSum.of (fun i => (\u2a02[R]^i) M) n) (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\nrw [tprod_apply, AlgHom.map_list_prod, List.map_ofFn]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 List.prod (List.ofFn (\u2191toDirectSum \u2218 fun i => \u2191(\u03b9 R) (x i))) =\n    \u2191(DirectSum.of (fun i => (\u2a02[R]^i) M) n) (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\nsimp_rw [Function.comp, toDirectSum_\u03b9]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 List.prod (List.ofFn fun x_1 => \u2191(DirectSum.of (fun n => (\u2a02[R]^n) M) 1) (\u2a02\u209c[R] (x_2 : Fin 1), x x_1)) =\n    \u2191(DirectSum.of (fun i => (\u2a02[R]^i) M) n) (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\nrw [DirectSum.list_prod_ofFn_of_eq_dProd]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 \u2191(DirectSum.of (fun n => (\u2a02[R]^n) M) (List.dProdIndex (List.finRange n) fun x => 1))\n      (List.dProd (List.finRange n) (fun x => 1) fun x_1 => \u2a02\u209c[R] (x_2 : Fin 1), x x_1) =\n    \u2191(DirectSum.of (fun i => (\u2a02[R]^i) M) n) (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\napply DirectSum.of_eq_of_gradedMonoid_eq\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 GradedMonoid.mk (List.dProdIndex (List.finRange n) fun x => 1)\n      (List.dProd (List.finRange n) (fun x => 1) fun x_1 => \u2a02\u209c[R] (x_2 : Fin 1), x x_1) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\nrw [GradedMonoid.mk_list_dProd]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx : Fin n \u2192 M\n\u22a2 List.prod (List.map (fun a => GradedMonoid.mk 1 (\u2a02\u209c[R] (x_1 : Fin 1), x a)) (List.finRange n)) =\n    GradedMonoid.mk n (\u2191(PiTensorProduct.tprod R) x)\n[PROOFSTEP]\nrw [TensorPower.list_prod_gradedMonoid_mk_single]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 AlgHom.comp toDirectSum ofDirectSum = AlgHom.id R (\u2a01 (n : \u2115), (\u2a02[R]^n) M)\n[PROOFSTEP]\next\n[GOAL]\ncase h.H.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ni\u271d : \u2115\nx\u271d : Fin i\u271d \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (LinearMap.comp (AlgHom.toLinearMap (AlgHom.comp toDirectSum ofDirectSum))\n            (DirectSum.lof R \u2115 (fun i => (\u2a02[R]^i) M) i\u271d))\n          (PiTensorProduct.tprod R))\n      x\u271d =\n    \u2191(LinearMap.compMultilinearMap\n          (LinearMap.comp (AlgHom.toLinearMap (AlgHom.id R (\u2a01 (n : \u2115), (\u2a02[R]^n) M)))\n            (DirectSum.lof R \u2115 (fun i => (\u2a02[R]^i) M) i\u271d))\n          (PiTensorProduct.tprod R))\n      x\u271d\n[PROOFSTEP]\nsimp [DirectSum.lof_eq_of, -tprod_apply, toDirectSum_tensorPower_tprod]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower", "llama_tokens": 8041, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677468516187, "lm_q2_score": 0.6992544273261175, "lm_q1q2_score": 0.5929452012157468}}
{"text": "[GOAL]\nx y : \u2115\n\u22a2 gcd x y = if x = 0 then y else gcd (y % x) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase zero\ny : \u2115\n\u22a2 gcd zero y = if zero = 0 then y else gcd (y % zero) zero\n[PROOFSTEP]\nsimp [Nat.gcd_succ]\n[GOAL]\ncase succ\ny n\u271d : \u2115\n\u22a2 gcd (succ n\u271d) y = if succ n\u271d = 0 then y else gcd (y % succ n\u271d) (succ n\u271d)\n[PROOFSTEP]\nsimp [Nat.gcd_succ]\n", "meta": {"mathlib_filename": "Mathlib.Init.Data.Nat.GCD", "llama_tokens": 180, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8856314798554445, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.5923814148384714}}
{"text": "[GOAL]\n\u03b1 : Type u\nb : Bool\na : \u03b1\n\u22a2 (bif b then a else a) = a\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u\na : \u03b1\n\u22a2 (bif false then a else a) = a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u\na : \u03b1\n\u22a2 (bif true then a else a) = a\n[PROOFSTEP]\nrfl\n[GOAL]\nb : Bool\n\u22a2 xor b b = false\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 xor false false = false\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 xor true true = false\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 xor b true = !b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 xor false true = !false\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 xor true true = !true\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 xor b false = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 xor false false = false\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 xor true false = true\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 xor true b = !b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 xor true false = !false\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 xor true true = !true\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 xor false b = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 xor false false = false\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 xor false true = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u00actrue = false\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u00acfalse = true\n[PROOFSTEP]\ndecide\n[GOAL]\nb : Bool\n\u22a2 (\u00acb = true) = (b = false)\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 (\u00acb = false) = (b = true)\n[PROOFSTEP]\nsimp\n[GOAL]\na b : Bool\n\u22a2 ((a && b) = true) = (a = true \u2227 b = true)\n[PROOFSTEP]\nsimp\n[GOAL]\na b : Bool\n\u22a2 ((a || b) = true) = (a = true \u2228 b = true)\n[PROOFSTEP]\nsimp\n[GOAL]\na : Bool\n\u22a2 ((!a) = true) = (a = false)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u22a2 ((!false) = true) = (false = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 ((!true) = true) = (true = false)\n[PROOFSTEP]\nsimp\n[GOAL]\na b : Bool\n\u22a2 ((a && b) = false) = (a = false \u2228 b = false)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\n\u22a2 ((false && b) = false) = (false = false \u2228 b = false)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb : Bool\n\u22a2 ((true && b) = false) = (true = false \u2228 b = false)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\n\u22a2 ((false && false) = false) = (false = false \u2228 false = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true\n\u22a2 ((false && true) = false) = (false = false \u2228 true = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false\n\u22a2 ((true && false) = false) = (true = false \u2228 false = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true\n\u22a2 ((true && true) = false) = (true = false \u2228 true = false)\n[PROOFSTEP]\nsimp\n[GOAL]\na b : Bool\n\u22a2 ((a || b) = false) = (a = false \u2227 b = false)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\n\u22a2 ((false || b) = false) = (false = false \u2227 b = false)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb : Bool\n\u22a2 ((true || b) = false) = (true = false \u2227 b = false)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\n\u22a2 ((false || false) = false) = (false = false \u2227 false = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true\n\u22a2 ((false || true) = false) = (false = false \u2227 true = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false\n\u22a2 ((true || false) = false) = (true = false \u2227 false = false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true\n\u22a2 ((true || true) = false) = (true = false \u2227 true = false)\n[PROOFSTEP]\nsimp\n[GOAL]\na : Bool\n\u22a2 ((!a) = false) = (a = true)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u22a2 ((!false) = false) = (false = true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 ((!true) = false) = (true = true)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 (false = true) = False\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 (true = true) = True\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 (false = true) = False\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 (true = true) = True\n[PROOFSTEP]\nsimp\n[GOAL]\np : Prop\nd : Decidable p\n\u22a2 decide p = true \u2194 p\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 \u00acb = true \u2194 b = false\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 \u00acfalse = true \u2194 false = false\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u22a2 \u00acfalse = true \u2194 false = false\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true\n\u22a2 \u00actrue = true \u2194 true = false\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u22a2 \u00actrue = true \u2194 true = false\n[PROOFSTEP]\ndecide\n[GOAL]\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p \u2194 q\n\u22a2 decide p = decide q\n[PROOFSTEP]\ncases h' : decide q with\n| false => exact decide_false (mt h.1 <| of_decide_false h')\n| true => exact decide_true (h.2 <| of_decide_true h')\n[GOAL]\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p \u2194 q\nx\u271d : Bool\nh' : decide q = x\u271d\n\u22a2 decide p = x\u271d\n[PROOFSTEP]\ncases h' : decide q with\n| false => exact decide_false (mt h.1 <| of_decide_false h')\n| true => exact decide_true (h.2 <| of_decide_true h')\n[GOAL]\ncase false\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p \u2194 q\nh' : decide q = false\n\u22a2 decide p = false\n[PROOFSTEP]\n\n| false => exact decide_false (mt h.1 <| of_decide_false h')\n[GOAL]\ncase false\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p \u2194 q\nh' : decide q = false\n\u22a2 decide p = false\n[PROOFSTEP]\nexact decide_false (mt h.1 <| of_decide_false h')\n[GOAL]\ncase true\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p \u2194 q\nh' : decide q = true\n\u22a2 decide p = true\n[PROOFSTEP]\n\n| true => exact decide_true (h.2 <| of_decide_true h')\n[GOAL]\ncase true\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p \u2194 q\nh' : decide q = true\n\u22a2 decide p = true\n[PROOFSTEP]\nexact decide_true (h.2 <| of_decide_true h')\n[GOAL]\na b : Bool\n\u22a2 (a || b) = true \u2194 a = true \u2228 b = true\n[PROOFSTEP]\nsimp\n[GOAL]\na b : Bool\n\u22a2 (a && b) = true \u2194 a = true \u2227 b = true\n[PROOFSTEP]\nsimp\n[GOAL]\na b : Bool\n\u22a2 xor a b = true \u2194 Xor' (a = true) (b = true)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\n\u22a2 xor false b = true \u2194 Xor' (false = true) (b = true)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb : Bool\n\u22a2 xor true b = true \u2194 Xor' (true = true) (b = true)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\n\u22a2 xor false false = true \u2194 Xor' (false = true) (false = true)\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u22a2 xor false false = true \u2194 Xor' (false = true) (false = true)\n[PROOFSTEP]\ndecide\n[GOAL]\ncase false.true\n\u22a2 xor false true = true \u2194 Xor' (false = true) (true = true)\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u22a2 xor false true = true \u2194 Xor' (false = true) (true = true)\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true.false\n\u22a2 xor true false = true \u2194 Xor' (true = true) (false = true)\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u22a2 xor true false = true \u2194 Xor' (true = true) (false = true)\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true.true\n\u22a2 xor true true = true \u2194 Xor' (true = true) (true = true)\n[PROOFSTEP]\nexact by decide\n[GOAL]\n\u22a2 xor true true = true \u2194 Xor' (true = true) (true = true)\n[PROOFSTEP]\ndecide\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Bool\n\u22a2 ((if c then a else b) = true) = if c then a = true else b = true\n[PROOFSTEP]\nby_cases c\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Bool\n\u22a2 ((if c then a else b) = true) = if c then a = true else b = true\n[PROOFSTEP]\nby_cases c\n[GOAL]\ncase pos\nc : Prop\ninst\u271d : Decidable c\na b : Bool\nh : c\n\u22a2 ((if c then a else b) = true) = if c then a = true else b = true\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nc : Prop\ninst\u271d : Decidable c\na b : Bool\nh : \u00acc\n\u22a2 ((if c then a else b) = true) = if c then a = true else b = true\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Bool\n\u22a2 ((if c then a else b) = false) = if c then a = false else b = false\n[PROOFSTEP]\nby_cases c\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Bool\n\u22a2 ((if c then a else b) = false) = if c then a = false else b = false\n[PROOFSTEP]\nby_cases c\n[GOAL]\ncase pos\nc : Prop\ninst\u271d : Decidable c\na b : Bool\nh : c\n\u22a2 ((if c then a else b) = false) = if c then a = false else b = false\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nc : Prop\ninst\u271d : Decidable c\na b : Bool\nh : \u00acc\n\u22a2 ((if c then a else b) = false) = if c then a = false else b = false\n[PROOFSTEP]\nsimp [*]\n", "meta": {"mathlib_filename": "Mathlib.Init.Data.Bool.Lemmas", "llama_tokens": 3364, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.793105941403651, "lm_q2_score": 0.7461389873857264, "lm_q1q2_score": 0.5917672640085234}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : AddMonoid R\nl : List R\n\u22a2 trop (sum l) = prod (map trop l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nR : Type u_1\nS : Type u_2\ninst\u271d : AddMonoid R\n\u22a2 trop (sum []) = prod (map trop [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u_1\nS : Type u_2\ninst\u271d : AddMonoid R\nhd : R\ntl : List R\nIH : trop (sum tl) = prod (map trop tl)\n\u22a2 trop (sum (hd :: tl)) = prod (map trop (hd :: tl))\n[PROOFSTEP]\nsimp [\u2190 IH]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Multiset R\n\u22a2 \u2200 (a : List R), trop (sum (Quotient.mk (List.isSetoid R) a)) = prod (map trop (Quotient.mk (List.isSetoid R) a))\n[PROOFSTEP]\nsimpa using List.trop_sum\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Finset S\nf : S \u2192 R\n\u22a2 trop (\u2211 i in s, f i) = \u220f i in s, trop (f i)\n[PROOFSTEP]\nconvert Multiset.trop_sum (s.val.map f)\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Finset S\nf : S \u2192 R\n\u22a2 \u220f i in s, trop (f i) = Multiset.prod (Multiset.map trop (Multiset.map f s.val))\n[PROOFSTEP]\nsimp only [Multiset.map_map, Function.comp_apply]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Finset S\nf : S \u2192 R\n\u22a2 \u220f i in s, trop (f i) = Multiset.prod (Multiset.map (fun i => trop (f i)) s.val)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : AddMonoid R\nl : List (Tropical R)\n\u22a2 untrop (prod l) = sum (map untrop l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nR : Type u_1\nS : Type u_2\ninst\u271d : AddMonoid R\n\u22a2 untrop (prod []) = sum (map untrop [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u_1\nS : Type u_2\ninst\u271d : AddMonoid R\nhd : Tropical R\ntl : List (Tropical R)\nIH : untrop (prod tl) = sum (map untrop tl)\n\u22a2 untrop (prod (hd :: tl)) = sum (map untrop (hd :: tl))\n[PROOFSTEP]\nsimp [\u2190 IH]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Multiset (Tropical R)\n\u22a2 \u2200 (a : List (Tropical R)),\n    untrop (prod (Quotient.mk (List.isSetoid (Tropical R)) a)) =\n      sum (map untrop (Quotient.mk (List.isSetoid (Tropical R)) a))\n[PROOFSTEP]\nsimpa using List.untrop_prod\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Finset S\nf : S \u2192 Tropical R\n\u22a2 untrop (\u220f i in s, f i) = \u2211 i in s, untrop (f i)\n[PROOFSTEP]\nconvert Multiset.untrop_prod (s.val.map f)\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Finset S\nf : S \u2192 Tropical R\n\u22a2 \u2211 i in s, untrop (f i) = Multiset.sum (Multiset.map untrop (Multiset.map f s.val))\n[PROOFSTEP]\nsimp only [Multiset.map_map, Function.comp_apply]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d : AddCommMonoid R\ns : Finset S\nf : S \u2192 Tropical R\n\u22a2 \u2211 i in s, untrop (f i) = Multiset.sum (Multiset.map (fun i => untrop (f i)) s.val)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : LinearOrder R\nl : List R\n\u22a2 trop (minimum l) = sum (map (trop \u2218 WithTop.some) l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nR : Type u_1\nS : Type u_2\ninst\u271d : LinearOrder R\n\u22a2 trop (minimum []) = sum (map (trop \u2218 WithTop.some) [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u_1\nS : Type u_2\ninst\u271d : LinearOrder R\nhd : R\ntl : List R\nIH : trop (minimum tl) = sum (map (trop \u2218 WithTop.some) tl)\n\u22a2 trop (minimum (hd :: tl)) = sum (map (trop \u2218 WithTop.some) (hd :: tl))\n[PROOFSTEP]\nsimp [List.minimum_cons, \u2190 IH]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Multiset R\n\u22a2 trop (inf s) = sum (map trop s)\n[PROOFSTEP]\ninduction' s using Multiset.induction with s x IH\n[GOAL]\ncase empty\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\n\u22a2 trop (inf 0) = sum (map trop 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : R\nx : Multiset R\nIH : trop (inf x) = sum (map trop x)\n\u22a2 trop (inf (s ::\u2098 x)) = sum (map trop (s ::\u2098 x))\n[PROOFSTEP]\nsimp [\u2190 IH]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Finset S\nf : S \u2192 R\n\u22a2 trop (inf s f) = \u2211 i in s, trop (f i)\n[PROOFSTEP]\nconvert Multiset.trop_inf (s.val.map f)\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Finset S\nf : S \u2192 R\n\u22a2 \u2211 i in s, trop (f i) = Multiset.sum (Multiset.map trop (Multiset.map f s.val))\n[PROOFSTEP]\nsimp only [Multiset.map_map, Function.comp_apply]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Finset S\nf : S \u2192 R\n\u22a2 \u2211 i in s, trop (f i) = Multiset.sum (Multiset.map (fun i => trop (f i)) s.val)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S \u2192 WithTop R\n\u22a2 trop (sInf (f '' \u2191s)) = \u2211 i in s, trop (f i)\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | h)\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nf : S \u2192 WithTop R\n\u22a2 trop (sInf (f '' \u2191\u2205)) = \u2211 i in \u2205, trop (f i)\n[PROOFSTEP]\nsimp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, trop_top]\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S \u2192 WithTop R\nh : Finset.Nonempty s\n\u22a2 trop (sInf (f '' \u2191s)) = \u2211 i in s, trop (f i)\n[PROOFSTEP]\nrw [\u2190 inf'_eq_csInf_image _ h, inf'_eq_inf, s.trop_inf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder R\ninst\u271d : Fintype S\nf : S \u2192 WithTop R\n\u22a2 trop (\u2a05 (i : S), f i) = \u2211 i : S, trop (f i)\n[PROOFSTEP]\nrw [iInf, \u2190 Set.image_univ, \u2190 coe_univ, trop_sInf_image]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Multiset (Tropical R)\n\u22a2 untrop (sum s) = inf (map untrop s)\n[PROOFSTEP]\ninduction' s using Multiset.induction with s x IH\n[GOAL]\ncase empty\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\n\u22a2 untrop (sum 0) = inf (map untrop 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Tropical R\nx : Multiset (Tropical R)\nIH : untrop (sum x) = inf (map untrop x)\n\u22a2 untrop (sum (s ::\u2098 x)) = inf (map untrop (s ::\u2098 x))\n[PROOFSTEP]\nsimp only [sum_cons, ge_iff_le, untrop_add, untrop_le_iff, map_cons, inf_cons, \u2190 IH]\n[GOAL]\ncase cons\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Tropical R\nx : Multiset (Tropical R)\nIH : untrop (sum x) = inf (map untrop x)\n\u22a2 min (untrop s) (untrop (sum x)) = untrop s \u2293 untrop (sum x)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Finset S\nf : S \u2192 Tropical R\n\u22a2 untrop (\u2211 i in s, f i) = inf s (untrop \u2218 f)\n[PROOFSTEP]\nconvert Multiset.untrop_sum (s.val.map f)\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Finset S\nf : S \u2192 Tropical R\n\u22a2 inf s (untrop \u2218 f) = Multiset.inf (Multiset.map untrop (Multiset.map f s.val))\n[PROOFSTEP]\nsimp only [Multiset.map_map, Function.comp_apply]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : LinearOrder R\ninst\u271d : OrderTop R\ns : Finset S\nf : S \u2192 Tropical R\n\u22a2 inf s (untrop \u2218 f) = Multiset.inf (Multiset.map (fun x => untrop (f x)) s.val)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S \u2192 Tropical (WithTop R)\n\u22a2 untrop (\u2211 i in s, f i) = sInf (untrop \u2218 f '' \u2191s)\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | h)\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nf : S \u2192 Tropical (WithTop R)\n\u22a2 untrop (\u2211 i in \u2205, f i) = sInf (untrop \u2218 f '' \u2191\u2205)\n[PROOFSTEP]\nsimp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, untrop_zero]\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S \u2192 Tropical (WithTop R)\nh : Finset.Nonempty s\n\u22a2 untrop (\u2211 i in s, f i) = sInf (untrop \u2218 f '' \u2191s)\n[PROOFSTEP]\nrw [\u2190 inf'_eq_csInf_image _ h, inf'_eq_inf, Finset.untrop_sum']\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder R\ninst\u271d : Fintype S\nf : S \u2192 Tropical (WithTop R)\n\u22a2 untrop (\u2211 i : S, f i) = \u2a05 (i : S), untrop (f i)\n[PROOFSTEP]\nrw [iInf, \u2190 Set.image_univ, \u2190 coe_univ, untrop_sum_eq_sInf_image]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : ConditionallyCompleteLinearOrder R\ninst\u271d : Fintype S\nf : S \u2192 Tropical (WithTop R)\n\u22a2 sInf ((untrop \u2218 fun i => f i) '' \u2191univ) = sInf ((fun i => untrop (f i)) '' \u2191univ)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S \u2192 Tropical (WithTop R)\n\u22a2 untrop (\u2211 i in s, f i) = \u2a05 (i : { x // x \u2208 s }), untrop (f \u2191i)\n[PROOFSTEP]\nsimpa [\u2190 _root_.untrop_sum] using sum_attach.symm\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Tropical.BigOperators", "llama_tokens": 4070, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711680567799, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.5915445104475238}}
{"text": "[GOAL]\nn m : \u2115\n\u22a2 succ n * m = 0 \u2192 succ n = 0 \u2228 m = 0\n[PROOFSTEP]\nrw [succ_mul]\n[GOAL]\nn m : \u2115\n\u22a2 n * m + m = 0 \u2192 succ n = 0 \u2228 m = 0\n[PROOFSTEP]\nintro h\n[GOAL]\nn m : \u2115\nh : n * m + m = 0\n\u22a2 succ n = 0 \u2228 m = 0\n[PROOFSTEP]\nexact Or.inr (Nat.eq_zero_of_add_eq_zero_left h)\n[GOAL]\nm : \u2115\nh : bit0 0 = bit0 (m + 1)\n\u22a2 0 = m + 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\nh : bit0 (n + 1) = bit0 0\n\u22a2 n + 1 = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\n\u22a2 n + 1 = m + 1\n[PROOFSTEP]\nhave : succ (succ (n + n)) = succ (succ (m + m)) :=\n  by\n  unfold bit0 at h ; simp [add_one, add_succ, succ_add] at h \n  have aux : n + n = m + m := h; rw [aux]\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))\n[PROOFSTEP]\nunfold bit0 at h \n[GOAL]\nn m : \u2115\nh : n + 1 + (n + 1) = m + 1 + (m + 1)\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))\n[PROOFSTEP]\nsimp [add_one, add_succ, succ_add] at h \n[GOAL]\nn m : \u2115\nh : n + n = m + m\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))\n[PROOFSTEP]\nhave aux : n + n = m + m := h\n[GOAL]\nn m : \u2115\nh aux : n + n = m + m\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))\n[PROOFSTEP]\nrw [aux]\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\nthis : succ (succ (n + n)) = succ (succ (m + m))\n\u22a2 n + 1 = m + 1\n[PROOFSTEP]\nhave : n + n = m + m := by repeat injection this with this\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\nthis : succ (succ (n + n)) = succ (succ (m + m))\n\u22a2 n + n = m + m\n[PROOFSTEP]\nrepeat injection this with this\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\nthis : succ (succ (n + n)) = succ (succ (m + m))\n\u22a2 n + n = m + m\n[PROOFSTEP]\ninjection this with this\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\nthis : succ (n + n) = succ (m + m)\n\u22a2 n + n = m + m\n[PROOFSTEP]\ninjection this with this\n[GOAL]\n\n[PROOFSTEP]\ninjection this with this\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\nthis\u271d : succ (succ (n + n)) = succ (succ (m + m))\nthis : n + n = m + m\n\u22a2 n + 1 = m + 1\n[PROOFSTEP]\nhave : n = m := Nat.bit0_inj this\n[GOAL]\nn m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\nthis\u271d\u00b9 : succ (succ (n + n)) = succ (succ (m + m))\nthis\u271d : n + n = m + m\nthis : n = m\n\u22a2 n + 1 = m + 1\n[PROOFSTEP]\nrw [this]\n[GOAL]\nn m : \u2115\nh : bit1 n = bit1 m\n\u22a2 succ (bit0 n) = succ (bit0 m)\n[PROOFSTEP]\nsimp [Nat.bit1_eq_succ_bit0] at h \n[GOAL]\nn m : \u2115\nh : bit0 n = bit0 m\n\u22a2 succ (bit0 n) = succ (bit0 m)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn m : \u2115\nh : bit1 n = bit1 m\nthis : succ (bit0 n) = succ (bit0 m)\n\u22a2 bit0 n = bit0 m\n[PROOFSTEP]\ninjection this\n[GOAL]\nh : 0 \u2260 0\n\u22a2 1 < bit1 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\n_h : succ n \u2260 0\n\u22a2 1 < bit1 (succ n)\n[PROOFSTEP]\nrw [Nat.bit1_succ_eq]\n[GOAL]\nn : \u2115\n_h : succ n \u2260 0\n\u22a2 1 < succ (succ (bit1 n))\n[PROOFSTEP]\napply succ_lt_succ\n[GOAL]\ncase a\nn : \u2115\n_h : succ n \u2260 0\n\u22a2 0 < bit1 n + 1\n[PROOFSTEP]\napply zero_lt_succ\n[GOAL]\nh : 0 \u2260 0\n\u22a2 1 < bit0 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\n_h : succ n \u2260 0\n\u22a2 1 < bit0 (succ n)\n[PROOFSTEP]\nrw [Nat.bit0_succ_eq]\n[GOAL]\nn : \u2115\n_h : succ n \u2260 0\n\u22a2 1 < succ (succ (bit0 n))\n[PROOFSTEP]\napply succ_lt_succ\n[GOAL]\ncase a\nn : \u2115\n_h : succ n \u2260 0\n\u22a2 0 < bit0 n + 1\n[PROOFSTEP]\napply zero_lt_succ\n[GOAL]\nn m : \u2115\nh : n < succ m\nthis\u271d : n \u2264 m\nthis : succ (n + n) \u2264 succ (m + m)\n\u22a2 succ (n + n) \u2264 succ m + m\n[PROOFSTEP]\nrw [succ_add]\n[GOAL]\nn m : \u2115\nh : n < succ m\nthis\u271d : n \u2264 m\nthis : succ (n + n) \u2264 succ (m + m)\n\u22a2 succ (n + n) \u2264 succ (m + m)\n[PROOFSTEP]\nassumption\n[GOAL]\nB : Sort u\nn : \u2115\nH1 : n = 0 \u2192 B\nH2 : (m : \u2115) \u2192 n = succ m \u2192 B\n\u22a2 B\n[PROOFSTEP]\ninduction' h : n\n[GOAL]\ncase zero\nB : Sort u\nn : \u2115\nH1 : n = 0 \u2192 B\nH2 : (m : \u2115) \u2192 n = succ m \u2192 B\nx\u271d : \u2115\nh\u271d : n = x\u271d\nh : n = zero\n\u22a2 B\n[PROOFSTEP]\nexact H1 h\n[GOAL]\ncase succ\nB : Sort u\nn : \u2115\nH1 : n = 0 \u2192 B\nH2 : (m : \u2115) \u2192 n = succ m \u2192 B\nx\u271d : \u2115\nh\u271d : n = x\u271d\nn\u271d : \u2115\nn_ih\u271d : n = n\u271d \u2192 B\nh : n = succ n\u271d\n\u22a2 B\n[PROOFSTEP]\nexact H2 _ h\n[GOAL]\nx y k : \u2115\nh : k \u2264 y\n\u22a2 x \u2264 y - k \u2194 x + k \u2264 y\n[PROOFSTEP]\nrw [\u2190 Nat.add_sub_cancel x k, Nat.sub_le_sub_iff_right h, Nat.add_sub_cancel]\n[GOAL]\nm n k : \u2115\n\u22a2 m - n - k = m - k - n\n[PROOFSTEP]\nrw [Nat.sub_sub, Nat.sub_sub, Nat.add_comm]\n[GOAL]\nx : \u2115\nd : Decidable (x % 2 = 1)\n\u22a2 (bif decide (x % 2 = 1) then 1 else 0) = x % 2\n[PROOFSTEP]\nby_cases h : x % 2 = 1\n[GOAL]\ncase pos\nx : \u2115\nd : Decidable (x % 2 = 1)\nh : x % 2 = 1\n\u22a2 (bif decide (x % 2 = 1) then 1 else 0) = x % 2\n[PROOFSTEP]\nsimp! [*]\n[GOAL]\ncase neg\nx : \u2115\nd : Decidable (x % 2 = 1)\nh : \u00acx % 2 = 1\n\u22a2 (bif decide (x % 2 = 1) then 1 else 0) = x % 2\n[PROOFSTEP]\ncases mod_two_eq_zero_or_one x\n[GOAL]\ncase neg.inl\nx : \u2115\nd : Decidable (x % 2 = 1)\nh : \u00acx % 2 = 1\nh\u271d : x % 2 = 0\n\u22a2 (bif decide (x % 2 = 1) then 1 else 0) = x % 2\n[PROOFSTEP]\nsimp! [*, Nat.zero_ne_one]\n[GOAL]\ncase neg.inr\nx : \u2115\nd : Decidable (x % 2 = 1)\nh : \u00acx % 2 = 1\nh\u271d : x % 2 = 1\n\u22a2 (bif decide (x % 2 = 1) then 1 else 0) = x % 2\n[PROOFSTEP]\nsimp! [*, Nat.zero_ne_one]\n[GOAL]\nm n k : \u2115\nH : 0 < m\n\u22a2 m * n / (m * k) = n / k\n[PROOFSTEP]\nrw [\u2190 Nat.div_div_eq_div_mul, Nat.mul_div_cancel_left _ H]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nn : \u2115\npn : p n\nm\u271d m : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc Nat.lbp k\nk : \u2115\nkn : n \u2264 k + succ m\ny : \u2115\nr : Nat.lbp y k\n_a : \u2200 (k_1 : \u2115), k_1 \u2264 k \u2192 \u00acp k_1\n\u22a2 n \u2264 k + 1 + m\n[PROOFSTEP]\nrw [Nat.add_right_comm]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nn : \u2115\npn : p n\nm\u271d m : \u2115\nIH : \u2200 (k : \u2115), n \u2264 k + m \u2192 Acc Nat.lbp k\nk : \u2115\nkn : n \u2264 k + succ m\ny : \u2115\nr : Nat.lbp y k\n_a : \u2200 (k_1 : \u2115), k_1 \u2264 k \u2192 \u00acp k_1\n\u22a2 n \u2264 k + m + 1\n[PROOFSTEP]\nexact kn\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp y m \u2192 (fun k => (\u2200 (n : \u2115), n < k \u2192 \u00acp n) \u2192 { n // p n \u2227 \u2200 (m : \u2115), m < n \u2192 \u00acp m }) y\nal : \u2200 (n : \u2115), n < m \u2192 \u00acp n\npm : \u00acp m\nn : \u2115\nh : n \u2264 m\ne : n = m\n\u22a2 \u00acp n\n[PROOFSTEP]\nrw [e]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nm : \u2115\nIH : (y : \u2115) \u2192 Nat.lbp y m \u2192 (fun k => (\u2200 (n : \u2115), n < k \u2192 \u00acp n) \u2192 { n // p n \u2227 \u2200 (m : \u2115), m < n \u2192 \u00acp m }) y\nal : \u2200 (n : \u2115), n < m \u2192 \u00acp n\npm : \u00acp m\nn : \u2115\nh : n \u2264 m\ne : n = m\n\u22a2 \u00acp m\n[PROOFSTEP]\nexact pm\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\n\u22a2 \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\n[PROOFSTEP]\ninduction f with (simp only [Nat.toDigitsCore, List.length]; intro n l1 l2 hlen)\n| zero => assumption\n| succ f ih =>\n  by_cases hx : n / b = 0\n  case pos => simp only [hx, if_true, List.length, congrArg (fun l \u21a6 l + 1) hlen]\n  case neg =>\n    simp only [hx, if_false]\n    specialize ih (n / b) (Nat.digitChar (n % b) :: l1) (Nat.digitChar (n % b) :: l2)\n    simp only [List.length, congrArg (fun l \u21a6 l + 1) hlen] at ih \n    exact ih trivial\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\n\u22a2 \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\n[PROOFSTEP]\ninduction f with (simp only [Nat.toDigitsCore, List.length]; intro n l1 l2 hlen)\n| zero => assumption\n| succ f ih =>\n  by_cases hx : n / b = 0\n  case pos => simp only [hx, if_true, List.length, congrArg (fun l \u21a6 l + 1) hlen]\n  case neg =>\n    simp only [hx, if_false]\n    specialize ih (n / b) (Nat.digitChar (n % b) :: l1) (Nat.digitChar (n % b) :: l2)\n    simp only [List.length, congrArg (fun l \u21a6 l + 1) hlen] at ih \n    exact ih trivial\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\n\u22a2 \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b zero n l1) = List.length (toDigitsCore b zero n l2)\n[PROOFSTEP]\nsimp only [Nat.toDigitsCore, List.length]\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\n\u22a2 \u2115 \u2192 \u2200 (l1 l2 : List Char), List.length l1 = List.length l2 \u2192 List.length l1 = List.length l2\n[PROOFSTEP]\nintro n l1 l2 hlen\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb n : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\n\u22a2 List.length l1 = List.length l2\n[PROOFSTEP]\n\n| zero => assumption\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb n : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\n\u22a2 List.length l1 = List.length l2\n[PROOFSTEP]\nassumption\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\n\u22a2 \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192\n      List.length (toDigitsCore b (succ f) n l1) = List.length (toDigitsCore b (succ f) n l2)\n[PROOFSTEP]\nsimp only [Nat.toDigitsCore, List.length]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\n\u22a2 \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192\n      List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n        List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nintro n l1 l2 hlen\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\n\n| succ f ih =>\n  by_cases hx : n / b = 0\n  case pos => simp only [hx, if_true, List.length, congrArg (fun l \u21a6 l + 1) hlen]\n  case neg =>\n    simp only [hx, if_false]\n    specialize ih (n / b) (Nat.digitChar (n % b) :: l1) (Nat.digitChar (n % b) :: l2)\n    simp only [List.length, congrArg (fun l \u21a6 l + 1) hlen] at ih \n    exact ih trivial\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nby_cases hx : n / b = 0\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : n / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\ncase pos => simp only [hx, if_true, List.length, congrArg (fun l \u21a6 l + 1) hlen]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : n / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\ncase pos => simp only [hx, if_true, List.length, congrArg (fun l \u21a6 l + 1) hlen]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : n / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nsimp only [hx, if_true, List.length, congrArg (fun l \u21a6 l + 1) hlen]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\ncase neg =>\n  simp only [hx, if_false]\n  specialize ih (n / b) (Nat.digitChar (n % b) :: l1) (Nat.digitChar (n % b) :: l2)\n  simp only [List.length, congrArg (fun l \u21a6 l + 1) hlen] at ih \n  exact ih trivial\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\ncase neg =>\n  simp only [hx, if_false]\n  specialize ih (n / b) (Nat.digitChar (n % b) :: l1) (Nat.digitChar (n % b) :: l2)\n  simp only [List.length, congrArg (fun l \u21a6 l + 1) hlen] at ih \n  exact ih trivial\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then digitChar (n % b) :: l1 else toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: l2 else toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nsimp only [hx, if_false]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (l1 l2 : List Char),\n    List.length l1 = List.length l2 \u2192 List.length (toDigitsCore b f n l1) = List.length (toDigitsCore b f n l2)\nn : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\n\u22a2 List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nspecialize ih (n / b) (Nat.digitChar (n % b) :: l1) (Nat.digitChar (n % b) :: l2)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\nih :\n  List.length (digitChar (n % b) :: l1) = List.length (digitChar (n % b) :: l2) \u2192\n    List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n      List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n\u22a2 List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nsimp only [List.length, congrArg (fun l \u21a6 l + 1) hlen] at ih \n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nl1 l2 : List Char\nhlen : List.length l1 = List.length l2\nhx : \u00acn / b = 0\nih :\n  True \u2192\n    List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n      List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n\u22a2 List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l1)) =\n    List.length (toDigitsCore b f (n / b) (digitChar (n % b) :: l2))\n[PROOFSTEP]\nexact ih trivial\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\n\u22a2 \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\n[PROOFSTEP]\ninduction f with (intro n c tl; simp only [Nat.toDigitsCore, List.length])\n| succ f ih =>\n  by_cases hnb : (n / b) = 0\n  case pos => simp only [hnb, if_true, List.length]\n  case neg =>\n    generalize hx : Nat.digitChar (n % b) = x\n    simp only [hx, hnb, if_false] at ih \n    simp only [hnb, if_false]\n    specialize ih (n / b) c (x :: tl)\n    rw [\u2190 ih]\n    have lens_eq : (x :: (c :: tl)).length = (c :: x :: tl).length := by simp\n    apply to_digits_core_lens_eq_aux\n    exact lens_eq\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\n\u22a2 \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\n[PROOFSTEP]\ninduction f with (intro n c tl; simp only [Nat.toDigitsCore, List.length])\n| succ f ih =>\n  by_cases hnb : (n / b) = 0\n  case pos => simp only [hnb, if_true, List.length]\n  case neg =>\n    generalize hx : Nat.digitChar (n % b) = x\n    simp only [hx, hnb, if_false] at ih \n    simp only [hnb, if_false]\n    specialize ih (n / b) c (x :: tl)\n    rw [\u2190 ih]\n    have lens_eq : (x :: (c :: tl)).length = (c :: x :: tl).length := by simp\n    apply to_digits_core_lens_eq_aux\n    exact lens_eq\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\n\u22a2 \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b (succ f) n (c :: tl)) = List.length (toDigitsCore b (succ f) n tl) + 1\n[PROOFSTEP]\nintro n c tl\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\n\u22a2 List.length (toDigitsCore b (succ f) n (c :: tl)) = List.length (toDigitsCore b (succ f) n tl) + 1\n[PROOFSTEP]\nsimp only [Nat.toDigitsCore, List.length]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\n\n| succ f ih =>\n  by_cases hnb : (n / b) = 0\n  case pos => simp only [hnb, if_true, List.length]\n  case neg =>\n    generalize hx : Nat.digitChar (n % b) = x\n    simp only [hx, hnb, if_false] at ih \n    simp only [hnb, if_false]\n    specialize ih (n / b) c (x :: tl)\n    rw [\u2190 ih]\n    have lens_eq : (x :: (c :: tl)).length = (c :: x :: tl).length := by simp\n    apply to_digits_core_lens_eq_aux\n    exact lens_eq\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\nby_cases hnb : (n / b) = 0\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : n / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\ncase pos => simp only [hnb, if_true, List.length]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : n / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\ncase pos => simp only [hnb, if_true, List.length]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : n / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\nsimp only [hnb, if_true, List.length]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\ncase neg =>\n  generalize hx : Nat.digitChar (n % b) = x\n  simp only [hx, hnb, if_false] at ih \n  simp only [hnb, if_false]\n  specialize ih (n / b) c (x :: tl)\n  rw [\u2190 ih]\n  have lens_eq : (x :: (c :: tl)).length = (c :: x :: tl).length := by simp\n  apply to_digits_core_lens_eq_aux\n  exact lens_eq\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\ncase neg =>\n  generalize hx : Nat.digitChar (n % b) = x\n  simp only [hx, hnb, if_false] at ih \n  simp only [hnb, if_false]\n  specialize ih (n / b) c (x :: tl)\n  rw [\u2190 ih]\n  have lens_eq : (x :: (c :: tl)).length = (c :: x :: tl).length := by simp\n  apply to_digits_core_lens_eq_aux\n  exact lens_eq\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\n\u22a2 List.length\n      (if n / b = 0 then digitChar (n % b) :: c :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: c :: tl)) =\n    List.length (if n / b = 0 then digitChar (n % b) :: tl else toDigitsCore b f (n / b) (digitChar (n % b) :: tl)) + 1\n[PROOFSTEP]\ngeneralize hx : Nat.digitChar (n % b) = x\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\n\u22a2 List.length (if n / b = 0 then x :: c :: tl else toDigitsCore b f (n / b) (x :: c :: tl)) =\n    List.length (if n / b = 0 then x :: tl else toDigitsCore b f (n / b) (x :: tl)) + 1\n[PROOFSTEP]\nsimp only [hx, hnb, if_false] at ih \n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\n\u22a2 List.length (if n / b = 0 then x :: c :: tl else toDigitsCore b f (n / b) (x :: c :: tl)) =\n    List.length (if n / b = 0 then x :: tl else toDigitsCore b f (n / b) (x :: tl)) + 1\n[PROOFSTEP]\nsimp only [hnb, if_false]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f : \u2115\nih :\n  \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b f n (c :: tl)) = List.length (toDigitsCore b f n tl) + 1\nn : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\n\u22a2 List.length (toDigitsCore b f (n / b) (x :: c :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\n[PROOFSTEP]\nspecialize ih (n / b) c (x :: tl)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\nih : List.length (toDigitsCore b f (n / b) (c :: x :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\n\u22a2 List.length (toDigitsCore b f (n / b) (x :: c :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\n[PROOFSTEP]\nrw [\u2190 ih]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\nih : List.length (toDigitsCore b f (n / b) (c :: x :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\n\u22a2 List.length (toDigitsCore b f (n / b) (x :: c :: tl)) = List.length (toDigitsCore b f (n / b) (c :: x :: tl))\n[PROOFSTEP]\nhave lens_eq : (x :: (c :: tl)).length = (c :: x :: tl).length := by simp\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\nih : List.length (toDigitsCore b f (n / b) (c :: x :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\n\u22a2 List.length (x :: c :: tl) = List.length (c :: x :: tl)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\nih : List.length (toDigitsCore b f (n / b) (c :: x :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\nlens_eq : List.length (x :: c :: tl) = List.length (c :: x :: tl)\n\u22a2 List.length (toDigitsCore b f (n / b) (x :: c :: tl)) = List.length (toDigitsCore b f (n / b) (c :: x :: tl))\n[PROOFSTEP]\napply to_digits_core_lens_eq_aux\n[GOAL]\ncase a\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb f n : \u2115\nc : Char\ntl : List Char\nhnb : \u00acn / b = 0\nx : Char\nhx : digitChar (n % b) = x\nih : List.length (toDigitsCore b f (n / b) (c :: x :: tl)) = List.length (toDigitsCore b f (n / b) (x :: tl)) + 1\nlens_eq : List.length (x :: c :: tl) = List.length (c :: x :: tl)\n\u22a2 List.length (x :: c :: tl) = List.length (c :: x :: tl)\n[PROOFSTEP]\nexact lens_eq\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\n\u22a2 \u2200 (n : \u2115) (c : Char) (tl : List Char),\n    List.length (toDigitsCore b zero n (c :: tl)) = List.length (toDigitsCore b zero n tl) + 1\n[PROOFSTEP]\nintro n c tl\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb n : \u2115\nc : Char\ntl : List Char\n\u22a2 List.length (toDigitsCore b zero n (c :: tl)) = List.length (toDigitsCore b zero n tl) + 1\n[PROOFSTEP]\nsimp only [Nat.toDigitsCore, List.length]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nn b e : \u2115\nh_b_pos : 0 < b\n\u22a2 n < b ^ succ e \u2192 n / b < b ^ e\n[PROOFSTEP]\nsimp only [Nat.pow_succ]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nn b e : \u2115\nh_b_pos : 0 < b\n\u22a2 n < b ^ e * b \u2192 n / b < b ^ e\n[PROOFSTEP]\nexact (@Nat.div_lt_iff_lt_mul b n (b ^ e) h_b_pos).mpr\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (toDigitsCore b f n []) \u2264 e\n[PROOFSTEP]\ninduction f generalizing n e hlt h_e_pos with simp only [Nat.toDigitsCore, List.length, Nat.zero_le]\n| succ f ih =>\n  cases e with\n  | zero => exact False.elim (Nat.lt_irrefl 0 h_e_pos)\n  | succ e =>\n    by_cases h_pred_pos : 0 < e\n    case pos =>\n      have _ : 0 < b := Nat.lt_trans (by decide) h\n      specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n      by_cases hdiv_ten : n / b = 0\n      case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n      case neg =>\n        simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n        exact Nat.succ_le_succ ih\n    case neg =>\n      have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n      rw [\u2039e = 0\u203a]\n      have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n      have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n      simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (toDigitsCore b f n []) \u2264 e\n[PROOFSTEP]\ninduction f generalizing n e hlt h_e_pos with simp only [Nat.toDigitsCore, List.length, Nat.zero_le]\n| succ f ih =>\n  cases e with\n  | zero => exact False.elim (Nat.lt_irrefl 0 h_e_pos)\n  | succ e =>\n    by_cases h_pred_pos : 0 < e\n    case pos =>\n      have _ : 0 < b := Nat.lt_trans (by decide) h\n      specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n      by_cases hdiv_ten : n / b = 0\n      case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n      case neg =>\n        simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n        exact Nat.succ_le_succ ih\n    case neg =>\n      have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n      rw [\u2039e = 0\u203a]\n      have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n      have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n      simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (toDigitsCore b (succ f) n []) \u2264 e\n[PROOFSTEP]\nsimp only [Nat.toDigitsCore, List.length, Nat.zero_le]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 e\n[PROOFSTEP]\n\n| succ f ih =>\n  cases e with\n  | zero => exact False.elim (Nat.lt_irrefl 0 h_e_pos)\n  | succ e =>\n    by_cases h_pred_pos : 0 < e\n    case pos =>\n      have _ : 0 < b := Nat.lt_trans (by decide) h\n      specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n      by_cases hdiv_ten : n / b = 0\n      case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n      case neg =>\n        simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n        exact Nat.succ_le_succ ih\n    case neg =>\n      have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n      rw [\u2039e = 0\u203a]\n      have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n      have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n      simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 e\n[PROOFSTEP]\ncases e with\n| zero => exact False.elim (Nat.lt_irrefl 0 h_e_pos)\n| succ e =>\n  by_cases h_pred_pos : 0 < e\n  case pos =>\n    have _ : 0 < b := Nat.lt_trans (by decide) h\n    specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n    by_cases hdiv_ten : n / b = 0\n    case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n    case neg =>\n      simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n      exact Nat.succ_le_succ ih\n  case neg =>\n    have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n    rw [\u2039e = 0\u203a]\n    have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n    have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n    simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 e\n[PROOFSTEP]\ncases e with\n| zero => exact False.elim (Nat.lt_irrefl 0 h_e_pos)\n| succ e =>\n  by_cases h_pred_pos : 0 < e\n  case pos =>\n    have _ : 0 < b := Nat.lt_trans (by decide) h\n    specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n    by_cases hdiv_ten : n / b = 0\n    case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n    case neg =>\n      simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n      exact Nat.succ_le_succ ih\n  case neg =>\n    have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n    rw [\u2039e = 0\u203a]\n    have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n    have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n    simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\ncase succ.zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn : \u2115\nhlt : n < b ^ zero\nh_e_pos : 0 < zero\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 zero\n[PROOFSTEP]\n\n| zero => exact False.elim (Nat.lt_irrefl 0 h_e_pos)\n[GOAL]\ncase succ.zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn : \u2115\nhlt : n < b ^ zero\nh_e_pos : 0 < zero\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 zero\n[PROOFSTEP]\nexact False.elim (Nat.lt_irrefl 0 h_e_pos)\n[GOAL]\ncase succ.succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\n\n| succ e =>\n  by_cases h_pred_pos : 0 < e\n  case pos =>\n    have _ : 0 < b := Nat.lt_trans (by decide) h\n    specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n    by_cases hdiv_ten : n / b = 0\n    case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n    case neg =>\n      simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n      exact Nat.succ_le_succ ih\n  case neg =>\n    have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n    rw [\u2039e = 0\u203a]\n    have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n    have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n    simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\ncase succ.succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nby_cases h_pred_pos : 0 < e\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase pos =>\n  have _ : 0 < b := Nat.lt_trans (by decide) h\n  specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n  by_cases hdiv_ten : n / b = 0\n  case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n  case neg =>\n    simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n    exact Nat.succ_le_succ ih\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase pos =>\n  have _ : 0 < b := Nat.lt_trans (by decide) h\n  specialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n  by_cases hdiv_ten : n / b = 0\n  case pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n  case neg =>\n    simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n    exact Nat.succ_le_succ ih\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nhave _ : 0 < b := Nat.lt_trans (by decide) h\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\n\u22a2 0 < 1\n[PROOFSTEP]\ndecide\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nspecialize ih (n / b) e (nat_repr_len_aux n b e \u20390 < b\u203a hlt) h_pred_pos\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nby_cases hdiv_ten : n / b = 0\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : n / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : n / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase pos => simp only [hdiv_ten]; exact Nat.le.step h_pred_pos\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : n / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nsimp only [hdiv_ten]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : n / b = 0\n\u22a2 List.length (if True then [digitChar (n % b)] else toDigitsCore b f 0 [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nexact Nat.le.step h_pred_pos\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase neg =>\n  simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n  exact Nat.succ_le_succ ih\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase neg =>\n  simp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n  exact Nat.succ_le_succ ih\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : \u00acn / b = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nsimp only [hdiv_ten, to_digits_core_lens_eq b f (n / b) (Nat.digitChar $ n % b), if_false]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf n e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : 0 < e\nx\u271d : 0 < b\nih : List.length (toDigitsCore b f (n / b) []) \u2264 e\nhdiv_ten : \u00acn / b = 0\n\u22a2 List.length (toDigitsCore b f (n / b) []) + 1 \u2264 succ e\n[PROOFSTEP]\nexact Nat.succ_le_succ ih\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase neg =>\n  have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n  rw [\u2039e = 0\u203a]\n  have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n  have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n  simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\ncase neg =>\n  have _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n  rw [\u2039e = 0\u203a]\n  have _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n  have _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n  simp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nhave _ : e = 0 := Nat.eq_zero_of_nonpos e h_pred_pos\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\nx\u271d : e = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ e\n[PROOFSTEP]\nrw [\u2039e = 0\u203a]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\nx\u271d : e = 0\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ 0\n[PROOFSTEP]\nhave _ : b ^ 1 = b := by simp only [pow_succ, pow_zero, Nat.one_mul]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\nx\u271d : e = 0\n\u22a2 b ^ 1 = b\n[PROOFSTEP]\nsimp only [pow_succ, pow_zero, Nat.one_mul]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\nx\u271d\u00b9 : e = 0\nx\u271d : b ^ 1 = b\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ 0\n[PROOFSTEP]\nhave _ : n < b := \u2039b ^ 1 = b\u203a \u25b8 (\u2039e = 0\u203a \u25b8 hlt : n < b ^ Nat.succ 0)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nf : \u2115\nih : \u2200 (n e : \u2115), n < b ^ e \u2192 0 < e \u2192 List.length (toDigitsCore b f n []) \u2264 e\nn e : \u2115\nhlt : n < b ^ succ e\nh_e_pos : 0 < succ e\nh_pred_pos : \u00ac0 < e\nx\u271d\u00b2 : e = 0\nx\u271d\u00b9 : b ^ 1 = b\nx\u271d : n < b\n\u22a2 List.length (if n / b = 0 then [digitChar (n % b)] else toDigitsCore b f (n / b) [digitChar (n % b)]) \u2264 succ 0\n[PROOFSTEP]\nsimp only [(@Nat.div_eq_of_lt n b \u2039n < b\u203a : n / b = 0), if_true, List.length]\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nb : \u2115\nh : 2 \u2264 b\nn e : \u2115\nhlt : n < b ^ e\nh_e_pos : 0 < e\n\u22a2 List.length (toDigitsCore b zero n []) \u2264 e\n[PROOFSTEP]\nsimp only [Nat.toDigitsCore, List.length, Nat.zero_le]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nn e : \u2115\n\u22a2 0 < e \u2192 n < 10 ^ e \u2192 String.length (Nat.repr n) \u2264 e\n[PROOFSTEP]\ncases n with (intro e0 he; simp only [Nat.repr, Nat.toDigits, String.length, List.asString])\n| zero => assumption\n| succ n =>\n  by_cases hterm : n.succ / 10 = 0\n  case pos => simp only [hterm, Nat.toDigitsCore]; assumption\n  case neg => exact to_digits_core_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\nn e : \u2115\n\u22a2 0 < e \u2192 n < 10 ^ e \u2192 String.length (Nat.repr n) \u2264 e\n[PROOFSTEP]\ncases n with (intro e0 he; simp only [Nat.repr, Nat.toDigits, String.length, List.asString])\n| zero => assumption\n| succ n =>\n  by_cases hterm : n.succ / 10 = 0\n  case pos => simp only [hterm, Nat.toDigitsCore]; assumption\n  case neg => exact to_digits_core_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne : \u2115\n\u22a2 0 < e \u2192 zero < 10 ^ e \u2192 String.length (Nat.repr zero) \u2264 e\n[PROOFSTEP]\nintro e0 he\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne : \u2115\ne0 : 0 < e\nhe : zero < 10 ^ e\n\u22a2 String.length (Nat.repr zero) \u2264 e\n[PROOFSTEP]\nsimp only [Nat.repr, Nat.toDigits, String.length, List.asString]\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne : \u2115\ne0 : 0 < e\nhe : zero < 10 ^ e\n\u22a2 List.length (toDigitsCore 10 (zero + 1) zero []) \u2264 e\n[PROOFSTEP]\n\n| zero => assumption\n[GOAL]\ncase zero\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne : \u2115\ne0 : 0 < e\nhe : zero < 10 ^ e\n\u22a2 List.length (toDigitsCore 10 (zero + 1) zero []) \u2264 e\n[PROOFSTEP]\nassumption\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\n\u22a2 0 < e \u2192 succ n < 10 ^ e \u2192 String.length (Nat.repr (succ n)) \u2264 e\n[PROOFSTEP]\nintro e0 he\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\n\u22a2 String.length (Nat.repr (succ n)) \u2264 e\n[PROOFSTEP]\nsimp only [Nat.repr, Nat.toDigits, String.length, List.asString]\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\n\n| succ n =>\n  by_cases hterm : n.succ / 10 = 0\n  case pos => simp only [hterm, Nat.toDigitsCore]; assumption\n  case neg => exact to_digits_core_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0\n[GOAL]\ncase succ\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\nby_cases hterm : n.succ / 10 = 0\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : succ n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : \u00acsucc n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\ncase pos => simp only [hterm, Nat.toDigitsCore]; assumption\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : succ n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\ncase pos => simp only [hterm, Nat.toDigitsCore]; assumption\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : succ n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\nsimp only [hterm, Nat.toDigitsCore]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : succ n / 10 = 0\n\u22a2 List.length\n      (if True then [digitChar (succ n % 10)]\n      else\n        if True then [digitChar (0 % 10), digitChar (succ n % 10)]\n        else toDigitsCore 10 n (0 / 10) [digitChar (0 % 10), digitChar (succ n % 10)]) \u2264\n    e\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : \u00acsucc n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\ncase neg => exact to_digits_core_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : \u00acsucc n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\ncase neg => exact to_digits_core_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : \u00acsucc n / 10 = 0\n\u22a2 List.length (toDigitsCore 10 (succ n + 1) (succ n) []) \u2264 e\n[PROOFSTEP]\nexact to_digits_core_length 10 (by decide) (Nat.succ n + 1) (Nat.succ n) e he e0\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nH : \u2203 n, p n\ne n : \u2115\ne0 : 0 < e\nhe : succ n < 10 ^ e\nhterm : \u00acsucc n / 10 = 0\n\u22a2 2 \u2264 10\n[PROOFSTEP]\ndecide\n", "meta": {"mathlib_filename": "Mathlib.Init.Data.Nat.Lemmas", "llama_tokens": 25915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.7662936484231889, "lm_q1q2_score": 0.5914587699708201}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NonUnitalNonAssocRing \u03b1\nk : \u03b1\nh : \u2200 (x : \u03b1), k * x = 0 \u2192 x = 0\n\u22a2 IsLeftRegular k\n[PROOFSTEP]\nrefine' fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NonUnitalNonAssocRing \u03b1\nk : \u03b1\nh : \u2200 (x : \u03b1), k * x = 0 \u2192 x = 0\nx y : \u03b1\nh' : k * x = k * y\n\u22a2 k * (x - y) = 0\n[PROOFSTEP]\nrw [mul_sub, sub_eq_zero, h']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NonUnitalNonAssocRing \u03b1\nk : \u03b1\nh : \u2200 (x : \u03b1), x * k = 0 \u2192 x = 0\n\u22a2 IsRightRegular k\n[PROOFSTEP]\nrefine' fun x y (h' : x * k = y * k) => sub_eq_zero.mp (h _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : NonUnitalNonAssocRing \u03b1\nk : \u03b1\nh : \u2200 (x : \u03b1), x * k = 0 \u2192 x = 0\nx y : \u03b1\nh' : x * k = y * k\n\u22a2 (x - y) * k = 0\n[PROOFSTEP]\nrw [sub_mul, sub_eq_zero, h']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Nontrivial \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : NoZeroDivisors \u03b1\nk : \u03b1\nh : IsRegular k\n\u22a2 k \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Nontrivial \u03b1\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : NoZeroDivisors \u03b1\nh : IsRegular 0\n\u22a2 False\n[PROOFSTEP]\nexact not_not.mpr h.left not_isLeftRegular_zero\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : NoZeroDivisors \u03b1\n\u22a2 MonoidWithZero \u03b1\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Regular", "llama_tokens": 620, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267118111485244, "lm_q2_score": 0.7154239836484143, "lm_q1q2_score": 0.5914494572610729}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : CharP R p\nI : Ideal R\nh : \u2200 (x : \u2115), \u2191x \u2208 I \u2192 \u2191x = 0\nx : \u2115\n\u22a2 \u2191x = 0 \u2194 p \u2223 x\n[PROOFSTEP]\nrw [\u2190 cast_eq_zero_iff R p x, \u2190 map_natCast (Ideal.Quotient.mk I)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : CharP R p\nI : Ideal R\nh : \u2200 (x : \u2115), \u2191x \u2208 I \u2192 \u2191x = 0\nx : \u2115\n\u22a2 \u2191(Ideal.Quotient.mk I) \u2191x = 0 \u2194 \u2191x = 0\n[PROOFSTEP]\nrefine' Ideal.Quotient.eq.trans (_ : \u2191x - 0 \u2208 I \u2194 _)\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : CharP R p\nI : Ideal R\nh : \u2200 (x : \u2115), \u2191x \u2208 I \u2192 \u2191x = 0\nx : \u2115\n\u22a2 \u2191x - 0 \u2208 I \u2194 \u2191x = 0\n[PROOFSTEP]\nrw [sub_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\np : \u2115\ninst\u271d : CharP R p\nI : Ideal R\nh : \u2200 (x : \u2115), \u2191x \u2208 I \u2192 \u2191x = 0\nx : \u2115\n\u22a2 \u2191x \u2208 I \u2194 \u2191x = 0\n[PROOFSTEP]\nexact \u27e8h x, fun h' => h'.symm \u25b8 I.zero_mem\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2191(AddSubgroup.index (Submodule.toAddSubgroup I)) = 0\n[PROOFSTEP]\nrw [AddSubgroup.index, Nat.card_eq]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2191(if h : Finite (R \u29f8 Submodule.toAddSubgroup I) then Fintype.card (R \u29f8 Submodule.toAddSubgroup I) else 0) = 0\n[PROOFSTEP]\nsplit_ifs with hq\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\ncase neg R : Type u_1 inst\u271d : CommRing R I : Ideal R hq : \u00acFinite (R \u29f8 Submodule.toAddSubgroup I) \u22a2 \u21910 = 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : \u00acFinite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u21910 = 0\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\n\u22a2 \u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\nh : \u00ac\u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\n\u22a2 False\n[PROOFSTEP]\nletI : Fintype (R \u29f8 I) := @Fintype.ofFinite _ hq\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\nh : \u00ac\u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\nthis : Fintype (R \u29f8 I) := Fintype.ofFinite (R \u29f8 I)\n\u22a2 False\n[PROOFSTEP]\nhave h : (Fintype.card (R \u29f8 I) : R \u29f8 I) \u2260 0 := h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nhq : Finite (R \u29f8 Submodule.toAddSubgroup I)\nh\u271d : \u00ac\u2191(Fintype.card (R \u29f8 Submodule.toAddSubgroup I)) = 0\nthis : Fintype (R \u29f8 I) := Fintype.ofFinite (R \u29f8 I)\nh : \u2191(Fintype.card (R \u29f8 I)) \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.Quotient", "llama_tokens": 1465, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256432832333, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.5914078174925256}}
{"text": "[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : One R\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na : \u03b1\ne : Sym2 \u03b1\n\u22a2 incMatrix R G a e = if e \u2208 incidenceSet G a then 1 else 0\n[PROOFSTEP]\nunfold incMatrix Set.indicator\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Zero R\ninst\u271d\u00b2 : One R\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na : \u03b1\ne : Sym2 \u03b1\n\u22a2 (if e \u2208 incidenceSet G a then OfNat.ofNat 1 e else 0) = if e \u2208 incidenceSet G a then 1 else 0\n[PROOFSTEP]\nsimp only [Pi.one_apply]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\n\u22a2 incMatrix R G a e * incMatrix R G b e = Set.indicator (incidenceSet G a \u2229 incidenceSet G b) 1 e\n[PROOFSTEP]\nclassical simp only [incMatrix, Set.indicator_apply, \u2190 ite_and_mul_zero, Pi.one_apply, mul_one, Set.mem_inter_iff]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\n\u22a2 incMatrix R G a e * incMatrix R G b e = Set.indicator (incidenceSet G a \u2229 incidenceSet G b) 1 e\n[PROOFSTEP]\nsimp only [incMatrix, Set.indicator_apply, \u2190 ite_and_mul_zero, Pi.one_apply, mul_one, Set.mem_inter_iff]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\nhab : a \u2260 b\nh : \u00acAdj G a b\n\u22a2 incMatrix R G a e * incMatrix R G b e = 0\n[PROOFSTEP]\nrw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem]\n[GOAL]\ncase h\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\nhab : a \u2260 b\nh : \u00acAdj G a b\n\u22a2 \u00ace \u2208 incidenceSet G a \u2229 incidenceSet G b\n[PROOFSTEP]\nrw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]\n[GOAL]\ncase h\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\nhab : a \u2260 b\nh : \u00acAdj G a b\n\u22a2 \u00ace \u2208 \u2205\n[PROOFSTEP]\nexact Set.not_mem_empty e\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\nh : \u00ace \u2208 incidenceSet G a\n\u22a2 incMatrix R G a e = 0\n[PROOFSTEP]\nrw [incMatrix_apply, Set.indicator_of_not_mem h]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\nh : e \u2208 incidenceSet G a\n\u22a2 incMatrix R G a e = 1\n[PROOFSTEP]\nrw [incMatrix_apply, Set.indicator_of_mem h, Pi.one_apply]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\n\u22a2 incMatrix R G a e = 0 \u2194 \u00ace \u2208 incidenceSet G a\n[PROOFSTEP]\nsimp only [incMatrix_apply, Set.indicator_apply_eq_zero, Pi.one_apply, one_ne_zero]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\n\u22a2 incMatrix R G a e = 1 \u2194 e \u2208 incidenceSet G a\n[PROOFSTEP]\nunfold incMatrix Set.indicator\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\n\u22a2 (if e \u2208 incidenceSet G a then OfNat.ofNat 1 e else 0) = 1 \u2194 e \u2208 incidenceSet G a\n[PROOFSTEP]\nsimp only [Pi.one_apply]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\n\u22a2 (if e \u2208 incidenceSet G a then 1 else 0) = 1 \u2194 e \u2208 incidenceSet G a\n[PROOFSTEP]\napply Iff.intro\n[GOAL]\ncase mp\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\n\u22a2 (if e \u2208 incidenceSet G a then 1 else 0) = 1 \u2192 e \u2208 incidenceSet G a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\n\u22a2 e \u2208 incidenceSet G a \u2192 (if e \u2208 incidenceSet G a then 1 else 0) = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\nh : (if e \u2208 incidenceSet G a then 1 else 0) = 1\n\u22a2 e \u2208 incidenceSet G a\n[PROOFSTEP]\nsplit at h \n[GOAL]\ncase mp.inl\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\nh\u271d : e \u2208 incidenceSet G a\nh : 1 = 1\n\u22a2 e \u2208 incidenceSet G a\n[PROOFSTEP]\nsimp_all only [zero_ne_one]\n[GOAL]\ncase mp.inr\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\nh\u271d : \u00ace \u2208 incidenceSet G a\nh : 0 = 1\n\u22a2 e \u2208 incidenceSet G a\n[PROOFSTEP]\nsimp_all only [zero_ne_one]\n[GOAL]\ncase mpr\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : MulZeroOneClass R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : Nontrivial R\nh : e \u2208 incidenceSet G a\n\u22a2 (if e \u2208 incidenceSet G a then 1 else 0) = 1\n[PROOFSTEP]\nsimp_all only [ite_true]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 \u2211 e : Sym2 \u03b1, incMatrix R G a e = \u2191(degree G a)\n[PROOFSTEP]\nsimp [incMatrix_apply', sum_boole, Set.filter_mem_univ_eq_toFinset]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a a = \u2191(degree G a)\n[PROOFSTEP]\nrw [\u2190 sum_incMatrix_apply]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a a = \u2211 e : Sym2 \u03b1, incMatrix R G a e\n[PROOFSTEP]\nsimp only [mul_apply, incMatrix_apply', transpose_apply, mul_ite, mul_one, mul_zero]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 (\u2211 x : Sym2 \u03b1, if x \u2208 incidenceSet G a then if x \u2208 incidenceSet G a then 1 else 0 else 0) =\n    \u2211 x : Sym2 \u03b1, if x \u2208 incidenceSet G a then 1 else 0\n[PROOFSTEP]\nsimp_all only [ite_true, sum_boole]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\n\u22a2 e \u2208 edgeSet G \u2192 \u2211 a : \u03b1, incMatrix R G a e = 2\n[PROOFSTEP]\nclassical\nrefine' e.ind _\nintro a b h\nrw [mem_edgeSet] at h \nrw [\u2190 Nat.cast_two, \u2190 card_doubleton h.ne]\nsimp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]\ncongr 2\next e\nsimp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\n\u22a2 e \u2208 edgeSet G \u2192 \u2211 a : \u03b1, incMatrix R G a e = 2\n[PROOFSTEP]\nrefine' e.ind _\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\n\u22a2 \u2200 (x y : \u03b1),\n    Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 edgeSet G \u2192 \u2211 a : \u03b1, incMatrix R G a (Quotient.mk (Rel.setoid \u03b1) (x, y)) = 2\n[PROOFSTEP]\nintro a b h\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\na b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) \u2208 edgeSet G\n\u22a2 \u2211 a_1 : \u03b1, incMatrix R G a_1 (Quotient.mk (Rel.setoid \u03b1) (a, b)) = 2\n[PROOFSTEP]\nrw [mem_edgeSet] at h \n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\na b : \u03b1\nh : Adj G a b\n\u22a2 \u2211 a_1 : \u03b1, incMatrix R G a_1 (Quotient.mk (Rel.setoid \u03b1) (a, b)) = 2\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, \u2190 card_doubleton h.ne]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\na b : \u03b1\nh : Adj G a b\n\u22a2 \u2211 a_1 : \u03b1, incMatrix R G a_1 (Quotient.mk (Rel.setoid \u03b1) (a, b)) = \u2191(card {a, b})\n[PROOFSTEP]\nsimp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\na b : \u03b1\nh : Adj G a b\n\u22a2 \u2191(card (filter (fun x => x = a \u2228 x = b) univ)) = \u2191(card {a, b})\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_s\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\na b : \u03b1\nh : Adj G a b\n\u22a2 filter (fun x => x = a \u2228 x = b) univ = {a, b}\n[PROOFSTEP]\next e\n[GOAL]\ncase e_a.e_s.a\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : NonAssocSemiring R\na\u271d b\u271d : \u03b1\ne\u271d : Sym2 \u03b1\na b : \u03b1\nh : Adj G a b\ne : \u03b1\n\u22a2 e \u2208 filter (fun x => x = a \u2228 x = b) univ \u2194 e \u2208 {a, b}\n[PROOFSTEP]\nsimp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 ((incMatrix R G)\u1d40 * incMatrix R G) e e = if e \u2208 edgeSet G then 2 else 0\n[PROOFSTEP]\nclassical\nsimp only [Matrix.mul_apply, incMatrix_apply', transpose_apply, \u2190 ite_and_mul_zero, one_mul, sum_boole, and_self_iff]\nsplit_ifs with h\n\u00b7 revert h\n  refine' e.ind _\n  intro v w h\n  rw [\u2190 Nat.cast_two, \u2190 card_doubleton (G.ne_of_adj h)]\n  simp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.mp h]\n  congr 2\n  ext u\n  simp\n\u00b7 revert h\n  refine' e.ind _\n  intro v w h\n  simp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.not.mp h]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 ((incMatrix R G)\u1d40 * incMatrix R G) e e = if e \u2208 edgeSet G then 2 else 0\n[PROOFSTEP]\nsimp only [Matrix.mul_apply, incMatrix_apply', transpose_apply, \u2190 ite_and_mul_zero, one_mul, sum_boole, and_self_iff]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 \u2191(card (filter (fun x => e \u2208 incidenceSet G x) univ)) = if e \u2208 edgeSet G then 2 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nh : e \u2208 edgeSet G\n\u22a2 \u2191(card (filter (fun x => e \u2208 incidenceSet G x) univ)) = 2\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 e \u2208 edgeSet G \u2192 \u2191(card (filter (fun x => e \u2208 incidenceSet G x) univ)) = 2\n[PROOFSTEP]\nrefine' e.ind _\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 \u2200 (x y : \u03b1),\n    Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 edgeSet G \u2192\n      \u2191(card (filter (fun x_1 => Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 incidenceSet G x_1) univ)) = 2\n[PROOFSTEP]\nintro v w h\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nv w : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 edgeSet G\n\u22a2 \u2191(card (filter (fun x => Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 incidenceSet G x) univ)) = 2\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, \u2190 card_doubleton (G.ne_of_adj h)]\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nv w : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 edgeSet G\n\u22a2 \u2191(card (filter (fun x => Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 incidenceSet G x) univ)) =\n    \u2191(card {(v, w).fst, (v, w).snd})\n[PROOFSTEP]\nsimp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.mp h]\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nv w : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 edgeSet G\n\u22a2 \u2191(card (filter (fun x => x = v \u2228 x = w) univ)) = \u2191(card {v, w})\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase pos.e_a.e_s\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nv w : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 edgeSet G\n\u22a2 filter (fun x => x = v \u2228 x = w) univ = {v, w}\n[PROOFSTEP]\next u\n[GOAL]\ncase pos.e_a.e_s.a\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nv w : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 edgeSet G\nu : \u03b1\n\u22a2 u \u2208 filter (fun x => x = v \u2228 x = w) univ \u2194 u \u2208 {v, w}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nh : \u00ace \u2208 edgeSet G\n\u22a2 \u2191(card (filter (fun x => e \u2208 incidenceSet G x) univ)) = 0\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase neg\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 \u00ace \u2208 edgeSet G \u2192 \u2191(card (filter (fun x => e \u2208 incidenceSet G x) univ)) = 0\n[PROOFSTEP]\nrefine' e.ind _\n[GOAL]\ncase neg\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 \u2200 (x y : \u03b1),\n    \u00acQuotient.mk (Rel.setoid \u03b1) (x, y) \u2208 edgeSet G \u2192\n      \u2191(card (filter (fun x_1 => Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 incidenceSet G x_1) univ)) = 0\n[PROOFSTEP]\nintro v w h\n[GOAL]\ncase neg\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : NonAssocSemiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d : DecidableRel G.Adj\nv w : \u03b1\nh : \u00acQuotient.mk (Rel.setoid \u03b1) (v, w) \u2208 edgeSet G\n\u22a2 \u2191(card (filter (fun x => Quotient.mk (Rel.setoid \u03b1) (v, w) \u2208 incidenceSet G x) univ)) = 0\n[PROOFSTEP]\nsimp [mk'_mem_incidenceSet_iff, G.mem_edgeSet.not.mp h]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype (Sym2 \u03b1)\ninst\u271d : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\nh : Adj G a b\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a b = 1\n[PROOFSTEP]\nclassical\nsimp_rw [Matrix.mul_apply, Matrix.transpose_apply, incMatrix_apply_mul_incMatrix_apply, Set.indicator_apply,\n  Pi.one_apply, sum_boole]\nconvert @Nat.cast_one R _\nconvert card_singleton \u27e6(a, b)\u27e7\nrw [\u2190 coe_eq_singleton, coe_filter_univ]\nexact G.incidenceSet_inter_incidenceSet_of_adj h\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype (Sym2 \u03b1)\ninst\u271d : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\nh : Adj G a b\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a b = 1\n[PROOFSTEP]\nsimp_rw [Matrix.mul_apply, Matrix.transpose_apply, incMatrix_apply_mul_incMatrix_apply, Set.indicator_apply,\n  Pi.one_apply, sum_boole]\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype (Sym2 \u03b1)\ninst\u271d : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\nh : Adj G a b\n\u22a2 \u2191(card (filter (fun x => x \u2208 incidenceSet G a \u2229 incidenceSet G b) univ)) = 1\n[PROOFSTEP]\nconvert @Nat.cast_one R _\n[GOAL]\ncase h.e'_2.h.e'_3\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype (Sym2 \u03b1)\ninst\u271d : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\nh : Adj G a b\n\u22a2 card (filter (fun x => x \u2208 incidenceSet G a \u2229 incidenceSet G b) univ) = 1\n[PROOFSTEP]\nconvert card_singleton \u27e6(a, b)\u27e7\n[GOAL]\ncase h.e'_2.h.e'_2.h\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype (Sym2 \u03b1)\ninst\u271d : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\nh : Adj G a b\ne_1\u271d : Sym2 \u03b1 = Quotient (Rel.setoid \u03b1)\n\u22a2 filter (fun x => x \u2208 incidenceSet G a \u2229 incidenceSet G b) univ = {Quotient.mk (Rel.setoid \u03b1) (a, b)}\n[PROOFSTEP]\nrw [\u2190 coe_eq_singleton, coe_filter_univ]\n[GOAL]\ncase h.e'_2.h.e'_2.h\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : Fintype (Sym2 \u03b1)\ninst\u271d : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\nh : Adj G a b\ne_1\u271d : Sym2 \u03b1 = Quotient (Rel.setoid \u03b1)\n\u22a2 {x | x \u2208 incidenceSet G a \u2229 incidenceSet G b} = {Quotient.mk (Rel.setoid \u03b1) (a, b)}\n[PROOFSTEP]\nexact G.incidenceSet_inter_incidenceSet_of_adj h\n[GOAL]\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u2074 : Fintype (Sym2 \u03b1)\ninst\u271d\u00b3 : Semiring R\na b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\n\u22a2 incMatrix R G * (incMatrix R G)\u1d40 = fun a b => if a = b then \u2191(degree G a) else if Adj G a b then 1 else 0\n[PROOFSTEP]\next a b\n[GOAL]\ncase a.h\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u2074 : Fintype (Sym2 \u03b1)\ninst\u271d\u00b3 : Semiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na b : \u03b1\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a b = if a = b then \u2191(degree G a) else if Adj G a b then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h h'\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u2074 : Fintype (Sym2 \u03b1)\ninst\u271d\u00b3 : Semiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na b : \u03b1\nh : a = b\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a b = \u2191(degree G a)\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u2074 : Fintype (Sym2 \u03b1)\ninst\u271d\u00b3 : Semiring R\na\u271d b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na : \u03b1\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a a = \u2191(degree G a)\n[PROOFSTEP]\nrename Semiring R => sr\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u00b3 : Fintype (Sym2 \u03b1)\nsr : Semiring R\na\u271d b : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na : \u03b1\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a a = \u2191(degree G a)\n[PROOFSTEP]\nconvert @incMatrix_mul_transpose_diag _ _ _ _ sr.toNonAssocSemiring _ _ _\n[GOAL]\ncase pos\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u2074 : Fintype (Sym2 \u03b1)\ninst\u271d\u00b3 : Semiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na b : \u03b1\nh : \u00aca = b\nh' : Adj G a b\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a b = 1\n[PROOFSTEP]\nexact G.incMatrix_mul_transpose_apply_of_adj h'\n[GOAL]\ncase neg\nR : Type u_1\n\u03b1 : Type u_2\nG : SimpleGraph \u03b1\ninst\u271d\u2074 : Fintype (Sym2 \u03b1)\ninst\u271d\u00b3 : Semiring R\na\u271d b\u271d : \u03b1\ne : Sym2 \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableRel G.Adj\na b : \u03b1\nh : \u00aca = b\nh' : \u00acAdj G a b\n\u22a2 (incMatrix R G * (incMatrix R G)\u1d40) a b = 0\n[PROOFSTEP]\nsimp only [Matrix.mul_apply, Matrix.transpose_apply, G.incMatrix_apply_mul_incMatrix_apply_of_not_adj h h',\n  sum_const_zero]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.IncMatrix", "llama_tokens": 8677, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388252252041, "lm_q2_score": 0.712232184238947, "lm_q1q2_score": 0.5903969100906138}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\na\u271d b\u271d : \u211d\nh : dist a\u271d b\u271d < \u03b5\n\u22a2 dist (-a\u271d) (-b\u271d) < \u03b5\n[PROOFSTEP]\nrw [dist_comm] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\na\u271d b\u271d : \u211d\nh : dist b\u271d a\u271d < \u03b5\n\u22a2 dist (-a\u271d) (-b\u271d) < \u03b5\n[PROOFSTEP]\nsimpa only [Real.dist_eq, neg_sub_neg] using h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 TopologicalAddGroup \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nx r : \u211d\n\u22a2 IsCompact (closedBall x r)\n[PROOFSTEP]\nrw [Real.closedBall_eq_Icc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nx r : \u211d\n\u22a2 IsCompact (Icc (x - r) (x + r))\n[PROOFSTEP]\napply isCompact_Icc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 \u2200 (u : Set \u211d), u \u2208 \u22c3 (a : \u211a) (b : \u211a) (_ : a < b), {Ioo \u2191a \u2191b} \u2192 IsOpen u\n[PROOFSTEP]\nsimp (config := { contextual := true }) [isOpen_Ioo]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nv : Set \u211d\nhav : a \u2208 v\nhv : IsOpen v\nl u : \u211d\nhl : l < a\nhu : a < u\nh : Ioo l u \u2286 v\nq : \u211a\nhlq : l < \u2191q\nhqa : \u2191q < a\np : \u211a\nhap : a < \u2191p\nhpu : \u2191p < u\n\u22a2 Ioo \u2191q \u2191p \u2208 \u22c3 (a : \u211a) (b : \u211a) (_ : a < b), {Ioo \u2191a \u2191b}\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nv : Set \u211d\nhav : a \u2208 v\nhv : IsOpen v\nl u : \u211d\nhl : l < a\nhu : a < u\nh : Ioo l u \u2286 v\nq : \u211a\nhlq : l < \u2191q\nhqa : \u2191q < a\np : \u211a\nhap : a < \u2191p\nhpu : \u2191p < u\n\u22a2 \u2203 i i_1 i_2, Ioo \u2191q \u2191p \u2208 {Ioo \u2191i \u2191i_1}\n[PROOFSTEP]\nexact \u27e8q, p, Rat.cast_lt.1 <| hqa.trans hap, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 cocompact \u211d = atBot \u2294 atTop\n[PROOFSTEP]\nsimp only [\u2190 comap_dist_right_atTop_eq_cocompact (0 : \u211d), Real.dist_eq, sub_zero, comap_abs_atTop]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nx : \u211d\n\u22a2 x \u2208 closure s \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 y, y \u2208 s \u2227 |y - x| < \u03b5\n[PROOFSTEP]\nsimp [mem_closure_iff_nhds_basis nhds_basis_ball, Real.dist_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 CompleteSpace \u211d\n[PROOFSTEP]\napply complete_of_cauchySeq_tendsto\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 \u2200 (u : \u2115 \u2192 \u211d), CauchySeq u \u2192 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nintro u hu\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nu : \u2115 \u2192 \u211d\nhu : CauchySeq u\n\u22a2 \u2203 a, Tendsto u atTop (\ud835\udcdd a)\n[PROOFSTEP]\nlet c : CauSeq \u211d abs := \u27e8u, Metric.cauchySeq_iff'.1 hu\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nu : \u2115 \u2192 \u211d\nhu : CauchySeq u\nc : CauSeq \u211d abs := { 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 (\ud835\udcdd a)\n[PROOFSTEP]\nrefine' \u27e8c.lim, fun s h => _\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nu : \u2115 \u2192 \u211d\nhu : CauchySeq u\nc : CauSeq \u211d abs := { 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 \u211d\nh : s \u2208 \ud835\udcdd (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\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nu : \u2115 \u2192 \u211d\nhu : CauchySeq u\nc : CauSeq \u211d abs := { 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 \u211d\nh : s \u2208 \ud835\udcdd (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\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nu : \u2115 \u2192 \u211d\nhu : CauchySeq u\nc : CauSeq \u211d abs := { 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 \u211d\nh : s \u2208 \ud835\udcdd (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 |\u2191(c - CauSeq.const abs (CauSeq.lim c)) j| < \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\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nu : \u2115 \u2192 \u211d\nhu : CauchySeq u\nc : CauSeq \u211d abs := { 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 \u211d\nh : s \u2208 \ud835\udcdd (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 |\u2191(c - CauSeq.const abs (CauSeq.lim c)) j| < \u03b5\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 b \u2208 u \u207b\u00b9' s\n[PROOFSTEP]\nrefine' this.imp fun N hN n hn => h\u03b5 (hN n hn)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nx \u03b5 : \u211d\n\u22a2 TotallyBounded (ball x \u03b5)\n[PROOFSTEP]\nrw [Real.ball_eq_Ioo]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nx \u03b5 : \u211d\n\u22a2 TotallyBounded (Ioo (x - \u03b5) (x + \u03b5))\n[PROOFSTEP]\napply totallyBounded_Ioo\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nq : \u211a\nx : \u211d\nhx : x \u2208 {r | \u2191q \u2264 r}\nt : Set \u211d\nht : t \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 t\np : \u211a\nh\u2081 : x < \u2191p\nh\u2082 : \u2191p < x + \u03b5\n\u22a2 \u2191p \u2208 ball x \u03b5\n[PROOFSTEP]\nrwa [mem_ball, Real.dist_eq, abs_of_pos (sub_pos.2 h\u2081), sub_lt_iff_lt_add']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\n\u22a2 Metric.Bounded s \u2192 BddBelow s \u2227 BddAbove s\n[PROOFSTEP]\nintro bdd\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : Metric.Bounded s\n\u22a2 BddBelow s \u2227 BddAbove s\n[PROOFSTEP]\nrcases(bounded_iff_subset_ball 0).1 bdd with\n  \u27e8r, hr\u27e9\n    -- hr : s \u2286 closed_ball 0 r\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : Metric.Bounded s\nr : \u211d\nhr : s \u2286 closedBall 0 r\n\u22a2 BddBelow s \u2227 BddAbove s\n[PROOFSTEP]\nrw [Real.closedBall_eq_Icc] at hr \n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : Metric.Bounded s\nr : \u211d\nhr : s \u2286 Icc (0 - r) (0 + r)\n\u22a2 BddBelow s \u2227 BddAbove s\n[PROOFSTEP]\nexact \u27e8bddBelow_Icc.mono hr, bddAbove_Icc.mono hr\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d : TopologicalSpace \u03b1\nf : \u211d \u2192 \u03b1\nc : \u211d\nhp : Periodic f c\nhc : c \u2260 0\nhf : Continuous f\n\u22a2 IsCompact (range f)\n[PROOFSTEP]\nrw [\u2190 hp.image_uIcc hc 0]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d : TopologicalSpace \u03b1\nf : \u211d \u2192 \u03b1\nc : \u211d\nhp : Periodic f c\nhc : c \u2260 0\nhf : Continuous f\n\u22a2 IsCompact (f '' [[0, 0 + c]])\n[PROOFSTEP]\nexact isCompact_uIcc.image hf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\n\u22a2 DiscreteTopology { x // x \u2208 AddSubgroup.zmultiples a }\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 DiscreteTopology { x // x \u2208 AddSubgroup.zmultiples 0 }\n[PROOFSTEP]\nrw [AddSubgroup.zmultiples_zero_eq_bot]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 DiscreteTopology { x // x \u2208 \u22a5 }\n[PROOFSTEP]\nexact Subsingleton.discreteTopology (\u03b1 := (\u22a5 : Submodule \u2124 \u211d))\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\n\u22a2 DiscreteTopology { x // x \u2208 AddSubgroup.zmultiples a }\n[PROOFSTEP]\nrw [discreteTopology_iff_open_singleton_zero, isOpen_induced_iff]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\n\u22a2 \u2203 t, IsOpen t \u2227 Subtype.val \u207b\u00b9' t = {0}\n[PROOFSTEP]\nrefine' \u27e8ball 0 |a|, isOpen_ball, _\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\n\u22a2 Subtype.val \u207b\u00b9' ball 0 |a| = {0}\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase inr.h.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\nx : \u211d\nhx : x \u2208 AddSubgroup.zmultiples a\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' ball 0 |a| \u2194 { val := x, property := hx } \u2208 {0}\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := AddSubgroup.mem_zmultiples_iff.mp hx\n[GOAL]\ncase inr.h.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\nk : \u2124\nhx : k \u2022 a \u2208 AddSubgroup.zmultiples a\n\u22a2 { val := k \u2022 a, property := hx } \u2208 Subtype.val \u207b\u00b9' ball 0 |a| \u2194 { val := k \u2022 a, property := hx } \u2208 {0}\n[PROOFSTEP]\nsimp [ha, Real.dist_eq, abs_mul, (by norm_cast : |(k : \u211d)| < 1 \u2194 |k| < 1)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\nk : \u2124\nhx : k \u2022 a \u2208 AddSubgroup.zmultiples a\n\u22a2 |\u2191k| < 1 \u2194 |k| < 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 Tendsto Int.cast cofinite (cocompact \u211d)\n[PROOFSTEP]\napply (castAddHom \u211d).tendsto_coe_cofinite_of_discrete cast_injective\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 DiscreteTopology { x // x \u2208 AddMonoidHom.range (castAddHom \u211d) }\n[PROOFSTEP]\nrw [range_castAddHom]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 DiscreteTopology { x // x \u2208 AddSubgroup.zmultiples 1 }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\n\u22a2 Tendsto (\u2191(\u2191(zmultiplesHom \u211d) a)) cofinite (cocompact \u211d)\n[PROOFSTEP]\napply (zmultiplesHom \u211d a).tendsto_coe_cofinite_of_discrete $ smul_left_injective \u2124 ha\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\n\u22a2 DiscreteTopology { x // x \u2208 AddMonoidHom.range (\u2191(zmultiplesHom \u211d) a) }\n[PROOFSTEP]\nrw [AddSubgroup.range_zmultiplesHom]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u211d\nha : a \u2260 0\n\u22a2 DiscreteTopology { x // x \u2208 AddSubgroup.zmultiples a }\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.Real", "llama_tokens": 4379, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.828938825225204, "lm_q2_score": 0.7122321720225278, "lm_q1q2_score": 0.5903968999639496}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nx : \u2124\u02e3\n\u22a2 x \u2208 {1, -1}\n[PROOFSTEP]\ncases Int.units_eq_one_or x\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nx : \u2124\u02e3\nh\u271d : x = 1\n\u22a2 x \u2208 {1, -1}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nx : \u2124\u02e3\nh\u271d : x = -1\n\u22a2 x \u2208 {1, -1}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : GroupWithZero \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card \u03b1 = card \u03b1\u02e3 + 1\n[PROOFSTEP]\nrw [eq_comm, Fintype.card_congr (unitsEquivNeZero \u03b1)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : GroupWithZero \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card { a // a \u2260 0 } + 1 = card \u03b1\n[PROOFSTEP]\nhave := Fintype.card_congr (Equiv.sumCompl (\u00b7 = (0 : \u03b1)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : GroupWithZero \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nthis : card ({ a // a = 0 } \u2295 { a // \u00aca = 0 }) = card \u03b1\n\u22a2 card { a // a \u2260 0 } + 1 = card \u03b1\n[PROOFSTEP]\nrwa [Fintype.card_sum, add_comm, Fintype.card_subtype_eq] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : GroupWithZero \u03b1\ninst\u271d : Finite \u03b1\n\u22a2 Nat.card \u03b1 = Nat.card \u03b1\u02e3 + 1\n[PROOFSTEP]\nhave : Fintype \u03b1 := Fintype.ofFinite \u03b1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : GroupWithZero \u03b1\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 = Nat.card \u03b1\u02e3 + 1\n[PROOFSTEP]\nclassical rw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, Fintype.card_eq_card_units_add_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : GroupWithZero \u03b1\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 Nat.card \u03b1 = Nat.card \u03b1\u02e3 + 1\n[PROOFSTEP]\nrw [Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, Fintype.card_eq_card_units_add_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : GroupWithZero \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 card \u03b1\u02e3 = card \u03b1 - 1\n[PROOFSTEP]\nrw [@Fintype.card_eq_card_units_add_one \u03b1, Nat.add_sub_cancel]\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Units", "llama_tokens": 855, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428947, "lm_q2_score": 0.7490872243177518, "lm_q1q2_score": 0.5902291922975034}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nn\u271d n : \u2115\nihn : id^[n] = id\n\u22a2 id^[Nat.succ n] = id\n[PROOFSTEP]\nrw [iterate_succ, ihn, comp.left_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\n\u22a2 f^[m + Nat.succ n] = f^[m] \u2218 f^[Nat.succ n]\n[PROOFSTEP]\nrw [Nat.add_succ, iterate_succ, iterate_succ, iterate_add m n]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\n\u22a2 (f^[m] \u2218 f^[n]) \u2218 f = f^[m] \u2218 f^[n] \u2218 f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\nx : \u03b1\n\u22a2 f^[m + n] x = f^[m] (f^[n] x)\n[PROOFSTEP]\nrw [iterate_add f m n]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\nx : \u03b1\n\u22a2 (f^[m] \u2218 f^[n]) x = f^[m] (f^[n] x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm : \u2115\n\u22a2 f^[m * 0] = f^[m]^[0]\n[PROOFSTEP]\nsimp only [Nat.mul_zero, iterate_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\n\u22a2 f^[m * (n + 1)] = f^[m]^[n + 1]\n[PROOFSTEP]\nsimp only [Nat.mul_succ, Nat.mul_one, iterate_one, iterate_add, iterate_mul m n]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nx : \u03b1\nh : f x = x\nn\u271d n : \u2115\nihn : f^[n] x = x\n\u22a2 f^[Nat.succ n] x = x\n[PROOFSTEP]\nrw [iterate_succ_apply, h, ihn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\ng : \u2115 \u2192 \u03b1 \u2192 \u03b1\nH : \u2200 (n : \u2115), Semiconj f (g n) (g (n + 1))\nn k : \u2115\n\u22a2 Semiconj f^[n] (g k) (g (n + k))\n[PROOFSTEP]\ninduction' n with n ihn generalizing k\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\ng : \u2115 \u2192 \u03b1 \u2192 \u03b1\nH : \u2200 (n : \u2115), Semiconj f (g n) (g (n + 1))\nk\u271d k : \u2115\n\u22a2 Semiconj f^[Nat.zero] (g k) (g (Nat.zero + k))\n[PROOFSTEP]\nrw [Nat.zero_add]\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\ng : \u2115 \u2192 \u03b1 \u2192 \u03b1\nH : \u2200 (n : \u2115), Semiconj f (g n) (g (n + 1))\nk\u271d k : \u2115\n\u22a2 Semiconj f^[Nat.zero] (g k) (g k)\n[PROOFSTEP]\nexact id_left\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\ng : \u2115 \u2192 \u03b1 \u2192 \u03b1\nH : \u2200 (n : \u2115), Semiconj f (g n) (g (n + 1))\nk\u271d n : \u2115\nihn : \u2200 (k : \u2115), Semiconj f^[n] (g k) (g (n + k))\nk : \u2115\n\u22a2 Semiconj f^[Nat.succ n] (g k) (g (Nat.succ n + k))\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, Nat.add_right_comm, Nat.add_assoc]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\ng : \u2115 \u2192 \u03b1 \u2192 \u03b1\nH : \u2200 (n : \u2115), Semiconj f (g n) (g (n + 1))\nk\u271d n : \u2115\nihn : \u2200 (k : \u2115), Semiconj f^[n] (g k) (g (n + k))\nk : \u2115\n\u22a2 Semiconj f^[n + 1] (g k) (g (n + (k + 1)))\n[PROOFSTEP]\nexact (H k).comp_left (ihn (k + 1))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\nn\u271d : \u2115\nx : \u03b1\nhx : f x = g x\nn : \u2115\nihn : f^[n] x = g^[n] x\n\u22a2 f^[Nat.succ n] x = g^[Nat.succ n] x\n[PROOFSTEP]\nsimp only [iterate_succ_apply, hx, (h.iterate_left n).eq, ihn, ((refl g).iterate_right n).eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\nn : \u2115\n\u22a2 (f \u2218 g)^[n] = f^[n] \u2218 g^[n]\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\n\u22a2 (f \u2218 g)^[Nat.zero] = f^[Nat.zero] \u2218 g^[Nat.zero]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\nn : \u2115\nihn : (f \u2218 g)^[n] = f^[n] \u2218 g^[n]\n\u22a2 (f \u2218 g)^[Nat.succ n] = f^[Nat.succ n] \u2218 g^[Nat.succ n]\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase succ.h\n\u03b1 : Type u\n\u03b2 : Type v\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\nn : \u2115\nihn : (f \u2218 g)^[n] = f^[n] \u2218 g^[n]\nx : \u03b1\n\u22a2 (f \u2218 g)^[Nat.succ n] x = (f^[Nat.succ n] \u2218 g^[Nat.succ n]) x\n[PROOFSTEP]\nsimp only [ihn, (h.iterate_right n).eq, iterate_succ, comp_apply]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nn : \u2115\n\u22a2 f^[Nat.succ n] = f \u2218 f^[n]\n[PROOFSTEP]\nrw [iterate_succ, (Commute.self_iterate f n).comp_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 f^[Nat.succ n] x = f (f^[n] x)\n[PROOFSTEP]\nrw [iterate_succ']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 (f \u2218 f^[n]) x = f (f^[n] x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nhn : 0 < n\n\u22a2 f^[Nat.pred n] \u2218 f = f^[n]\n[PROOFSTEP]\nrw [\u2190 iterate_succ, Nat.succ_pred_eq_of_pos hn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nhn : 0 < n\n\u22a2 f \u2218 f^[Nat.pred n] = f^[n]\n[PROOFSTEP]\nrw [\u2190 iterate_succ', Nat.succ_pred_eq_of_pos hn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n\u271d\u00b9 : \u2115\na : \u03b1\ng : \u03b1 \u2192 \u03b1\nhg : LeftInverse g f\nn\u271d n : \u2115\nihn : LeftInverse g^[n] f^[n]\n\u22a2 LeftInverse g^[Nat.succ n] f^[Nat.succ n]\n[PROOFSTEP]\nrw [iterate_succ', iterate_succ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n\u271d\u00b9 : \u2115\na : \u03b1\ng : \u03b1 \u2192 \u03b1\nhg : LeftInverse g f\nn\u271d n : \u2115\nihn : LeftInverse g^[n] f^[n]\n\u22a2 LeftInverse (g \u2218 g^[n]) (f^[n] \u2218 f)\n[PROOFSTEP]\nexact ihn.comp hg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u03b1 \u2192 \u03b1\nm\u271d n\u271d : \u2115\na : \u03b1\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\n\u22a2 f^[n * m] = f^[m * n]\n[PROOFSTEP]\nrw [Nat.mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\n\u22a2 f^[m + n] a = f^[n] a \u2194 f^[n] (f^[m] a) = f^[n] a\n[PROOFSTEP]\nrw [\u2190 iterate_add_apply, Nat.add_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\n\u22a2 f^[m - n] a = a\n[PROOFSTEP]\nobtain h | h := le_total m n\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\nh : m \u2264 n\n\u22a2 f^[m - n] a = a\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\nh : n \u2264 m\n\u22a2 f^[m - n] a = a\n[PROOFSTEP]\n{simp [Nat.sub_eq_zero_of_le h]\n}\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\nh : m \u2264 n\n\u22a2 f^[m - n] a = a\n[PROOFSTEP]\nsimp [Nat.sub_eq_zero_of_le h]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\nh : n \u2264 m\n\u22a2 f^[m - n] a = a\n[PROOFSTEP]\n{exact iterate_cancel_of_add hf (by rwa [Nat.sub_add_cancel h])\n}\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\nh : n \u2264 m\n\u22a2 f^[m - n] a = a\n[PROOFSTEP]\nexact iterate_cancel_of_add hf (by rwa [Nat.sub_add_cancel h])\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\nm n : \u2115\na : \u03b1\nhf : Injective f\nha : f^[m] a = f^[n] a\nh : n \u2264 m\n\u22a2 f^[m - n + ?m.6123] a = f^[?m.6123] a\n[PROOFSTEP]\nrwa [Nat.sub_add_cancel h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\na : \u03b1\nl : List \u03b2\n\u22a2 foldl (fun b x => f b) a l = f^[length l] a\n[PROOFSTEP]\ninduction' l with b l H generalizing a\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\na\u271d a : \u03b1\n\u22a2 foldl (fun b x => f b) a [] = f^[length []] a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b1\na\u271d : \u03b1\nb : \u03b2\nl : List \u03b2\nH : \u2200 (a : \u03b1), foldl (fun b x => f b) a l = f^[length l] a\na : \u03b1\n\u22a2 foldl (fun b x => f b) a (b :: l) = f^[length (b :: l)] a\n[PROOFSTEP]\nrw [length_cons, foldl, iterate_succ_apply, H]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b2 \u2192 \u03b2\nb : \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 foldr (fun x => f) b (a :: l) = f^[length (a :: l)] b\n[PROOFSTEP]\nrw [length_cons, foldr, foldr_const f b l, iterate_succ_apply']\n", "meta": {"mathlib_filename": "Mathlib.Logic.Function.Iterate", "llama_tokens": 3699, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104788995148792, "lm_q2_score": 0.727975443004307, "lm_q1q2_score": 0.5900087359199874}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\nha : 0 < a\n\u22a2 a * 0 < a * a\u207b\u00b9\n[PROOFSTEP]\nsimp [ne_of_gt ha, zero_lt_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 0 \u2264 a\u207b\u00b9 \u2194 0 \u2264 a\n[PROOFSTEP]\nsimp only [le_iff_eq_or_lt, inv_pos, zero_eq_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a\u207b\u00b9 < 0 \u2194 a < 0\n[PROOFSTEP]\nsimp only [\u2190 not_le, inv_nonneg]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a\u207b\u00b9 \u2264 0 \u2194 a \u2264 0\n[PROOFSTEP]\nsimp only [\u2190 not_lt, inv_pos]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 0 < a / b\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 0 < a * b\u207b\u00b9\n[PROOFSTEP]\nexact mul_pos ha (inv_pos.2 hb)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a / b\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a * b\u207b\u00b9\n[PROOFSTEP]\nexact mul_nonneg ha (inv_nonneg.2 hb)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : a \u2264 0\nhb : 0 \u2264 b\n\u22a2 a / b \u2264 0\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : a \u2264 0\nhb : 0 \u2264 b\n\u22a2 a * b\u207b\u00b9 \u2264 0\n[PROOFSTEP]\nexact mul_nonpos_of_nonpos_of_nonneg ha (inv_nonneg.2 hb)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhb : b \u2264 0\n\u22a2 a / b \u2264 0\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhb : b \u2264 0\n\u22a2 a * b\u207b\u00b9 \u2264 0\n[PROOFSTEP]\nexact mul_nonpos_of_nonneg_of_nonpos ha (inv_nonpos.2 hb)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a ^ n\n[PROOFSTEP]\nexact pow_nonneg ha _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a ^ (-\u2191(n + 1))\n[PROOFSTEP]\nrw [zpow_neg, inv_nonneg, zpow_ofNat]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a ^ (n + 1)\n[PROOFSTEP]\nexact pow_nonneg ha _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 < a\nn : \u2115\n\u22a2 0 < a ^ \u2191n\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 < a\nn : \u2115\n\u22a2 0 < a ^ n\n[PROOFSTEP]\nexact pow_pos ha _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 < a\nn : \u2115\n\u22a2 0 < a ^ (-\u2191(n + 1))\n[PROOFSTEP]\nrw [zpow_neg, inv_pos, zpow_ofNat]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n\u271d : \u2124\nha : 0 < a\nn : \u2115\n\u22a2 0 < a ^ (n + 1)\n[PROOFSTEP]\nexact pow_pos ha _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 < c\n\u22a2 a \u2264 b / c \u2194 c * a \u2264 b\n[PROOFSTEP]\nrw [mul_comm, le_div_iff hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\nh : a / b \u2264 c\n\u22a2 a = a / b * b\n[PROOFSTEP]\nrw [div_mul_cancel _ (ne_of_lt hb).symm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\nh : a \u2264 c * b\n\u22a2 c * b / b = c\n[PROOFSTEP]\nrefine' (div_eq_iff (ne_of_gt hb)).mpr rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\n\u22a2 a / b \u2264 c \u2194 a \u2264 b * c\n[PROOFSTEP]\nrw [mul_comm, div_le_iff hb]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 < c\n\u22a2 a < b / c \u2194 c * a < b\n[PROOFSTEP]\nrw [mul_comm, lt_div_iff hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 < c\n\u22a2 b / c < a \u2194 b < c * a\n[PROOFSTEP]\nrw [mul_comm, div_lt_iff hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 b\u207b\u00b9 * a \u2264 c \u2194 a \u2264 b * c\n[PROOFSTEP]\nrw [inv_eq_one_div, mul_comm, \u2190 div_eq_mul_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 a / b \u2264 c \u2194 a \u2264 b * c\n[PROOFSTEP]\nexact div_le_iff' h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 b\u207b\u00b9 * a \u2264 c \u2194 a \u2264 c * b\n[PROOFSTEP]\nrw [inv_mul_le_iff h, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 b * c\n[PROOFSTEP]\nrw [mul_comm, inv_mul_le_iff h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 a * b\u207b\u00b9 \u2264 c \u2194 a \u2264 c * b\n[PROOFSTEP]\nrw [mul_comm, inv_mul_le_iff' h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\nh : a = 0\n\u22a2 a / a \u2264 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\nh : \u00aca = 0\n\u22a2 a / a \u2264 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 b\u207b\u00b9 * a < c \u2194 a < b * c\n[PROOFSTEP]\nrw [inv_eq_one_div, mul_comm, \u2190 div_eq_mul_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 a / b < c \u2194 a < b * c\n[PROOFSTEP]\nexact div_lt_iff' h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 b\u207b\u00b9 * a < c \u2194 a < c * b\n[PROOFSTEP]\nrw [inv_mul_lt_iff h, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 a * b\u207b\u00b9 < c \u2194 a < b * c\n[PROOFSTEP]\nrw [mul_comm, inv_mul_lt_iff h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 a * b\u207b\u00b9 < c \u2194 a < c * b\n[PROOFSTEP]\nrw [mul_comm, inv_mul_lt_iff' h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 a\u207b\u00b9 \u2264 b \u2194 1 \u2264 b * a\n[PROOFSTEP]\nrw [inv_eq_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 1 / a \u2264 b \u2194 1 \u2264 b * a\n[PROOFSTEP]\nexact div_le_iff ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 a\u207b\u00b9 \u2264 b \u2194 1 \u2264 a * b\n[PROOFSTEP]\nrw [inv_eq_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 1 / a \u2264 b \u2194 1 \u2264 a * b\n[PROOFSTEP]\nexact div_le_iff' ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 a\u207b\u00b9 < b \u2194 1 < b * a\n[PROOFSTEP]\nrw [inv_eq_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 1 / a < b \u2194 1 < b * a\n[PROOFSTEP]\nexact div_lt_iff ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 a\u207b\u00b9 < b \u2194 1 < a * b\n[PROOFSTEP]\nrw [inv_eq_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 1 / a < b \u2194 1 < a * b\n[PROOFSTEP]\nexact div_lt_iff' ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\n\u22a2 a / b \u2264 c\n[PROOFSTEP]\nrcases eq_or_lt_of_le hb with (rfl | hb')\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nhb : 0 \u2264 0\nh : a \u2264 c * 0\n\u22a2 a / 0 \u2264 c\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\nhb' : 0 < b\n\u22a2 a / b \u2264 c\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\nhb' : 0 < b\n\u22a2 a / b \u2264 c\n[PROOFSTEP]\nrwa [div_le_iff hb']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 b / c\n\u22a2 a * c \u2264 b\n[PROOFSTEP]\nobtain rfl | hc := hc.eq_or_lt\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 0\nh : a \u2264 b / 0\n\u22a2 a * 0 \u2264 b\n[PROOFSTEP]\nsimpa using hb\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc\u271d : 0 \u2264 c\nh : a \u2264 b / c\nhc : 0 < c\n\u22a2 a * c \u2264 b\n[PROOFSTEP]\nrwa [le_div_iff hc] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : a \u2264 b\nhb : 0 \u2264 b\n\u22a2 a \u2264 1 * b\n[PROOFSTEP]\nrwa [one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nh : a \u2264 b\n\u22a2 b\u207b\u00b9 \u2264 a\u207b\u00b9\n[PROOFSTEP]\nrwa [\u2190 one_div a, le_div_iff' ha, \u2190 div_eq_mul_inv, div_le_iff (ha.trans_le h), one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 a\u207b\u00b9 \u2264 b\u207b\u00b9 \u2194 b \u2264 a\n[PROOFSTEP]\nrw [\u2190 one_div, div_le_iff ha, \u2190 div_eq_inv_mul, le_div_iff hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 a\u207b\u00b9 \u2264 b \u2194 b\u207b\u00b9 \u2264 a\n[PROOFSTEP]\nrw [\u2190 inv_le_inv hb (inv_pos.2 ha), inv_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 a \u2264 b\u207b\u00b9 \u2194 b \u2264 a\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 inv_le_inv (inv_pos.2 hb) ha, inv_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 1 < a\n\u22a2 a\u207b\u00b9 < 1\n[PROOFSTEP]\nrwa [inv_lt (zero_lt_one.trans ha) zero_lt_one, inv_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh\u2081 : 0 < a\nh\u2082 : a < 1\n\u22a2 1 < a\u207b\u00b9\n[PROOFSTEP]\nrwa [lt_inv (@zero_lt_one \u03b1 _ _ _ _ _) h\u2081, inv_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 1 \u2264 a\n\u22a2 a\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrwa [inv_le (zero_lt_one.trans_le ha) zero_lt_one, inv_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh\u2081 : 0 < a\nh\u2082 : a \u2264 1\n\u22a2 1 \u2264 a\u207b\u00b9\n[PROOFSTEP]\nrwa [le_inv (@zero_lt_one \u03b1 _ _ _ _ _) h\u2081, inv_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a\u207b\u00b9 < 1 \u2194 a \u2264 0 \u2228 1 < a\n[PROOFSTEP]\ncases' le_or_lt a 0 with ha ha\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : a \u2264 0\n\u22a2 a\u207b\u00b9 < 1 \u2194 a \u2264 0 \u2228 1 < a\n[PROOFSTEP]\nsimp [ha, (inv_nonpos.2 ha).trans_lt zero_lt_one]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\n\u22a2 a\u207b\u00b9 < 1 \u2194 a \u2264 0 \u2228 1 < a\n[PROOFSTEP]\nsimp only [ha.not_le, false_or_iff, inv_lt_one_iff_of_pos ha]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a\u207b\u00b9 \u2264 1 \u2194 a \u2264 0 \u2228 1 \u2264 a\n[PROOFSTEP]\nrcases em (a = 1) with (rfl | ha)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\nb c d e : \u03b1\nm n : \u2124\n\u22a2 1\u207b\u00b9 \u2264 1 \u2194 1 \u2264 0 \u2228 1 \u2264 1\n[PROOFSTEP]\nsimp [le_rfl]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : \u00aca = 1\n\u22a2 a\u207b\u00b9 \u2264 1 \u2194 a \u2264 0 \u2228 1 \u2264 a\n[PROOFSTEP]\nsimp only [Ne.le_iff_lt (Ne.symm ha), Ne.le_iff_lt (mt inv_eq_one.1 ha), inv_lt_one_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nh : a \u2264 b\n\u22a2 a / c \u2264 b / c\n[PROOFSTEP]\nrw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nh : a \u2264 b\n\u22a2 a * (1 / c) \u2264 b * (1 / c)\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right h (one_div_nonneg.2 hc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhc : 0 < c\nh : c \u2264 b\n\u22a2 a / b \u2264 a / c\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhc : 0 < c\nh : c \u2264 b\n\u22a2 a * b\u207b\u00b9 \u2264 a * c\u207b\u00b9\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left ((inv_le_inv (hc.trans_le h) hc).mpr h) ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 < c\nh : a < b\n\u22a2 a / c < b / c\n[PROOFSTEP]\nrw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 < c\nh : a < b\n\u22a2 a * (1 / c) < b * (1 / c)\n[PROOFSTEP]\nexact mul_lt_mul_of_pos_right h (one_div_pos.2 hc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n\u22a2 a / b < a / c \u2194 c < b\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv hb hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nb0 : 0 < b\nd0 : 0 < d\n\u22a2 a / b < c / d \u2194 a * d < c * b\n[PROOFSTEP]\nrw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nb0 : 0 < b\nd0 : 0 < d\n\u22a2 a / b \u2264 c / d \u2194 a * d \u2264 c * b\n[PROOFSTEP]\nrw [le_div_iff d0, div_mul_eq_mul_div, div_le_iff b0]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nhac : a \u2264 c\nhd : 0 < d\nhbd : d \u2264 b\n\u22a2 a / b \u2264 c / d\n[PROOFSTEP]\nrw [div_le_div_iff (hd.trans_le hbd) hd]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nhac : a \u2264 c\nhd : 0 < d\nhbd : d \u2264 b\n\u22a2 a * d \u2264 c * b\n[PROOFSTEP]\nexact mul_le_mul hac hbd hd.le hc\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhb : 1 \u2264 b\n\u22a2 a / b \u2264 a\n[PROOFSTEP]\nsimpa only [div_one] using div_le_div_of_le_left ha zero_lt_one hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 1 < b\n\u22a2 a / b < a\n[PROOFSTEP]\nsimpa only [div_one] using div_lt_div_of_lt_left ha zero_lt_one hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 \u2264 a\nhb\u2080 : 0 < b\nhb\u2081 : b \u2264 1\n\u22a2 a \u2264 a / b\n[PROOFSTEP]\nsimpa only [div_one] using div_le_div_of_le_left ha hb\u2080 hb\u2081\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\n\u22a2 1 \u2264 a / b \u2194 b \u2264 a\n[PROOFSTEP]\nrw [le_div_iff hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\n\u22a2 a / b \u2264 1 \u2194 a \u2264 b\n[PROOFSTEP]\nrw [div_le_iff hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\n\u22a2 1 < a / b \u2194 b < a\n[PROOFSTEP]\nrw [lt_div_iff hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 < b\n\u22a2 a / b < 1 \u2194 a < b\n[PROOFSTEP]\nrw [div_lt_iff hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 1 / a \u2264 b \u2194 1 / b \u2264 a\n[PROOFSTEP]\nsimpa using inv_le ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 1 / a < b \u2194 1 / b < a\n[PROOFSTEP]\nsimpa using inv_lt ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 a \u2264 1 / b \u2194 b \u2264 1 / a\n[PROOFSTEP]\nsimpa using le_inv ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 a < 1 / b \u2194 b < 1 / a\n[PROOFSTEP]\nsimpa using lt_inv ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nh : a \u2264 b\n\u22a2 1 / b \u2264 1 / a\n[PROOFSTEP]\nsimpa using inv_le_inv_of_le ha h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nh : a < b\n\u22a2 1 / b < 1 / a\n[PROOFSTEP]\nrwa [lt_div_iff' ha, \u2190 div_eq_mul_one_div, div_lt_one (ha.trans h)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh1 : 0 < a\nh2 : a < 1\n\u22a2 1 < 1 / a\n[PROOFSTEP]\nrwa [lt_one_div (@zero_lt_one \u03b1 _ _ _ _ _) h1, one_div_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh1 : 0 < a\nh2 : a \u2264 1\n\u22a2 1 \u2264 1 / a\n[PROOFSTEP]\nrwa [le_one_div (@zero_lt_one \u03b1 _ _ _ _ _) h1, one_div_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\n\u22a2 a / 2 + a / 2 = a\n[PROOFSTEP]\nrw [div_add_div_same, \u2190 two_mul, mul_div_cancel_left a two_ne_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\n\u22a2 (a + a) / 2 = a\n[PROOFSTEP]\nrw [\u2190 mul_two, mul_div_cancel a two_ne_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a / 2 \u2264 a \u2194 0 \u2264 a\n[PROOFSTEP]\nrw [div_le_iff (zero_lt_two' \u03b1), mul_two, le_add_iff_nonneg_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a / 2 < a \u2194 0 < a\n[PROOFSTEP]\nrw [div_lt_iff (zero_lt_two' \u03b1), mul_two, lt_add_iff_pos_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 a < (a + b) / 2 \u2194 a < b\n[PROOFSTEP]\nsimp [lt_div_iff, mul_two]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\n\u22a2 (a + b) / 2 < b \u2194 a < b\n[PROOFSTEP]\nsimp [div_lt_iff, mul_two]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\n\u22a2 a / 3 + a / 3 + a / 3 = a\n[PROOFSTEP]\nrw [div_add_div_same, div_add_div_same, \u2190 two_mul, \u2190 add_one_mul 2 a, two_add_one_eq_three,\n  mul_div_cancel_left a three_ne_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : a * (b / c) \u2264 d\nhc : 0 < c\n\u22a2 b * a \u2264 d * c\n[PROOFSTEP]\nrw [\u2190 mul_div_assoc] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : a * b / c \u2264 d\nhc : 0 < c\n\u22a2 b * a \u2264 d * c\n[PROOFSTEP]\nrwa [mul_comm b, \u2190 div_le_iff hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : a / b \u2264 c / d\nhe : 0 \u2264 e\n\u22a2 a / (b * e) \u2264 c / (d * e)\n[PROOFSTEP]\nrw [div_mul_eq_div_mul_one_div, div_mul_eq_div_mul_one_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : a / b \u2264 c / d\nhe : 0 \u2264 e\n\u22a2 a / b * (1 / e) \u2264 c / d * (1 / e)\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right h (one_div_nonneg.2 he)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b\u271d c d e : \u03b1\nm n : \u2124\na : \u03b1\nh : 0 < a\nb : \u03b1\n\u22a2 \u2203 c, 0 < c \u2227 b * c < a\n[PROOFSTEP]\nhave : 0 < a / max (b + 1) 1 := div_pos h (lt_max_iff.2 (Or.inr zero_lt_one))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b\u271d c d e : \u03b1\nm n : \u2124\na : \u03b1\nh : 0 < a\nb : \u03b1\nthis : 0 < a / max (b + 1) 1\n\u22a2 \u2203 c, 0 < c \u2227 b * c < a\n[PROOFSTEP]\nrefine' \u27e8a / max (b + 1) 1, this, _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b\u271d c d e : \u03b1\nm n : \u2124\na : \u03b1\nh : 0 < a\nb : \u03b1\nthis : 0 < a / max (b + 1) 1\n\u22a2 b * (a / max (b + 1) 1) < a\n[PROOFSTEP]\nrw [\u2190 lt_div_iff this, div_div_cancel' h.ne']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b\u271d c d e : \u03b1\nm n : \u2124\na : \u03b1\nh : 0 < a\nb : \u03b1\nthis : 0 < a / max (b + 1) 1\n\u22a2 b < max (b + 1) 1\n[PROOFSTEP]\nexact lt_max_iff.2 (Or.inl <| lt_add_one _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b\u271d c\u271d d e : \u03b1\nm n : \u2124\na : \u03b1\nh : 0 < a\nb c : \u03b1\nhc\u2080 : 0 < c\nhc : b * c < a\n\u22a2 b < c\u207b\u00b9 * a\n[PROOFSTEP]\nrwa [\u2190 div_eq_inv_mul, lt_div_iff hc\u2080]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\na b c\u271d d e : \u03b1\nm n : \u2124\n\u03b2 : Type u_4\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Monotone f\nc : \u03b1\nhc : 0 \u2264 c\n\u22a2 Monotone fun x => f x / c\n[PROOFSTEP]\nhaveI := @LinearOrder.decidableLE \u03b1 _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\na b c\u271d d e : \u03b1\nm n : \u2124\n\u03b2 : Type u_4\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Monotone f\nc : \u03b1\nhc : 0 \u2264 c\nthis : DecidableRel fun x x_1 => x \u2264 x_1\n\u22a2 Monotone fun x => f x / c\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using (monotone_mul_right_of_nonneg (inv_nonneg.2 hc)).comp hf\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2\u271d : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\na b c\u271d d e : \u03b1\nm n : \u2124\n\u03b2 : Type u_4\ninst\u271d : Preorder \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : StrictMono f\nc : \u03b1\nhc : 0 < c\n\u22a2 StrictMono fun x => f x / c\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hf.mul_const (inv_pos.2 hc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 \u2264 a\nm n : \u2115\nmn : m \u2264 n\n\u22a2 1 / a ^ n \u2264 1 / a ^ m\n[PROOFSTEP]\nrefine' (one_div_le_one_div _ _).mpr (pow_le_pow a1 mn)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 \u2264 a\nm n : \u2115\nmn : m \u2264 n\n\u22a2 0 < a ^ n\n[PROOFSTEP]\nexact pow_pos (zero_lt_one.trans_le a1) _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 \u2264 a\nm n : \u2115\nmn : m \u2264 n\n\u22a2 0 < a ^ m\n[PROOFSTEP]\nexact pow_pos (zero_lt_one.trans_le a1) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 < a\nm n : \u2115\nmn : m < n\n\u22a2 1 / a ^ n < 1 / a ^ m\n[PROOFSTEP]\nrefine' (one_div_lt_one_div _ _).mpr (pow_lt_pow a1 mn)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 < a\nm n : \u2115\nmn : m < n\n\u22a2 0 < a ^ n\n[PROOFSTEP]\nexact pow_pos (zero_lt_one.trans a1) _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 < a\nm n : \u2115\nmn : m < n\n\u22a2 0 < a ^ m\n[PROOFSTEP]\nexact pow_pos (zero_lt_one.trans a1) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 \u2264 a\nm n : \u2115\nmn : m \u2264 n\n\u22a2 (a ^ n)\u207b\u00b9 \u2264 (a ^ m)\u207b\u00b9\n[PROOFSTEP]\nconvert one_div_pow_le_one_div_pow_of_le a1 mn using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 \u2264 a\nm n : \u2115\nmn : m \u2264 n\n\u22a2 (a ^ n)\u207b\u00b9 = 1 / a ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 \u2264 a\nm n : \u2115\nmn : m \u2264 n\n\u22a2 (a ^ m)\u207b\u00b9 = 1 / a ^ m\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 < a\nm n : \u2115\nmn : m < n\n\u22a2 (a ^ n)\u207b\u00b9 < (a ^ m)\u207b\u00b9\n[PROOFSTEP]\nconvert one_div_pow_lt_one_div_pow_of_lt a1 mn using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 < a\nm n : \u2115\nmn : m < n\n\u22a2 (a ^ n)\u207b\u00b9 = 1 / a ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm\u271d n\u271d : \u2124\na1 : 1 < a\nm n : \u2115\nmn : m < n\n\u22a2 (a ^ m)\u207b\u00b9 = 1 / a ^ m\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\ns : Set \u03b1\nha : 0 \u2264 a\nhs : IsGLB s b\n\u22a2 IsGLB ((fun b => a * b) '' s) (a * b)\n[PROOFSTEP]\nrcases lt_or_eq_of_le ha with (ha | rfl)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\ns : Set \u03b1\nha\u271d : 0 \u2264 a\nhs : IsGLB s b\nha : 0 < a\n\u22a2 IsGLB ((fun b => a * b) '' s) (a * b)\n[PROOFSTEP]\nexact (OrderIso.mulLeft\u2080 _ ha).isGLB_image'.2 hs\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\nb c d e : \u03b1\nm n : \u2124\ns : Set \u03b1\nhs : IsGLB s b\nha : 0 \u2264 0\n\u22a2 IsGLB ((fun b => 0 * b) '' s) (0 * b)\n[PROOFSTEP]\nsimp_rw [zero_mul]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\nb c d e : \u03b1\nm n : \u2124\ns : Set \u03b1\nhs : IsGLB s b\nha : 0 \u2264 0\n\u22a2 IsGLB ((fun a => 0) '' s) 0\n[PROOFSTEP]\nrw [hs.nonempty.image_const]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\nb c d e : \u03b1\nm n : \u2124\ns : Set \u03b1\nhs : IsGLB s b\nha : 0 \u2264 0\n\u22a2 IsGLB {0} 0\n[PROOFSTEP]\nexact isGLB_singleton\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\ns : Set \u03b1\nha : 0 \u2264 a\nhs : IsGLB s b\n\u22a2 IsGLB ((fun b => b * a) '' s) (b * a)\n[PROOFSTEP]\nsimpa [mul_comm] using hs.mul_left ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 0 < a / b \u2194 0 < a \u2227 0 < b \u2228 a < 0 \u2227 b < 0\n[PROOFSTEP]\nsimp only [division_def, mul_pos_iff, inv_pos, inv_lt_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 a / b < 0 \u2194 0 < a \u2227 b < 0 \u2228 a < 0 \u2227 0 < b\n[PROOFSTEP]\nsimp [division_def, mul_neg_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 0 \u2264 a / b \u2194 0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0\n[PROOFSTEP]\nsimp [division_def, mul_nonneg_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 a / b \u2264 0 \u2194 0 \u2264 a \u2227 b \u2264 0 \u2228 a \u2264 0 \u2227 0 \u2264 b\n[PROOFSTEP]\nsimp [division_def, mul_nonpos_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\n\u22a2 b / c \u2264 a \u2194 c * a \u2264 b\n[PROOFSTEP]\nrw [mul_comm, div_le_iff_of_neg hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\n\u22a2 a \u2264 b / c \u2194 b \u2264 a * c\n[PROOFSTEP]\nrw [\u2190 neg_neg c, mul_neg, div_neg, le_neg, div_le_iff (neg_pos.2 hc), neg_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\n\u22a2 a \u2264 b / c \u2194 b \u2264 c * a\n[PROOFSTEP]\nrw [mul_comm, le_div_iff_of_neg hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\n\u22a2 b / c < a \u2194 c * a < b\n[PROOFSTEP]\nrw [mul_comm, div_lt_iff_of_neg hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\n\u22a2 a < b / c \u2194 b < c * a\n[PROOFSTEP]\nrw [mul_comm, lt_div_iff_of_neg hc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : b \u2264 a\nhb : b \u2264 0\n\u22a2 a / b \u2264 1\n[PROOFSTEP]\nsimpa only [neg_div_neg_eq] using div_le_one_of_le (neg_le_neg h) (neg_nonneg_of_nonpos hb)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 a\u207b\u00b9 \u2264 b\u207b\u00b9 \u2194 b \u2264 a\n[PROOFSTEP]\nrw [\u2190 one_div, div_le_iff_of_neg ha, \u2190 div_eq_inv_mul, div_le_iff_of_neg hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 a\u207b\u00b9 \u2264 b \u2194 b\u207b\u00b9 \u2264 a\n[PROOFSTEP]\nrw [\u2190 inv_le_inv_of_neg hb (inv_lt_zero.2 ha), inv_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 a \u2264 b\u207b\u00b9 \u2194 b \u2264 a\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 inv_le_inv_of_neg (inv_lt_zero.2 hb) ha, inv_inv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c \u2264 0\nh : b \u2264 a\n\u22a2 a / c \u2264 b / c\n[PROOFSTEP]\nrw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c \u2264 0\nh : b \u2264 a\n\u22a2 a * (1 / c) \u2264 b * (1 / c)\n[PROOFSTEP]\nexact mul_le_mul_of_nonpos_right h (one_div_nonpos.2 hc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\nh : b < a\n\u22a2 a / c < b / c\n[PROOFSTEP]\nrw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c < 0\nh : b < a\n\u22a2 a * (1 / c) < b * (1 / c)\n[PROOFSTEP]\nexact mul_lt_mul_of_neg_right h (one_div_neg.2 hc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 1 \u2264 a / b \u2194 a \u2264 b\n[PROOFSTEP]\nrw [le_div_iff_of_neg hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 a / b \u2264 1 \u2194 b \u2264 a\n[PROOFSTEP]\nrw [div_le_iff_of_neg hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 1 < a / b \u2194 a < b\n[PROOFSTEP]\nrw [lt_div_iff_of_neg hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 a / b < 1 \u2194 b < a\n[PROOFSTEP]\nrw [div_lt_iff_of_neg hb, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 1 / a \u2264 b \u2194 1 / b \u2264 a\n[PROOFSTEP]\nsimpa using inv_le_of_neg ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 1 / a < b \u2194 1 / b < a\n[PROOFSTEP]\nsimpa using inv_lt_of_neg ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 a \u2264 1 / b \u2194 b \u2264 1 / a\n[PROOFSTEP]\nsimpa using le_inv_of_neg ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 a < 1 / b \u2194 b < 1 / a\n[PROOFSTEP]\nsimpa using lt_inv_of_neg ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 1 < a / b \u2194 0 < b \u2227 b < a \u2228 b < 0 \u2227 a < b\n[PROOFSTEP]\nrcases lt_trichotomy b 0 with (hb | rfl | hb)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 1 < a / b \u2194 0 < b \u2227 b < a \u2228 b < 0 \u2227 a < b\n[PROOFSTEP]\nsimp [hb, hb.not_lt, one_lt_div_of_neg]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na c d : \u03b1\nn : \u2124\n\u22a2 1 < a / 0 \u2194 0 < 0 \u2227 0 < a \u2228 0 < 0 \u2227 a < 0\n[PROOFSTEP]\nsimp [lt_irrefl, zero_le_one]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : 0 < b\n\u22a2 1 < a / b \u2194 0 < b \u2227 b < a \u2228 b < 0 \u2227 a < b\n[PROOFSTEP]\nsimp [hb, hb.not_lt, one_lt_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b\n[PROOFSTEP]\nrcases lt_trichotomy b 0 with (hb | rfl | hb)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b\n[PROOFSTEP]\nsimp [hb, hb.not_lt, one_le_div_of_neg]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na c d : \u03b1\nn : \u2124\n\u22a2 1 \u2264 a / 0 \u2194 0 < 0 \u2227 0 \u2264 a \u2228 0 < 0 \u2227 a \u2264 0\n[PROOFSTEP]\nsimp [lt_irrefl, zero_lt_one.not_le, zero_lt_one]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : 0 < b\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b\n[PROOFSTEP]\nsimp [hb, hb.not_lt, one_le_div]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 a / b < 1 \u2194 0 < b \u2227 a < b \u2228 b = 0 \u2228 b < 0 \u2227 b < a\n[PROOFSTEP]\nrcases lt_trichotomy b 0 with (hb | rfl | hb)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 a / b < 1 \u2194 0 < b \u2227 a < b \u2228 b = 0 \u2228 b < 0 \u2227 b < a\n[PROOFSTEP]\nsimp [hb, hb.not_lt, hb.ne, div_lt_one_of_neg]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na c d : \u03b1\nn : \u2124\n\u22a2 a / 0 < 1 \u2194 0 < 0 \u2227 a < 0 \u2228 0 = 0 \u2228 0 < 0 \u2227 0 < a\n[PROOFSTEP]\nsimp [zero_lt_one]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : 0 < b\n\u22a2 a / b < 1 \u2194 0 < b \u2227 a < b \u2228 b = 0 \u2228 b < 0 \u2227 b < a\n[PROOFSTEP]\nsimp [hb, hb.not_lt, div_lt_one, hb.ne.symm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 a / b \u2264 1 \u2194 0 < b \u2227 a \u2264 b \u2228 b = 0 \u2228 b < 0 \u2227 b \u2264 a\n[PROOFSTEP]\nrcases lt_trichotomy b 0 with (hb | rfl | hb)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 a / b \u2264 1 \u2194 0 < b \u2227 a \u2264 b \u2228 b = 0 \u2228 b < 0 \u2227 b \u2264 a\n[PROOFSTEP]\nsimp [hb, hb.not_lt, hb.ne, div_le_one_of_neg]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na c d : \u03b1\nn : \u2124\n\u22a2 a / 0 \u2264 1 \u2194 0 < 0 \u2227 a \u2264 0 \u2228 0 = 0 \u2228 0 < 0 \u2227 0 \u2264 a\n[PROOFSTEP]\nsimp [zero_le_one]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : 0 < b\n\u22a2 a / b \u2264 1 \u2194 0 < b \u2227 a \u2264 b \u2228 b = 0 \u2228 b < 0 \u2227 b \u2264 a\n[PROOFSTEP]\nsimp [hb, hb.not_lt, div_le_one, hb.ne.symm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\nh : a \u2264 b\n\u22a2 1 / b \u2264 1 / a\n[PROOFSTEP]\nrwa [div_le_iff_of_neg' hb, \u2190 div_eq_mul_one_div, div_le_one_of_neg (h.trans_lt hb)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\nh : a < b\n\u22a2 1 / b < 1 / a\n[PROOFSTEP]\nrwa [div_lt_iff_of_neg' hb, \u2190 div_eq_mul_one_div, div_lt_one_of_neg (h.trans hb)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 1 / a \u2264 1 / b \u2194 b \u2264 a\n[PROOFSTEP]\nsimpa [one_div] using inv_le_inv_of_neg ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh1 : a < 0\nh2 : -1 < a\nthis : 1 / a < 1 / -1\n\u22a2 1 / a < -1\n[PROOFSTEP]\nrwa [one_div_neg_one_eq_neg_one] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh1 : a < 0\nh2 : -1 \u2264 a\nthis : 1 / a \u2264 1 / -1\n\u22a2 1 / a \u2264 -1\n[PROOFSTEP]\nrwa [one_div_neg_one_eq_neg_one] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 a - a / 2 = a / 2\n[PROOFSTEP]\nsuffices a / 2 + a / 2 - a / 2 = a / 2 by rwa [add_halves] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\nthis : a / 2 + a / 2 - a / 2 = a / 2\n\u22a2 a - a / 2 = a / 2\n[PROOFSTEP]\nrwa [add_halves] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 a / 2 + a / 2 - a / 2 = a / 2\n[PROOFSTEP]\nrw [add_sub_cancel]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 a / 2 - a = -(a / 2)\n[PROOFSTEP]\nsuffices a / 2 - (a / 2 + a / 2) = -(a / 2) by rwa [add_halves] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\nthis : a / 2 - (a / 2 + a / 2) = -(a / 2)\n\u22a2 a / 2 - a = -(a / 2)\n[PROOFSTEP]\nrwa [add_halves] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 a / 2 - (a / 2 + a / 2) = -(a / 2)\n[PROOFSTEP]\nrw [sub_add_eq_sub_sub, sub_self, zero_sub]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : a < b\n\u22a2 a + (b - a) / 2 < b\n[PROOFSTEP]\nrwa [\u2190 div_sub_div_same, sub_eq_add_neg, add_comm (b / 2), \u2190 add_assoc, \u2190 sub_eq_add_neg, \u2190 lt_sub_iff_add_lt,\n  sub_self_div_two, sub_self_div_two, div_lt_div_right (zero_lt_two' \u03b1)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\na2 : 2 \u2264 a\n\u22a2 (1 - 1 / a)\u207b\u00b9 \u2264 2\n[PROOFSTEP]\nrefine'\n  (inv_le_inv_of_le (inv_pos.2 <| zero_lt_two' \u03b1) _).trans_eq\n    (inv_inv (2 : \u03b1))\n      -- move `1 / a` to the left and `2\u207b\u00b9` to the right.\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\na2 : 2 \u2264 a\n\u22a2 2\u207b\u00b9 \u2264 1 - 1 / a\n[PROOFSTEP]\nrw [le_sub_iff_add_le, add_comm, \u2190 le_sub_iff_add_le]\n  -- take inverses on both sides and use the assumption `2 \u2264 a`.\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\na2 : 2 \u2264 a\n\u22a2 1 / a \u2264 1 - 2\u207b\u00b9\n[PROOFSTEP]\nconvert (one_div a).le.trans (inv_le_inv_of_le zero_lt_two a2) using 1\n  -- show `1 - 1 / 2 = 1 / 2`.\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\na2 : 2 \u2264 a\n\u22a2 1 - 2\u207b\u00b9 = 2\u207b\u00b9\n[PROOFSTEP]\nrw [sub_eq_iff_eq_add, \u2190 two_mul, mul_inv_cancel two_ne_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\ns : Set \u03b1\nha : 0 \u2264 a\nhs : IsLUB s b\n\u22a2 IsLUB ((fun b => a * b) '' s) (a * b)\n[PROOFSTEP]\nrcases lt_or_eq_of_le ha with (ha | rfl)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\ns : Set \u03b1\nha\u271d : 0 \u2264 a\nhs : IsLUB s b\nha : 0 < a\n\u22a2 IsLUB ((fun b => a * b) '' s) (a * b)\n[PROOFSTEP]\nexact (OrderIso.mulLeft\u2080 _ ha).isLUB_image'.2 hs\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\ns : Set \u03b1\nhs : IsLUB s b\nha : 0 \u2264 0\n\u22a2 IsLUB ((fun b => 0 * b) '' s) (0 * b)\n[PROOFSTEP]\nsimp_rw [zero_mul]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\ns : Set \u03b1\nhs : IsLUB s b\nha : 0 \u2264 0\n\u22a2 IsLUB ((fun a => 0) '' s) 0\n[PROOFSTEP]\nrw [hs.nonempty.image_const]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\ns : Set \u03b1\nhs : IsLUB s b\nha : 0 \u2264 0\n\u22a2 IsLUB {0} 0\n[PROOFSTEP]\nexact isLUB_singleton\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\ns : Set \u03b1\nha : 0 \u2264 a\nhs : IsLUB s b\n\u22a2 IsLUB ((fun b => b * a) '' s) (b * a)\n[PROOFSTEP]\nsimpa [mul_comm] using hs.mul_left ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c \u2260 0\nhd : d \u2260 0\n\u22a2 (a * d - b * c) / (c * d) < 0 \u2194 a / c < b / d\n[PROOFSTEP]\nrw [mul_comm b c, \u2190 div_sub_div _ _ hc hd, sub_lt_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhc : c \u2260 0\nhd : d \u2260 0\n\u22a2 (a * d - b * c) / (c * d) \u2264 0 \u2194 a / c \u2264 b / d\n[PROOFSTEP]\nrw [mul_comm b c, \u2190 div_sub_div _ _ hc hd, sub_nonpos]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : \u2200 (\u03b5 : \u03b1), \u03b5 > 0 \u2192 b - \u03b5 \u2264 a\n\u22a2 b \u2264 a\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : a < b\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 a < b - \u03b5\n[PROOFSTEP]\nsimpa only [@and_comm ((0 : \u03b1) < _), lt_sub_iff_add_lt, gt_iff_lt] using exists_add_lt_and_pos_of_lt h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\na0 : 0 \u2264 a\nb0 : 0 \u2264 b\nh : a = -b\n\u22a2 a = b\n[PROOFSTEP]\nsubst a\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\nb0 : 0 \u2264 b\na0 : 0 \u2264 -b\n\u22a2 -b = b\n[PROOFSTEP]\nhave : b = 0 := le_antisymm (neg_nonneg.1 a0) b0\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\nb0 : 0 \u2264 b\na0 : 0 \u2264 -b\nthis : b = 0\n\u22a2 -b = b\n[PROOFSTEP]\nrw [this, neg_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 |1 / a| = 1 / |a|\n[PROOFSTEP]\nrw [abs_div, abs_one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Field.Basic", "llama_tokens": 22625, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.7341195327172402, "lm_q1q2_score": 0.5896255720722603}}
{"text": "[GOAL]\n\u22a2 lift \ud835\udd20 = \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 two_power_aleph0, lift_two_power, lift_aleph0, two_power_aleph0]\n[GOAL]\nc : Cardinal.{u}\n\u22a2 \ud835\udd20 \u2264 lift c \u2194 \ud835\udd20 \u2264 c\n[PROOFSTEP]\nrw [\u2190 lift_continuum.{u, v}, lift_le]\n[GOAL]\nc : Cardinal.{u}\n\u22a2 lift c \u2264 \ud835\udd20 \u2194 c \u2264 \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 lift_continuum.{u, v}, lift_le]\n[GOAL]\nc : Cardinal.{u}\n\u22a2 \ud835\udd20 < lift c \u2194 \ud835\udd20 < c\n[PROOFSTEP]\nrw [\u2190 lift_continuum.{u, v}, lift_lt]\n[GOAL]\nc : Cardinal.{u}\n\u22a2 lift c < \ud835\udd20 \u2194 c < \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 lift_continuum.{u, v}, lift_lt]\n[GOAL]\n\u22a2 beth 1 = \ud835\udd20\n[PROOFSTEP]\nsimpa using beth_succ 0\n[GOAL]\n\u22a2 #(Set \u2115) = \ud835\udd20\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 aleph 1 \u2264 \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 succ_aleph0]\n[GOAL]\n\u22a2 Order.succ \u2135\u2080 \u2264 \ud835\udd20\n[PROOFSTEP]\nexact Order.succ_le_of_lt aleph0_lt_continuum\n[GOAL]\n\u22a2 \ud835\udd20 ^ \u2135\u2080 = \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 two_power_aleph0, \u2190 power_mul, mul_eq_left le_rfl le_rfl aleph0_ne_zero]\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Cardinal.Continuum", "llama_tokens": 512, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8991213691605412, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.5888298514884581}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type u_1\ninst\u271d\u00b3 : OrderedCommMonoid \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Pow \u03b2 \u2115\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\none : f 1 = 1\nmul : \u2200 (x y : \u03b2), f (x * y) = f x * f y\nnpow : \u2200 (x : \u03b2) (n : \u2115), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : PartialOrder \u03b2 := PartialOrder.lift f hf\nsrc\u271d : CommMonoid \u03b2 := Injective.commMonoid f hf one mul npow\na b : \u03b2\nab : a \u2264 b\nc : \u03b2\n\u22a2 f (c * a) \u2264 f (c * b)\n[PROOFSTEP]\nrw [mul, mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2\u271d : Type u_1\ninst\u271d\u00b3 : OrderedCommMonoid \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Pow \u03b2 \u2115\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\none : f 1 = 1\nmul : \u2200 (x y : \u03b2), f (x * y) = f x * f y\nnpow : \u2200 (x : \u03b2) (n : \u2115), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : PartialOrder \u03b2 := PartialOrder.lift f hf\nsrc\u271d : CommMonoid \u03b2 := Injective.commMonoid f hf one mul npow\na b : \u03b2\nab : a \u2264 b\nc : \u03b2\n\u22a2 f c * f a \u2264 f c * f b\n[PROOFSTEP]\napply mul_le_mul_left'\n[GOAL]\ncase bc\n\u03b1 : Type u\n\u03b2\u271d : Type u_1\ninst\u271d\u00b3 : OrderedCommMonoid \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : One \u03b2\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Pow \u03b2 \u2115\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\none : f 1 = 1\nmul : \u2200 (x y : \u03b2), f (x * y) = f x * f y\nnpow : \u2200 (x : \u03b2) (n : \u2115), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : PartialOrder \u03b2 := PartialOrder.lift f hf\nsrc\u271d : CommMonoid \u03b2 := Injective.commMonoid f hf one mul npow\na b : \u03b2\nab : a \u2264 b\nc : \u03b2\n\u22a2 f a \u2264 f b\n[PROOFSTEP]\nexact ab\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : LinearOrder \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 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b a0 b0 : \u03b1\nha : a0 \u2264 a\nhb : b0 \u2264 b\nab : a * b \u2264 a0 * b0\n\u22a2 a = a0 \u2227 b = b0\n[PROOFSTEP]\nhaveI := Mul.to_covariantClass_right \u03b1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : LinearOrder \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 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b a0 b0 : \u03b1\nha : a0 \u2264 a\nhb : b0 \u2264 b\nab : a * b \u2264 a0 * b0\nthis : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a = a0 \u2227 b = b0\n[PROOFSTEP]\nhave ha' : \u00aca0 * b0 < a * b \u2192 \u00aca0 < a := mt (mul_lt_mul_of_lt_of_le \u00b7 hb)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : LinearOrder \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 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b a0 b0 : \u03b1\nha : a0 \u2264 a\nhb : b0 \u2264 b\nab : a * b \u2264 a0 * b0\nthis : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nha' : \u00aca0 * b0 < a * b \u2192 \u00aca0 < a\n\u22a2 a = a0 \u2227 b = b0\n[PROOFSTEP]\nhave hb' : \u00aca0 * b0 < a * b \u2192 \u00acb0 < b := mt (mul_lt_mul_of_le_of_lt ha \u00b7)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : LinearOrder \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 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b a0 b0 : \u03b1\nha : a0 \u2264 a\nhb : b0 \u2264 b\nab : a * b \u2264 a0 * b0\nthis : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nha' : \u00aca0 * b0 < a * b \u2192 \u00aca0 < a\nhb' : \u00aca0 * b0 < a * b \u2192 \u00acb0 < b\n\u22a2 a = a0 \u2227 b = b0\n[PROOFSTEP]\npush_neg at ha' hb' \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d\u2074 : Mul \u03b1\ninst\u271d\u00b3 : LinearOrder \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 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b a0 b0 : \u03b1\nha : a0 \u2264 a\nhb : b0 \u2264 b\nab : a * b \u2264 a0 * b0\nthis : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nha' : a * b \u2264 a0 * b0 \u2192 a \u2264 a0\nhb' : a * b \u2264 a0 * b0 \u2192 b \u2264 b0\n\u22a2 a = a0 \u2227 b = b0\n[PROOFSTEP]\nexact \u27e8ha.antisymm' (ha' ab), hb.antisymm' (hb' ab)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Monoid.Basic", "llama_tokens": 2163, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267118068790619, "lm_q2_score": 0.7122321842389469, "lm_q1q2_score": 0.5888107559496006}}
{"text": "[GOAL]\n\u22a2 \u2200 (a b c : Game), a + b + c = a + (b + c)\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9 \u27e8z\u27e9\n[GOAL]\ncase mk.mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\nc\u271d : Game\nz : PGame\n\u22a2 Quot.mk Setoid.r x + Quot.mk Setoid.r y + Quot.mk Setoid.r z =\n    Quot.mk Setoid.r x + (Quot.mk Setoid.r y + Quot.mk Setoid.r z)\n[PROOFSTEP]\nexact Quot.sound add_assoc_equiv\n[GOAL]\n\u22a2 \u2200 (a : Game), 0 + a = a\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\na\u271d : Game\nx : PGame\n\u22a2 0 + Quot.mk Setoid.r x = Quot.mk Setoid.r x\n[PROOFSTEP]\nexact Quot.sound (zero_add_equiv x)\n[GOAL]\n\u22a2 \u2200 (a : Game), a + 0 = a\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\na\u271d : Game\nx : PGame\n\u22a2 Quot.mk Setoid.r x + 0 = Quot.mk Setoid.r x\n[PROOFSTEP]\nexact Quot.sound (add_zero_equiv x)\n[GOAL]\n\u22a2 \u2200 (a : Game), -a + a = 0\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\na\u271d : Game\nx : PGame\n\u22a2 -Quot.mk Setoid.r x + Quot.mk Setoid.r x = 0\n[PROOFSTEP]\nexact Quot.sound (add_left_neg_equiv x)\n[GOAL]\n\u22a2 \u2200 (a b : Game), a + b = b + a\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9\n[GOAL]\ncase mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\n\u22a2 Quot.mk Setoid.r x + Quot.mk Setoid.r y = Quot.mk Setoid.r y + Quot.mk Setoid.r x\n[PROOFSTEP]\nexact Quot.sound add_comm_equiv\n[GOAL]\n\u22a2 \u2200 (a : Game), a \u2264 a\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\na\u271d : Game\nx : PGame\n\u22a2 Quot.mk Setoid.r x \u2264 Quot.mk Setoid.r x\n[PROOFSTEP]\nexact le_refl x\n[GOAL]\n\u22a2 \u2200 (a b c : Game), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9 \u27e8z\u27e9\n[GOAL]\ncase mk.mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\nc\u271d : Game\nz : PGame\n\u22a2 Quot.mk Setoid.r x \u2264 Quot.mk Setoid.r y \u2192\n    Quot.mk Setoid.r y \u2264 Quot.mk Setoid.r z \u2192 Quot.mk Setoid.r x \u2264 Quot.mk Setoid.r z\n[PROOFSTEP]\nexact @le_trans _ _ x y z\n[GOAL]\n\u22a2 \u2200 (a b : Game), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9\n[GOAL]\ncase mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\n\u22a2 Quot.mk Setoid.r x < Quot.mk Setoid.r y \u2194\n    Quot.mk Setoid.r x \u2264 Quot.mk Setoid.r y \u2227 \u00acQuot.mk Setoid.r y \u2264 Quot.mk Setoid.r x\n[PROOFSTEP]\nexact @lt_iff_le_not_le _ _ x y\n[GOAL]\n\u22a2 \u2200 (a b : Game), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9 h\u2081 h\u2082\n[GOAL]\ncase mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\nh\u2081 : Quot.mk Setoid.r x \u2264 Quot.mk Setoid.r y\nh\u2082 : Quot.mk Setoid.r y \u2264 Quot.mk Setoid.r x\n\u22a2 Quot.mk Setoid.r x = Quot.mk Setoid.r y\n[PROOFSTEP]\napply Quot.sound\n[GOAL]\ncase mk.mk.a\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\nh\u2081 : Quot.mk Setoid.r x \u2264 Quot.mk Setoid.r y\nh\u2082 : Quot.mk Setoid.r y \u2264 Quot.mk Setoid.r x\n\u22a2 Setoid.r x y\n[PROOFSTEP]\nexact \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\n\u22a2 \u2200 {x y : Game}, \u00acx \u2264 y \u2194 y \u29cf x\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9\n[GOAL]\ncase mk.mk\nx\u271d : Game\nx : PGame\ny\u271d : Game\ny : PGame\n\u22a2 \u00acQuot.mk Setoid.r x \u2264 Quot.mk Setoid.r y \u2194 Quot.mk Setoid.r y \u29cf Quot.mk Setoid.r x\n[PROOFSTEP]\nexact PGame.not_le\n[GOAL]\n\u22a2 \u2200 {x y : Game}, \u00acx \u29cf y \u2194 y \u2264 x\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9\n[GOAL]\ncase mk.mk\nx\u271d : Game\nx : PGame\ny\u271d : Game\ny : PGame\n\u22a2 \u00acQuot.mk Setoid.r x \u29cf Quot.mk Setoid.r y \u2194 Quot.mk Setoid.r y \u2264 Quot.mk Setoid.r x\n[PROOFSTEP]\nexact PGame.not_lf\n[GOAL]\n\u22a2 \u2200 (a b : Game), a \u29cf b \u2228 a = b \u2228 b \u29cf a\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9\n[GOAL]\ncase mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\n\u22a2 Quot.mk Setoid.r x \u29cf Quot.mk Setoid.r y \u2228\n    Quot.mk Setoid.r x = Quot.mk Setoid.r y \u2228 Quot.mk Setoid.r y \u29cf Quot.mk Setoid.r x\n[PROOFSTEP]\nchange _ \u2228 \u27e6x\u27e7 = \u27e6y\u27e7 \u2228 _\n[GOAL]\ncase mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\n\u22a2 Quot.mk Setoid.r x \u29cf Quot.mk Setoid.r y \u2228\n    Quotient.mk setoid x = Quotient.mk setoid y \u2228 Quot.mk Setoid.r y \u29cf Quot.mk Setoid.r x\n[PROOFSTEP]\nrw [Quotient.eq]\n[GOAL]\ncase mk.mk\na\u271d : Game\nx : PGame\nb\u271d : Game\ny : PGame\n\u22a2 Quot.mk Setoid.r x \u29cf Quot.mk Setoid.r y \u2228 x \u2248 y \u2228 Quot.mk Setoid.r y \u29cf Quot.mk Setoid.r x\n[PROOFSTEP]\napply lf_or_equiv_or_gf\n[GOAL]\n\u22a2 Covariant Game Game (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 h\n[GOAL]\ncase mk.mk.mk\nm\u271d : Game\na : PGame\nn\u2081\u271d : Game\nb : PGame\nn\u2082\u271d : Game\nc : PGame\nh : Quot.mk Setoid.r b \u2264 Quot.mk Setoid.r c\n\u22a2 (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r b) \u2264\n    (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)\n[PROOFSTEP]\nexact @add_le_add_left _ _ _ _ b c h a\n[GOAL]\n\u22a2 Covariant Game Game (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 h\n[GOAL]\ncase mk.mk.mk\nm\u271d : Game\na : PGame\nn\u2081\u271d : Game\nb : PGame\nn\u2082\u271d : Game\nc : PGame\nh : Quot.mk Setoid.r b \u2264 Quot.mk Setoid.r c\n\u22a2 swap (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r b) \u2264\n    swap (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)\n[PROOFSTEP]\nexact @add_le_add_right _ _ _ _ b c h a\n[GOAL]\n\u22a2 Covariant Game Game (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 h\n[GOAL]\ncase mk.mk.mk\nm\u271d : Game\na : PGame\nn\u2081\u271d : Game\nb : PGame\nn\u2082\u271d : Game\nc : PGame\nh : Quot.mk Setoid.r b < Quot.mk Setoid.r c\n\u22a2 (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r b) <\n    (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)\n[PROOFSTEP]\nexact @add_lt_add_left _ _ _ _ b c h a\n[GOAL]\n\u22a2 Covariant Game Game (swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\n[PROOFSTEP]\nrintro \u27e8a\u27e9 \u27e8b\u27e9 \u27e8c\u27e9 h\n[GOAL]\ncase mk.mk.mk\nm\u271d : Game\na : PGame\nn\u2081\u271d : Game\nb : PGame\nn\u2082\u271d : Game\nc : PGame\nh : Quot.mk Setoid.r b < Quot.mk Setoid.r c\n\u22a2 swap (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r b) <\n    swap (fun x x_1 => x + x_1) (Quot.mk Setoid.r a) (Quot.mk Setoid.r c)\n[PROOFSTEP]\nexact @add_lt_add_right _ _ _ _ b c h a\n[GOAL]\n\u22a2 \u2200 {b c : Game}, b \u29cf c \u2192 \u2200 (a : Game), b + a \u29cf c + a\n[PROOFSTEP]\nrintro \u27e8b\u27e9 \u27e8c\u27e9 h \u27e8a\u27e9\n[GOAL]\ncase mk.mk.mk\nb\u271d : Game\nb : PGame\nc\u271d : Game\nc : PGame\nh : Quot.mk Setoid.r b \u29cf Quot.mk Setoid.r c\na\u271d : Game\na : PGame\n\u22a2 Quot.mk Setoid.r b + Quot.mk Setoid.r a \u29cf Quot.mk Setoid.r c + Quot.mk Setoid.r a\n[PROOFSTEP]\napply PGame.add_lf_add_right h\n[GOAL]\n\u22a2 \u2200 {b c : Game}, b \u29cf c \u2192 \u2200 (a : Game), a + b \u29cf a + c\n[PROOFSTEP]\nrintro \u27e8b\u27e9 \u27e8c\u27e9 h \u27e8a\u27e9\n[GOAL]\ncase mk.mk.mk\nb\u271d : Game\nb : PGame\nc\u271d : Game\nc : PGame\nh : Quot.mk Setoid.r b \u29cf Quot.mk Setoid.r c\na\u271d : Game\na : PGame\n\u22a2 Quot.mk Setoid.r a + Quot.mk Setoid.r b \u29cf Quot.mk Setoid.r a + Quot.mk Setoid.r c\n[PROOFSTEP]\napply PGame.add_lf_add_left h\n[GOAL]\nx y : PGame\nL : LeftMoves x \u2243 LeftMoves y\nR : RightMoves x \u2243 RightMoves y\nhl : \u2200 (i : LeftMoves x), Quotient.mk setoid (moveLeft x i) = Quotient.mk setoid (moveLeft y (\u2191L i))\nhr : \u2200 (j : RightMoves x), Quotient.mk setoid (moveRight x j) = Quotient.mk setoid (moveRight y (\u2191R j))\n\u22a2 Quotient.mk setoid x = Quotient.mk setoid y\n[PROOFSTEP]\nexact\n  Quot.sound\n    (equiv_of_mk_equiv L R (fun _ => Game.PGame.equiv_iff_game_eq.2 (hl _))\n      (fun _ => Game.PGame.equiv_iff_game_eq.2 (hr _)))\n[GOAL]\nx y : PGame\n\u22a2 PGame\n[PROOFSTEP]\ninduction' x with xl xr _ _ IHxl IHxr generalizing y\n[GOAL]\ncase mk\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\ny : PGame\n\u22a2 PGame\n[PROOFSTEP]\ninduction' y with yl yr yL yR IHyl IHyr\n[GOAL]\ncase mk.mk\ny : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\n\u22a2 PGame\n[PROOFSTEP]\nhave y := mk yl yr yL yR\n[GOAL]\ncase mk.mk\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\n\u22a2 PGame\n[PROOFSTEP]\nrefine' \u27e8Sum (xl \u00d7 yl) (xr \u00d7 yr), Sum (xl \u00d7 yr) (xr \u00d7 yl), _, _\u27e9\n[GOAL]\ncase mk.mk.refine'_1\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\n\u22a2 xl \u00d7 yl \u2295 xr \u00d7 yr \u2192 PGame\n[PROOFSTEP]\nrintro (\u27e8i, j\u27e9 | \u27e8i, j\u27e9)\n[GOAL]\ncase mk.mk.refine'_2\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\n\u22a2 xl \u00d7 yr \u2295 xr \u00d7 yl \u2192 PGame\n[PROOFSTEP]\nrintro (\u27e8i, j\u27e9 | \u27e8i, j\u27e9)\n[GOAL]\ncase mk.mk.refine'_1.inl.mk\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\ni : xl\nj : yl\n\u22a2 PGame\n[PROOFSTEP]\nexact IHxl i y + IHyl j - IHxl i (yL j)\n[GOAL]\ncase mk.mk.refine'_1.inr.mk\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\ni : xr\nj : yr\n\u22a2 PGame\n[PROOFSTEP]\nexact IHxr i y + IHyr j - IHxr i (yR j)\n[GOAL]\ncase mk.mk.refine'_2.inl.mk\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\ni : xl\nj : yr\n\u22a2 PGame\n[PROOFSTEP]\nexact IHxl i y + IHyr j - IHxl i (yR j)\n[GOAL]\ncase mk.mk.refine'_2.inr.mk\ny\u271d : PGame\nxl xr : Type u\na\u271d\u00b9 : xl \u2192 PGame\na\u271d : xr \u2192 PGame\nIHxl : xl \u2192 PGame \u2192 PGame\nIHxr : xr \u2192 PGame \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nIHyl : yl \u2192 PGame\nIHyr : yr \u2192 PGame\ny : PGame\ni : xr\nj : yl\n\u22a2 PGame\n[PROOFSTEP]\nexact IHxr i y + IHyl j - IHxr i (yL j)\n[GOAL]\nx y : PGame\ni : LeftMoves x\nj : LeftMoves y\n\u22a2 moveLeft (x * y) (\u2191toLeftMovesMul (Sum.inl (i, j))) =\n    moveLeft x i * y + x * moveLeft y j - moveLeft x i * moveLeft y j\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\ny : PGame\nj : LeftMoves y\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y) (\u2191toLeftMovesMul (Sum.inl (i, j))) =\n    moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * y + mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * moveLeft y j - moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * moveLeft y j\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\n\u03b1\u271d\u00b9 \u03b2\u271d\u00b9 : Type u_1\na\u271d\u00b3 : \u03b1\u271d\u00b9 \u2192 PGame\na\u271d\u00b2 : \u03b2\u271d\u00b9 \u2192 PGame\ni : LeftMoves (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nj : LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveLeft (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (\u2191toLeftMovesMul (Sum.inl (i, j))) =\n    moveLeft (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d + mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j -\n      moveLeft (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : PGame\ni : RightMoves x\nj : RightMoves y\n\u22a2 moveLeft (x * y) (\u2191toLeftMovesMul (Sum.inr (i, j))) =\n    moveRight x i * y + x * moveRight y j - moveRight x i * moveRight y j\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\ny : PGame\nj : RightMoves y\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y) (\u2191toLeftMovesMul (Sum.inr (i, j))) =\n    moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * y + mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * moveRight y j - moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * moveRight y j\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\n\u03b1\u271d\u00b9 \u03b2\u271d\u00b9 : Type u_1\na\u271d\u00b3 : \u03b1\u271d\u00b9 \u2192 PGame\na\u271d\u00b2 : \u03b2\u271d\u00b9 \u2192 PGame\ni : RightMoves (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nj : RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveLeft (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (\u2191toLeftMovesMul (Sum.inr (i, j))) =\n    moveRight (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d + mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j -\n      moveRight (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : PGame\ni : LeftMoves x\nj : RightMoves y\n\u22a2 moveRight (x * y) (\u2191toRightMovesMul (Sum.inl (i, j))) =\n    moveLeft x i * y + x * moveRight y j - moveLeft x i * moveRight y j\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\ny : PGame\nj : RightMoves y\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y) (\u2191toRightMovesMul (Sum.inl (i, j))) =\n    moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * y + mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * moveRight y j - moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * moveRight y j\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\n\u03b1\u271d\u00b9 \u03b2\u271d\u00b9 : Type u_1\na\u271d\u00b3 : \u03b1\u271d\u00b9 \u2192 PGame\na\u271d\u00b2 : \u03b2\u271d\u00b9 \u2192 PGame\ni : LeftMoves (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nj : RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveRight (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (\u2191toRightMovesMul (Sum.inl (i, j))) =\n    moveLeft (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d + mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j -\n      moveLeft (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : PGame\ni : RightMoves x\nj : LeftMoves y\n\u22a2 moveRight (x * y) (\u2191toRightMovesMul (Sum.inr (i, j))) =\n    moveRight x i * y + x * moveLeft y j - moveRight x i * moveLeft y j\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\ny : PGame\nj : LeftMoves y\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\ni : RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * y) (\u2191toRightMovesMul (Sum.inr (i, j))) =\n    moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * y + mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * moveLeft y j - moveRight (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i * moveLeft y j\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\n\u03b1\u271d\u00b9 \u03b2\u271d\u00b9 : Type u_1\na\u271d\u00b3 : \u03b1\u271d\u00b9 \u2192 PGame\na\u271d\u00b2 : \u03b2\u271d\u00b9 \u2192 PGame\ni : RightMoves (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\nj : LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 moveRight (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) (\u2191toRightMovesMul (Sum.inr (i, j))) =\n    moveRight (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d + mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2 * moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j -\n      moveRight (mk \u03b1\u271d\u00b9 \u03b2\u271d\u00b9 a\u271d\u00b3 a\u271d\u00b2) i * moveLeft (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) j\n[PROOFSTEP]\nrfl\n[GOAL]\nx y : PGame\nk : LeftMoves (x * y)\nP : LeftMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (iy : LeftMoves y), P (\u2191toLeftMovesMul (Sum.inl (ix, iy)))\nhr : \u2200 (jx : RightMoves x) (jy : RightMoves y), P (\u2191toLeftMovesMul (Sum.inr (jx, jy)))\n\u22a2 P k\n[PROOFSTEP]\nrw [\u2190 toLeftMovesMul.apply_symm_apply k]\n[GOAL]\nx y : PGame\nk : LeftMoves (x * y)\nP : LeftMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (iy : LeftMoves y), P (\u2191toLeftMovesMul (Sum.inl (ix, iy)))\nhr : \u2200 (jx : RightMoves x) (jy : RightMoves y), P (\u2191toLeftMovesMul (Sum.inr (jx, jy)))\n\u22a2 P (\u2191toLeftMovesMul (\u2191toLeftMovesMul.symm k))\n[PROOFSTEP]\nrcases toLeftMovesMul.symm k with (\u27e8ix, iy\u27e9 | \u27e8jx, jy\u27e9)\n[GOAL]\ncase inl.mk\nx y : PGame\nk : LeftMoves (x * y)\nP : LeftMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (iy : LeftMoves y), P (\u2191toLeftMovesMul (Sum.inl (ix, iy)))\nhr : \u2200 (jx : RightMoves x) (jy : RightMoves y), P (\u2191toLeftMovesMul (Sum.inr (jx, jy)))\nix : LeftMoves x\niy : LeftMoves y\n\u22a2 P (\u2191toLeftMovesMul (Sum.inl (ix, iy)))\n[PROOFSTEP]\napply hl\n[GOAL]\ncase inr.mk\nx y : PGame\nk : LeftMoves (x * y)\nP : LeftMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (iy : LeftMoves y), P (\u2191toLeftMovesMul (Sum.inl (ix, iy)))\nhr : \u2200 (jx : RightMoves x) (jy : RightMoves y), P (\u2191toLeftMovesMul (Sum.inr (jx, jy)))\njx : RightMoves x\njy : RightMoves y\n\u22a2 P (\u2191toLeftMovesMul (Sum.inr (jx, jy)))\n[PROOFSTEP]\napply hr\n[GOAL]\nx y : PGame\nk : RightMoves (x * y)\nP : RightMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (jy : RightMoves y), P (\u2191toRightMovesMul (Sum.inl (ix, jy)))\nhr : \u2200 (jx : RightMoves x) (iy : LeftMoves y), P (\u2191toRightMovesMul (Sum.inr (jx, iy)))\n\u22a2 P k\n[PROOFSTEP]\nrw [\u2190 toRightMovesMul.apply_symm_apply k]\n[GOAL]\nx y : PGame\nk : RightMoves (x * y)\nP : RightMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (jy : RightMoves y), P (\u2191toRightMovesMul (Sum.inl (ix, jy)))\nhr : \u2200 (jx : RightMoves x) (iy : LeftMoves y), P (\u2191toRightMovesMul (Sum.inr (jx, iy)))\n\u22a2 P (\u2191toRightMovesMul (\u2191toRightMovesMul.symm k))\n[PROOFSTEP]\nrcases toRightMovesMul.symm k with (\u27e8ix, iy\u27e9 | \u27e8jx, jy\u27e9)\n[GOAL]\ncase inl.mk\nx y : PGame\nk : RightMoves (x * y)\nP : RightMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (jy : RightMoves y), P (\u2191toRightMovesMul (Sum.inl (ix, jy)))\nhr : \u2200 (jx : RightMoves x) (iy : LeftMoves y), P (\u2191toRightMovesMul (Sum.inr (jx, iy)))\nix : LeftMoves x\niy : RightMoves y\n\u22a2 P (\u2191toRightMovesMul (Sum.inl (ix, iy)))\n[PROOFSTEP]\napply hl\n[GOAL]\ncase inr.mk\nx y : PGame\nk : RightMoves (x * y)\nP : RightMoves (x * y) \u2192 Prop\nhl : \u2200 (ix : LeftMoves x) (jy : RightMoves y), P (\u2191toRightMovesMul (Sum.inl (ix, jy)))\nhr : \u2200 (jx : RightMoves x) (iy : LeftMoves y), P (\u2191toRightMovesMul (Sum.inr (jx, iy)))\njx : RightMoves x\njy : LeftMoves y\n\u22a2 P (\u2191toRightMovesMul (Sum.inr (jx, jy)))\n[PROOFSTEP]\napply hr\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 mk xl xr xL xR * mk yl yr yL yR \u2261r mk yl yr yL yR * mk xl xr xL xR\n[PROOFSTEP]\nrefine'\n  \u27e8Equiv.sumCongr (Equiv.prodComm _ _) (Equiv.prodComm _ _),\n    (Equiv.sumComm _ _).trans (Equiv.sumCongr (Equiv.prodComm _ _) (Equiv.prodComm _ _)), _, _\u27e9\n[GOAL]\ncase refine'_1\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (i : LeftMoves (mk xl xr xL xR * mk yl yr yL yR)) \u2192\n    moveLeft (mk xl xr xL xR * mk yl yr yL yR) i \u2261r\n      moveLeft (mk yl yr yL yR * mk xl xr xL xR) (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) i)\n[PROOFSTEP]\nrintro (\u27e8i, j\u27e9 | \u27e8i, j\u27e9)\n[GOAL]\ncase refine'_2\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (j : RightMoves (mk xl xr xL xR * mk yl yr yL yR)) \u2192\n    moveRight (mk xl xr xL xR * mk yl yr yL yR) j \u2261r\n      moveRight (mk yl yr yL yR * mk xl xr xL xR)\n        (\u2191((Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)).trans (Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr))) j)\n[PROOFSTEP]\nrintro (\u27e8i, j\u27e9 | \u27e8i, j\u27e9)\n[GOAL]\ncase refine'_1.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 moveLeft (mk xl xr xL xR * mk yl yr yL yR) (Sum.inl (i, j)) \u2261r\n    moveLeft (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) (Sum.inl (i, j)))\n[PROOFSTEP]\n{ dsimp\n  exact\n    ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n      (mulCommRelabelling _ _)\n}\n[GOAL]\ncase refine'_1.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 moveLeft (mk xl xr xL xR * mk yl yr yL yR) (Sum.inl (i, j)) \u2261r\n    moveLeft (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) (Sum.inl (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 xL i * mk yl yr yL yR + mk xl xr xL xR * yL j - xL i * yL j \u2261r\n    moveLeft (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) (Sum.inl (i, j)))\n[PROOFSTEP]\nexact\n  ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n    (mulCommRelabelling _ _)\n[GOAL]\ncase refine'_1.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 moveLeft (mk xl xr xL xR * mk yl yr yL yR) (Sum.inr (i, j)) \u2261r\n    moveLeft (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) (Sum.inr (i, j)))\n[PROOFSTEP]\n{ dsimp\n  exact\n    ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n      (mulCommRelabelling _ _)\n}\n[GOAL]\ncase refine'_1.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 moveLeft (mk xl xr xL xR * mk yl yr yL yR) (Sum.inr (i, j)) \u2261r\n    moveLeft (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) (Sum.inr (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 xR i * mk yl yr yL yR + mk xl xr xL xR * yR j - xR i * yR j \u2261r\n    moveLeft (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xl yl) (Equiv.prodComm xr yr)) (Sum.inr (i, j)))\n[PROOFSTEP]\nexact\n  ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n    (mulCommRelabelling _ _)\n[GOAL]\ncase refine'_2.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 moveRight (mk xl xr xL xR * mk yl yr yL yR) (Sum.inl (i, j)) \u2261r\n    moveRight (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191((Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)).trans (Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr)))\n        (Sum.inl (i, j)))\n[PROOFSTEP]\n{ dsimp\n  exact\n    ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n      (mulCommRelabelling _ _)\n}\n[GOAL]\ncase refine'_2.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 moveRight (mk xl xr xL xR * mk yl yr yL yR) (Sum.inl (i, j)) \u2261r\n    moveRight (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191((Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)).trans (Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr)))\n        (Sum.inl (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 xL i * mk yl yr yL yR + mk xl xr xL xR * yR j - xL i * yR j \u2261r\n    moveRight (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr))\n        (\u2191(Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)) (Sum.inl (i, j))))\n[PROOFSTEP]\nexact\n  ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n    (mulCommRelabelling _ _)\n[GOAL]\ncase refine'_2.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 moveRight (mk xl xr xL xR * mk yl yr yL yR) (Sum.inr (i, j)) \u2261r\n    moveRight (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191((Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)).trans (Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr)))\n        (Sum.inr (i, j)))\n[PROOFSTEP]\n{ dsimp\n  exact\n    ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n      (mulCommRelabelling _ _)\n}\n[GOAL]\ncase refine'_2.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 moveRight (mk xl xr xL xR * mk yl yr yL yR) (Sum.inr (i, j)) \u2261r\n    moveRight (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191((Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)).trans (Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr)))\n        (Sum.inr (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 xR i * mk yl yr yL yR + mk xl xr xL xR * yL j - xR i * yL j \u2261r\n    moveRight (mk yl yr yL yR * mk xl xr xL xR)\n      (\u2191(Equiv.sumCongr (Equiv.prodComm xr yl) (Equiv.prodComm xl yr))\n        (\u2191(Equiv.sumComm (xl \u00d7 yr) (xr \u00d7 yl)) (Sum.inr (i, j))))\n[PROOFSTEP]\nexact\n  ((addCommRelabelling _ _).trans <| (mulCommRelabelling _ _).addCongr (mulCommRelabelling _ _)).subCongr\n    (mulCommRelabelling _ _)\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nx : PGame\n\u22a2 IsEmpty (LeftMoves (x * 0))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\n\u22a2 IsEmpty (LeftMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * 0))\n[PROOFSTEP]\nexact instIsEmptySum\n[GOAL]\nx : PGame\n\u22a2 IsEmpty (RightMoves (x * 0))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\n\u22a2 IsEmpty (RightMoves (mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d * 0))\n[PROOFSTEP]\napply instIsEmptySum\n[GOAL]\nx : PGame\n\u22a2 IsEmpty (LeftMoves (0 * x))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\n\u22a2 IsEmpty (LeftMoves (0 * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d))\n[PROOFSTEP]\napply instIsEmptySum\n[GOAL]\nx : PGame\n\u22a2 IsEmpty (RightMoves (0 * x))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1\u271d \u03b2\u271d : Type u\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\n\u22a2 IsEmpty (RightMoves (0 * mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d))\n[PROOFSTEP]\napply instIsEmptySum\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 -mk xl xr xL xR * mk yl yr yL yR \u2261r -(mk xl xr xL xR * mk yl yr yL yR)\n[PROOFSTEP]\nrefine' \u27e8Equiv.sumComm _ _, Equiv.sumComm _ _, _, _\u27e9\n[GOAL]\ncase refine'_1\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (i : LeftMoves (-mk xl xr xL xR * mk yl yr yL yR)) \u2192\n    moveLeft (-mk xl xr xL xR * mk yl yr yL yR) i \u2261r\n      moveLeft (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yl) (xl \u00d7 yr)) i)\n[PROOFSTEP]\nrintro (\u27e8i, j\u27e9 | \u27e8i, j\u27e9)\n[GOAL]\ncase refine'_2\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (j : RightMoves (-mk xl xr xL xR * mk yl yr yL yR)) \u2192\n    moveRight (-mk xl xr xL xR * mk yl yr yL yR) j \u2261r\n      moveRight (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yr) (xl \u00d7 yl)) j)\n[PROOFSTEP]\nrintro (\u27e8i, j\u27e9 | \u27e8i, j\u27e9)\n[GOAL]\ncase refine'_1.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 moveLeft (-mk xl xr xL xR * mk yl yr yL yR) (Sum.inl (i, j)) \u2261r\n    moveLeft (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yl) (xl \u00d7 yr)) (Sum.inl (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -xR i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j - -xR i * yL j \u2261r\n    moveLeft (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yl) (xl \u00d7 yr)) (Sum.inl (i, j)))\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -((fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xR a))\n            i (mk yl yr yL yR) +\n          (fun a =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a\n                        (fun val =>\n                          Prod.casesOn val fun i j =>\n                            (fun a =>\n                                    rec (motive := fun x => PGame \u2192 PGame)\n                                      (fun xl xr a a IHxl IHxr y =>\n                                        rec\n                                          (fun yl yr yL yR IHyl IHyr =>\n                                            let_fun y := mk yl yr yL yR;\n                                            mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                              (fun a =>\n                                                Sum.casesOn a\n                                                  (fun val =>\n                                                    Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                  fun val =>\n                                                  Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                              fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                          y)\n                                      (xL a))\n                                  i y +\n                                IHyl j -\n                              (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i (yL j))\n                        fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xR a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i (yR j))\n                    fun a =>\n                    Sum.casesOn a\n                      (fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i (yR j))\n                      fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i (yL j))\n                (yL a))\n            j) +\n      - -(fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xR a))\n            i (yL j) \u2261r\n    -xR i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j - -xR i * yL j\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans (Relabelling.addCongr _ _)).subCongr\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xR a))\n        i (yL j) \u2261r\n    -xR i * yL j\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xR a))\n        i (mk yl yr yL yR) \u2261r\n    -xR i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yL a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xR a))\n        i (mk yl yr yL yR) \u2261r\n    -xR i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yL a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yL a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nchange -(mk xl xr xL xR * _) \u2261r _\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 -(mk xl xr xL xR * yL j) \u2261r (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\ncase refine'_1.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 moveLeft (-mk xl xr xL xR * mk yl yr yL yR) (Sum.inr (i, j)) \u2261r\n    moveLeft (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yl) (xl \u00d7 yr)) (Sum.inr (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -xL i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j - -xL i * yR j \u2261r\n    moveLeft (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yl) (xl \u00d7 yr)) (Sum.inr (i, j)))\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -((fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xL a))\n            i (mk yl yr yL yR) +\n          (fun a =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a\n                        (fun val =>\n                          Prod.casesOn val fun i j =>\n                            (fun a =>\n                                    rec (motive := fun x => PGame \u2192 PGame)\n                                      (fun xl xr a a IHxl IHxr y =>\n                                        rec\n                                          (fun yl yr yL yR IHyl IHyr =>\n                                            let_fun y := mk yl yr yL yR;\n                                            mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                              (fun a =>\n                                                Sum.casesOn a\n                                                  (fun val =>\n                                                    Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                  fun val =>\n                                                  Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                              fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                          y)\n                                      (xL a))\n                                  i y +\n                                IHyl j -\n                              (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i (yL j))\n                        fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xR a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i (yR j))\n                    fun a =>\n                    Sum.casesOn a\n                      (fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i (yR j))\n                      fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i (yL j))\n                (yR a))\n            j) +\n      - -(fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xL a))\n            i (yR j) \u2261r\n    -xL i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j - -xL i * yR j\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans (Relabelling.addCongr _ _)).subCongr\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xL a))\n        i (yR j) \u2261r\n    -xL i * yR j\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xL a))\n        i (mk yl yr yL yR) \u2261r\n    -xL i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yR a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xL a))\n        i (mk yl yr yL yR) \u2261r\n    -xL i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yR a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yR a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nchange -(mk xl xr xL xR * _) \u2261r _\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 -(mk xl xr xL xR * yR j) \u2261r (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\ncase refine'_2.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 moveRight (-mk xl xr xL xR * mk yl yr yL yR) (Sum.inl (i, j)) \u2261r\n    moveRight (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yr) (xl \u00d7 yl)) (Sum.inl (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.inl.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -xR i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j - -xR i * yR j \u2261r\n    moveRight (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yr) (xl \u00d7 yl)) (Sum.inl (i, j)))\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -((fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xR a))\n            i (mk yl yr yL yR) +\n          (fun a =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a\n                        (fun val =>\n                          Prod.casesOn val fun i j =>\n                            (fun a =>\n                                    rec (motive := fun x => PGame \u2192 PGame)\n                                      (fun xl xr a a IHxl IHxr y =>\n                                        rec\n                                          (fun yl yr yL yR IHyl IHyr =>\n                                            let_fun y := mk yl yr yL yR;\n                                            mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                              (fun a =>\n                                                Sum.casesOn a\n                                                  (fun val =>\n                                                    Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                  fun val =>\n                                                  Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                              fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                          y)\n                                      (xL a))\n                                  i y +\n                                IHyl j -\n                              (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i (yL j))\n                        fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xR a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i (yR j))\n                    fun a =>\n                    Sum.casesOn a\n                      (fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i (yR j))\n                      fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i (yL j))\n                (yR a))\n            j) +\n      - -(fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xR a))\n            i (yR j) \u2261r\n    -xR i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j - -xR i * yR j\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans (Relabelling.addCongr _ _)).subCongr\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xR a))\n        i (yR j) \u2261r\n    -xR i * yR j\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xR a))\n        i (mk yl yr yL yR) \u2261r\n    -xR i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yR a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xR a))\n        i (mk yl yr yL yR) \u2261r\n    -xR i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yR a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yR a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nchange -(mk xl xr xL xR * _) \u2261r _\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 -(mk xl xr xL xR * yR j) \u2261r (mk xr xl (fun j => -xR j) fun i => -xL i) * yR j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\ncase refine'_2.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 moveRight (-mk xl xr xL xR * mk yl yr yL yR) (Sum.inr (i, j)) \u2261r\n    moveRight (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yr) (xl \u00d7 yl)) (Sum.inr (i, j)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2.inr.mk\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -xL i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j - -xL i * yL j \u2261r\n    moveRight (-(mk xl xr xL xR * mk yl yr yL yR)) (\u2191(Equiv.sumComm (xr \u00d7 yr) (xl \u00d7 yl)) (Sum.inr (i, j)))\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -((fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xL a))\n            i (mk yl yr yL yR) +\n          (fun a =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a\n                        (fun val =>\n                          Prod.casesOn val fun i j =>\n                            (fun a =>\n                                    rec (motive := fun x => PGame \u2192 PGame)\n                                      (fun xl xr a a IHxl IHxr y =>\n                                        rec\n                                          (fun yl yr yL yR IHyl IHyr =>\n                                            let_fun y := mk yl yr yL yR;\n                                            mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                              (fun a =>\n                                                Sum.casesOn a\n                                                  (fun val =>\n                                                    Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                  fun val =>\n                                                  Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                              fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                          y)\n                                      (xL a))\n                                  i y +\n                                IHyl j -\n                              (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i (yL j))\n                        fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xR a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i (yR j))\n                    fun a =>\n                    Sum.casesOn a\n                      (fun val =>\n                        Prod.casesOn val fun i j =>\n                          (fun a =>\n                                  rec (motive := fun x => PGame \u2192 PGame)\n                                    (fun xl xr a a IHxl IHxr y =>\n                                      rec\n                                        (fun yl yr yL yR IHyl IHyr =>\n                                          let_fun y := mk yl yr yL yR;\n                                          mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                            (fun a =>\n                                              Sum.casesOn a\n                                                (fun val =>\n                                                  Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                                fun val =>\n                                                Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                            fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                        y)\n                                    (xL a))\n                                i y +\n                              IHyr j -\n                            (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i (yR j))\n                      fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xR a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i (yL j))\n                (yL a))\n            j) +\n      - -(fun a =>\n              rec (motive := fun x => PGame \u2192 PGame)\n                (fun xl xr a a IHxl IHxr y =>\n                  rec\n                    (fun yl yr yL yR IHyl IHyr =>\n                      let_fun y := mk yl yr yL yR;\n                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                        (fun a =>\n                          Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                        fun a =>\n                        Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                    y)\n                (xL a))\n            i (yL j) \u2261r\n    -xL i * mk yl yr yL yR + (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j - -xL i * yL j\n[PROOFSTEP]\napply ((negAddRelabelling _ _).trans (Relabelling.addCongr _ _)).subCongr\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xL a))\n        i (yL j) \u2261r\n    -xL i * yL j\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xL a))\n        i (mk yl yr yL yR) \u2261r\n    -xL i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yL a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(fun a =>\n          rec (motive := fun x => PGame \u2192 PGame)\n            (fun xl xr a a IHxl IHxr y =>\n              rec\n                (fun yl yr yL yR IHyl IHyr =>\n                  let_fun y := mk yl yr yL yR;\n                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                    (fun a =>\n                      Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                    fun a =>\n                    Sum.casesOn a (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j)) fun val =>\n                      Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                y)\n            (xL a))\n        i (mk yl yr yL yR) \u2261r\n    -xL i * mk yl yr yL yR\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yL a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(fun a =>\n          rec\n            (fun yl yr yL yR IHyl IHyr =>\n              let_fun y := mk yl yr yL yR;\n              mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                (fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun i j =>\n                        (fun a =>\n                                rec (motive := fun x => PGame \u2192 PGame)\n                                  (fun xl xr a a IHxl IHxr y =>\n                                    rec\n                                      (fun yl yr yL yR IHyl IHyr =>\n                                        let_fun y := mk yl yr yL yR;\n                                        mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                          (fun a =>\n                                            Sum.casesOn a\n                                              (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                              fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                          fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                      y)\n                                  (xL a))\n                              i y +\n                            IHyl j -\n                          (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i (yL j))\n                    fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xR a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i (yR j))\n                fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun i j =>\n                      (fun a =>\n                              rec (motive := fun x => PGame \u2192 PGame)\n                                (fun xl xr a a IHxl IHxr y =>\n                                  rec\n                                    (fun yl yr yL yR IHyl IHyr =>\n                                      let_fun y := mk yl yr yL yR;\n                                      mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                        (fun a =>\n                                          Sum.casesOn a\n                                            (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                            fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                        fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                    y)\n                                (xL a))\n                            i y +\n                          IHyr j -\n                        (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xL a))\n                          i (yR j))\n                  fun val =>\n                  Prod.casesOn val fun i j =>\n                    (fun a =>\n                            rec (motive := fun x => PGame \u2192 PGame)\n                              (fun xl xr a a IHxl IHxr y =>\n                                rec\n                                  (fun yl yr yL yR IHyl IHyr =>\n                                    let_fun y := mk yl yr yL yR;\n                                    mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                      (fun a =>\n                                        Sum.casesOn a\n                                          (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                          fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                      fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                  y)\n                              (xR a))\n                          i y +\n                        IHyl j -\n                      (fun a =>\n                          rec (motive := fun x => PGame \u2192 PGame)\n                            (fun xl xr a a IHxl IHxr y =>\n                              rec\n                                (fun yl yr yL yR IHyl IHyr =>\n                                  let_fun y := mk yl yr yL yR;\n                                  mk (xl \u00d7 yl \u2295 xr \u00d7 yr) (xl \u00d7 yr \u2295 xr \u00d7 yl)\n                                    (fun a =>\n                                      Sum.casesOn a\n                                        (fun val => Prod.casesOn val fun i j => IHxl i y + IHyl j - IHxl i (yL j))\n                                        fun val => Prod.casesOn val fun i j => IHxr i y + IHyr j - IHxr i (yR j))\n                                    fun a =>\n                                    Sum.casesOn a\n                                      (fun val => Prod.casesOn val fun i j => IHxl i y + IHyr j - IHxl i (yR j))\n                                      fun val => Prod.casesOn val fun i j => IHxr i y + IHyl j - IHxr i (yL j))\n                                y)\n                            (xR a))\n                        i (yL j))\n            (yL a))\n        j \u2261r\n    (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nchange -(mk xl xr xL xR * _) \u2261r _\n[GOAL]\nx y : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 -(mk xl xr xL xR * yL j) \u2261r (mk xr xl (fun j => -xR j) fun i => -xL i) * yL j\n[PROOFSTEP]\nexact (negMulRelabelling _ _).symm\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := yR j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := mk yl yr yL yR } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => (x, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := yL j } { fst := mk xl xr xL xR, snd := mk yl yr yL yR }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nx y z : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nlet x := mk xl xr xL xR\n[GOAL]\nx\u271d y z : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nlet y := mk yl yr yL yR\n[GOAL]\nx\u271d y\u271d z : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nlet z := mk zl zr zL zR\n[GOAL]\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrefine' quot_eq_of_mk'_quot_eq _ _ _ _\n[GOAL]\ncase refine'_1\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) \u2243\n    LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase refine'_1.toFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) \u2192\n    LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrintro (\u27e8_, _ | _\u27e9 | \u27e8_, _ | _\u27e9)\n[GOAL]\ncase refine'_1.toFun.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : yl\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.toFun.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : zl\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.toFun.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : yr\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.toFun.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : zr\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR) \u2192\n    LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, _\u27e9 | \u27e8_, _\u27e9)\n[GOAL]\ncase refine'_1.invFun.inl.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : yl\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun.inl.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : yr\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun.inr.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : zl\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun.inr.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : zr\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.left_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n        fun val =>\n        Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n          Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val))\n[PROOFSTEP]\nrintro (\u27e8_, _ | _\u27e9 | \u27e8_, _ | _\u27e9)\n[GOAL]\ncase refine'_1.left_inv.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inl (fst\u271d, Sum.inl val\u271d))) =\n    Sum.inl (fst\u271d, Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.left_inv.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inl (fst\u271d, Sum.inr val\u271d))) =\n    Sum.inl (fst\u271d, Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.left_inv.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inr (fst\u271d, Sum.inl val\u271d))) =\n    Sum.inr (fst\u271d, Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.left_inv.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inr (fst\u271d, Sum.inr val\u271d))) =\n    Sum.inr (fst\u271d, Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 Function.RightInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n        fun val =>\n        Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n          Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, _\u27e9 | \u27e8_, _\u27e9)\n[GOAL]\ncase refine'_1.right_inv.inl.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inl (Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inl (Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv.inl.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inl (Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inl (Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv.inr.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inr (Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inr (Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv.inr.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inr (Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inr (Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) \u2243\n    RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase refine'_2.toFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) \u2192\n    RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrintro (\u27e8_, _ | _\u27e9 | \u27e8_, _ | _\u27e9)\n[GOAL]\ncase refine'_2.toFun.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : yr\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.toFun.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : zr\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.toFun.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : yl\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.toFun.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : zl\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR) \u2192\n    RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, _\u27e9 | \u27e8_, _\u27e9)\n[GOAL]\ncase refine'_2.invFun.inl.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : yr\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun.inl.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : yl\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun.inr.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : zr\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun.inr.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : zl\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 6 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.left_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n        fun val =>\n        Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n          Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val))\n[PROOFSTEP]\nrintro (\u27e8_, _ | _\u27e9 | \u27e8_, _ | _\u27e9)\n[GOAL]\ncase refine'_2.left_inv.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inl (fst\u271d, Sum.inl val\u271d))) =\n    Sum.inl (fst\u271d, Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.left_inv.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nval\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inl (fst\u271d, Sum.inr val\u271d))) =\n    Sum.inl (fst\u271d, Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.left_inv.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inr (fst\u271d, Sum.inl val\u271d))) =\n    Sum.inr (fst\u271d, Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.left_inv.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nval\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n          fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n        (Sum.inr (fst\u271d, Sum.inr val\u271d))) =\n    Sum.inr (fst\u271d, Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.right_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 Function.RightInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n            Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n        fun val =>\n        Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n          Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, _\u27e9 | \u27e8_, _\u27e9)\n[GOAL]\ncase refine'_2.right_inv.inl.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inl (Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inl (Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_left_distrib reference below, because otherwise the decreasing_by block\n      -- failed. Previously, each branch ended with: `simp [quot_left_distrib]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_2.right_inv.inl.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inl (Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inl (Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_left_distrib reference below, because otherwise the decreasing_by block\n      -- failed. Previously, each branch ended with: `simp [quot_left_distrib]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_2.right_inv.inr.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xl\nsnd\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inr (Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inr (Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_left_distrib reference below, because otherwise the decreasing_by block\n      -- failed. Previously, each branch ended with: `simp [quot_left_distrib]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_2.right_inv.inr.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d : xr\nsnd\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n            fun val =>\n            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n              Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n        (Sum.inr (Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inr (Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_left_distrib reference below, because otherwise the decreasing_by block\n      -- failed. Previously, each branch ended with: `simp [quot_left_distrib]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_3\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 \u2200 (i : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n    Quotient.mk setoid (moveLeft (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) i) =\n      Quotient.mk setoid\n        (moveLeft (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n          (\u2191{\n                toFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun fst snd =>\n                        Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                          Sum.inr (Sum.inl (fst, val)))\n                    fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n                invFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                        Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                    fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n                left_inv :=\n                  (_ :\n                    \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Prod.casesOn val fun fst snd =>\n                                    Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                      Sum.inr (Sum.inl (fst, val)))\n                                fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                    Sum.inr (Sum.inr (fst, val)))\n                            x) =\n                        x),\n                right_inv :=\n                  (_ :\n                    \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Sum.casesOn val\n                                    (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                                fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                            x) =\n                        x) }\n            i))\n[PROOFSTEP]\nrintro (\u27e8i, j | k\u27e9 | \u27e8i, j | k\u27e9)\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inl (i, Sum.inl j))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inl (i, Sum.inl j))))\n[PROOFSTEP]\nchange \u27e6xL i * (y + z) + x * (yL j + z) - xL i * (yL j + z)\u27e7 = \u27e6xL i * y + x * yL j - xL i * yL j + x * z\u27e7\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (y + z) + x * (yL j + z) - xL i * (yL j + z)) =\n    Quotient.mk setoid (xL i * y + x * yL j - xL i * yL j + x * z)\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (yL j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xL i * (yL j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xL i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (yL j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xL i * (yL j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xL i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yL j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      Quotient.mk setoid (xL i * (yL j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xL i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yL j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xL i * yL j) + Quotient.mk setoid (xL i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xL i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yL j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xL i * yL j) + Quotient.mk setoid (xL i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xL i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inl (i, Sum.inr k))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inl (i, Sum.inr k))))\n[PROOFSTEP]\nchange \u27e6xL i * (y + z) + x * (y + zL k) - xL i * (y + zL k)\u27e7 = \u27e6x * y + (xL i * z + x * zL k - xL i * zL k)\u27e7\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (xL i * (y + z) + x * (y + zL k) - xL i * (y + zL k)) =\n    Quotient.mk setoid (x * y + (xL i * z + x * zL k - xL i * zL k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zL k)) -\n      Quotient.mk setoid (xL i * (mk yl yr yL yR + zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xL i * zL k))\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zL k)) -\n      Quotient.mk setoid (xL i * (mk yl yr yL yR + zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xL i * zL k))\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zL k)) -\n      Quotient.mk setoid (xL i * (mk yl yr yL yR + zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xL i * zL k))\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zL k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xL i * zL k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zL k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xL i * zL k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inr (i, Sum.inl j))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inr (i, Sum.inl j))))\n[PROOFSTEP]\nchange \u27e6xR i * (y + z) + x * (yR j + z) - xR i * (yR j + z)\u27e7 = \u27e6xR i * y + x * yR j - xR i * yR j + x * z\u27e7\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (y + z) + x * (yR j + z) - xR i * (yR j + z)) =\n    Quotient.mk setoid (xR i * y + x * yR j - xR i * yR j + x * z)\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (yR j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xR i * (yR j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xR i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (yR j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xR i * (yR j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xR i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yR j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      Quotient.mk setoid (xR i * (yR j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xR i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yR j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xR i * yR j) + Quotient.mk setoid (xR i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xR i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yR j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xR i * yR j) + Quotient.mk setoid (xR i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xR i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inr (i, Sum.inr k))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inr (i, Sum.inr k))))\n[PROOFSTEP]\nchange \u27e6xR i * (y + z) + x * (y + zR k) - xR i * (y + zR k)\u27e7 = \u27e6x * y + (xR i * z + x * zR k - xR i * zR k)\u27e7\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (xR i * (y + z) + x * (y + zR k) - xR i * (y + zR k)) =\n    Quotient.mk setoid (x * y + (xR i * z + x * zR k - xR i * zR k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zR k)) -\n      Quotient.mk setoid (xR i * (mk yl yr yL yR + zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xR i * zR k))\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zR k)) -\n      Quotient.mk setoid (xR i * (mk yl yr yL yR + zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xR i * zR k))\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zR k)) -\n      Quotient.mk setoid (xR i * (mk yl yr yL yR + zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xR i * zR k))\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zR k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xR i * zR k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zR k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xR i * zR k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 \u2200 (j : RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n    Quotient.mk setoid (moveRight (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) j) =\n      Quotient.mk setoid\n        (moveRight (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n          (\u2191{\n                toFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun fst snd =>\n                        Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                          Sum.inr (Sum.inl (fst, val)))\n                    fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n                invFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                        Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                    fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n                left_inv :=\n                  (_ :\n                    \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Prod.casesOn val fun fst snd =>\n                                    Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                      Sum.inr (Sum.inl (fst, val)))\n                                fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                    Sum.inr (Sum.inr (fst, val)))\n                            x) =\n                        x),\n                right_inv :=\n                  (_ :\n                    \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Sum.casesOn val\n                                    (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                                fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                            x) =\n                        x) }\n            j))\n[PROOFSTEP]\nrintro (\u27e8i, j | k\u27e9 | \u27e8i, j | k\u27e9)\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inl (i, Sum.inl j))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inl (i, Sum.inl j))))\n[PROOFSTEP]\nchange \u27e6xL i * (y + z) + x * (yR j + z) - xL i * (yR j + z)\u27e7 = \u27e6xL i * y + x * yR j - xL i * yR j + x * z\u27e7\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (y + z) + x * (yR j + z) - xL i * (yR j + z)) =\n    Quotient.mk setoid (xL i * y + x * yR j - xL i * yR j + x * z)\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (yR j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xL i * (yR j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xL i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (yR j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xL i * (yR j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xL i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yR j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      Quotient.mk setoid (xL i * (yR j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xL i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yR j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xL i * yR j) + Quotient.mk setoid (xL i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xL i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inl.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yR j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xL i * yR j) + Quotient.mk setoid (xL i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yR j) -\n        Quotient.mk setoid (xL i * yR j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inl (i, Sum.inr k))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inl (i, Sum.inr k))))\n[PROOFSTEP]\nchange \u27e6xL i * (y + z) + x * (y + zR k) - xL i * (y + zR k)\u27e7 = \u27e6x * y + (xL i * z + x * zR k - xL i * zR k)\u27e7\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (xL i * (y + z) + x * (y + zR k) - xL i * (y + zR k)) =\n    Quotient.mk setoid (x * y + (xL i * z + x * zR k - xL i * zR k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zR k)) -\n      Quotient.mk setoid (xL i * (mk yl yr yL yR + zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xL i * zR k))\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zR k)) -\n      Quotient.mk setoid (xL i * (mk yl yr yL yR + zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xL i * zR k))\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zR k)) -\n      Quotient.mk setoid (xL i * (mk yl yr yL yR + zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xL i * zR k))\n[PROOFSTEP]\nrw [quot_left_distrib (xL i) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zR k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xL i * zR k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inl.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xl\nk : zr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zR k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR) + Quotient.mk setoid (xL i * zR k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xL i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zR k) -\n        Quotient.mk setoid (xL i * zR k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inr (i, Sum.inl j))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inr (i, Sum.inl j))))\n[PROOFSTEP]\nchange \u27e6xR i * (y + z) + x * (yL j + z) - xR i * (yL j + z)\u27e7 = \u27e6xR i * y + x * yL j - xR i * yL j + x * z\u27e7\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (y + z) + x * (yL j + z) - xR i * (yL j + z)) =\n    Quotient.mk setoid (xR i * y + x * yL j - xR i * yL j + x * z)\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (yL j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xR i * (yL j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xR i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (yL j + mk zl zr zL zR)) -\n      Quotient.mk setoid (xR i * (yL j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xR i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yL j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      Quotient.mk setoid (xR i * (yL j + mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xR i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yL j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xR i * yL j) + Quotient.mk setoid (xR i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xR i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inl\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * yL j) + Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)) -\n      (Quotient.mk setoid (xR i * yL j) + Quotient.mk setoid (xR i * mk zl zr zL zR)) =\n    Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * yL j) -\n        Quotient.mk setoid (xR i * yL j) +\n      Quotient.mk setoid (mk xl xr xL xR * mk zl zr zL zR)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR)) (Sum.inr (i, Sum.inr k))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val => Sum.inr (Sum.inl (fst, val)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val => Sum.inr (Sum.inr (fst, val)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd)) fun val =>\n                      Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                  fun val =>\n                  Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd)) fun val =>\n                    Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR + mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                            fun val =>\n                            Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                              fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                    Sum.inr (Sum.inl (fst, val)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                  Sum.inr (Sum.inr (fst, val)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR + mk xl xr xL xR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd (fun val => Sum.inl (Sum.inl (fst, val))) fun val =>\n                                  Sum.inr (Sum.inl (fst, val)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd (fun val => Sum.inl (Sum.inr (fst, val))) fun val =>\n                                Sum.inr (Sum.inr (fst, val)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inl snd))\n                                  fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inl snd))\n                              fun val =>\n                              Sum.casesOn val (fun val => Prod.casesOn val fun fst snd => Sum.inl (fst, Sum.inr snd))\n                                fun val => Prod.casesOn val fun fst snd => Sum.inr (fst, Sum.inr snd))\n                          x) =\n                      x) }\n          (Sum.inr (i, Sum.inr k))))\n[PROOFSTEP]\nchange \u27e6xR i * (y + z) + x * (y + zL k) - xR i * (y + zL k)\u27e7 = \u27e6x * y + (xR i * z + x * zL k - xR i * zL k)\u27e7\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (xR i * (y + z) + x * (y + zL k) - xR i * (y + zL k)) =\n    Quotient.mk setoid (x * y + (xR i * z + x * zL k - xR i * zL k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add]\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR + mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zL k)) -\n      Quotient.mk setoid (xR i * (mk yl yr yL yR + zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xR i * zL k))\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR + zL k)) -\n      Quotient.mk setoid (xR i * (mk yl yr yL yR + zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xR i * zL k))\n[PROOFSTEP]\nrw [quot_left_distrib (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zL k)) -\n      Quotient.mk setoid (xR i * (mk yl yr yL yR + zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xR i * zL k))\n[PROOFSTEP]\nrw [quot_left_distrib (xR i) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zL k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xR i * zL k))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inr\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\ni : xr\nk : zl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * mk zl zr zL zR) +\n        (Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) + Quotient.mk setoid (mk xl xr xL xR * zL k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR) + Quotient.mk setoid (xR i * zL k)) =\n    Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR) +\n      (Quotient.mk setoid (xR i * mk zl zr zL zR) + Quotient.mk setoid (mk xl xr xL xR * zL k) -\n        Quotient.mk setoid (xR i * zL k))\n[PROOFSTEP]\nabel\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xl\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * (y.2.1 + y.2.2)) = Quotient.mk setoid (y.1 * y.2.1) + Quotient.mk setoid (y.1 * y.2.2)\ni : xr\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nx y z : PGame\n\u22a2 Quotient.mk setoid (x * (y - z)) = Quotient.mk setoid (x * y) - Quotient.mk setoid (x * z)\n[PROOFSTEP]\nchange (\u27e6x * (y + -z)\u27e7 : Game) = \u27e6x * y\u27e7 + -\u27e6x * z\u27e7\n[GOAL]\nx y z : PGame\n\u22a2 Quotient.mk setoid (x * (y + -z)) = Quotient.mk setoid (x * y) + -Quotient.mk setoid (x * z)\n[PROOFSTEP]\nrw [quot_left_distrib, quot_mul_neg]\n[GOAL]\nx y z : PGame\n\u22a2 Quotient.mk setoid ((x + y) * z) = Quotient.mk setoid (x * z) + Quotient.mk setoid (y * z)\n[PROOFSTEP]\nsimp only [quot_mul_comm, quot_left_distrib]\n[GOAL]\nx y z : PGame\n\u22a2 Quotient.mk setoid ((y - z) * x) = Quotient.mk setoid (y * x) - Quotient.mk setoid (z * x)\n[PROOFSTEP]\nchange (\u27e6(y + -z) * x\u27e7 : Game) = \u27e6y * x\u27e7 + -\u27e6z * x\u27e7\n[GOAL]\nx y z : PGame\n\u22a2 Quotient.mk setoid ((y + -z) * x) = Quotient.mk setoid (y * x) + -Quotient.mk setoid (z * x)\n[PROOFSTEP]\nrw [quot_right_distrib, quot_neg_mul]\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 mk xl xr xL xR * 1 \u2261r mk xl xr xL xR\n[PROOFSTEP]\nshow _ * One.one \u2261r _\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 mk xl xr xL xR * One.one \u2261r mk xl xr xL xR\n[PROOFSTEP]\nunfold One.one\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 mk xl xr xL xR * instOnePGame.1 \u2261r mk xl xr xL xR\n[PROOFSTEP]\nunfold instOnePGame\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 mk xl xr xL xR * { one := mk PUnit PEmpty (fun x => 0) PEmpty.elim }.1 \u2261r mk xl xr xL xR\n[PROOFSTEP]\nchange\n  mk _ _ _ _ * mk _ _ _ _ \u2261r\n    _\n      -- Porting note: changed `refine'` to `refine`,\n          -- otherwise there are typeclass inference failures.\n[GOAL]\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim \u2261r mk xl xr xL xR\n[PROOFSTEP]\nrefine \u27e8(Equiv.sumEmpty _ _).trans (Equiv.prodPUnit _), (Equiv.emptySum _ _).trans (Equiv.prodPUnit _), ?_, ?_\u27e9\n[GOAL]\ncase refine_1\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 (i : LeftMoves (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim)) \u2192\n    moveLeft (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) i \u2261r\n      moveLeft (mk xl xr xL xR) (\u2191((Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)).trans (Equiv.prodPUnit xl)) i)\n[PROOFSTEP]\ntry rintro (\u27e8i, \u27e8\u27e9\u27e9 | \u27e8i, \u27e8\u27e9\u27e9)\n[GOAL]\ncase refine_1\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 (i : LeftMoves (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim)) \u2192\n    moveLeft (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) i \u2261r\n      moveLeft (mk xl xr xL xR) (\u2191((Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)).trans (Equiv.prodPUnit xl)) i)\n[PROOFSTEP]\nrintro (\u27e8i, \u27e8\u27e9\u27e9 | \u27e8i, \u27e8\u27e9\u27e9)\n[GOAL]\ncase refine_2\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 (j : RightMoves (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim)) \u2192\n    moveRight (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) j \u2261r\n      moveRight (mk xl xr xL xR) (\u2191((Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)).trans (Equiv.prodPUnit xr)) j)\n[PROOFSTEP]\ntry rintro (\u27e8i, \u27e8\u27e9\u27e9 | \u27e8i, \u27e8\u27e9\u27e9)\n[GOAL]\ncase refine_2\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 (j : RightMoves (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim)) \u2192\n    moveRight (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) j \u2261r\n      moveRight (mk xl xr xL xR) (\u2191((Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)).trans (Equiv.prodPUnit xr)) j)\n[PROOFSTEP]\nrintro (\u27e8i, \u27e8\u27e9\u27e9 | \u27e8i, \u27e8\u27e9\u27e9)\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 moveLeft (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inl (i, PUnit.unit)) \u2261r\n    moveLeft (mk xl xr xL xR)\n      (\u2191((Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)).trans (Equiv.prodPUnit xl)) (Sum.inl (i, PUnit.unit)))\n[PROOFSTEP]\n{ (try intro i)\n  dsimp\n  apply (Relabelling.subCongr (Relabelling.refl _) (mulZeroRelabelling _)).trans\n  rw [sub_zero]\n  exact\n    (addZeroRelabelling _).trans <|\n      (((mulOneRelabelling _).addCongr (mulZeroRelabelling _)).trans <| addZeroRelabelling _)\n}\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 moveLeft (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inl (i, PUnit.unit)) \u2261r\n    moveLeft (mk xl xr xL xR)\n      (\u2191((Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)).trans (Equiv.prodPUnit xl)) (Sum.inl (i, PUnit.unit)))\n[PROOFSTEP]\ntry intro i\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 moveLeft (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inl (i, PUnit.unit)) \u2261r\n    moveLeft (mk xl xr xL xR)\n      (\u2191((Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)).trans (Equiv.prodPUnit xl)) (Sum.inl (i, PUnit.unit)))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 moveLeft (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inl (i, PUnit.unit)) \u2261r\n    moveLeft (mk xl xr xL xR)\n      (\u2191((Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)).trans (Equiv.prodPUnit xl)) (Sum.inl (i, PUnit.unit)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 xL i * mk PUnit PEmpty (fun x => 0) PEmpty.elim + mk xl xr xL xR * 0 - xL i * 0 \u2261r\n    xL (\u2191(Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)) (Sum.inl (i, PUnit.unit))).fst\n[PROOFSTEP]\napply (Relabelling.subCongr (Relabelling.refl _) (mulZeroRelabelling _)).trans\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 xL i * mk PUnit PEmpty (fun x => 0) PEmpty.elim + mk xl xr xL xR * 0 - 0 \u2261r\n    xL (\u2191(Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)) (Sum.inl (i, PUnit.unit))).fst\n[PROOFSTEP]\nrw [sub_zero]\n[GOAL]\ncase refine_1.inl.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xl\n\u22a2 xL i * mk PUnit PEmpty (fun x => 0) PEmpty.elim + mk xl xr xL xR * 0 + 0 \u2261r\n    xL (\u2191(Equiv.sumEmpty (xl \u00d7 PUnit) (xr \u00d7 PEmpty)) (Sum.inl (i, PUnit.unit))).fst\n[PROOFSTEP]\nexact\n  (addZeroRelabelling _).trans <|\n    (((mulOneRelabelling _).addCongr (mulZeroRelabelling _)).trans <| addZeroRelabelling _)\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 moveRight (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inr (i, PUnit.unit)) \u2261r\n    moveRight (mk xl xr xL xR)\n      (\u2191((Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)).trans (Equiv.prodPUnit xr)) (Sum.inr (i, PUnit.unit)))\n[PROOFSTEP]\n{ (try intro i)\n  dsimp\n  apply (Relabelling.subCongr (Relabelling.refl _) (mulZeroRelabelling _)).trans\n  rw [sub_zero]\n  exact\n    (addZeroRelabelling _).trans <|\n      (((mulOneRelabelling _).addCongr (mulZeroRelabelling _)).trans <| addZeroRelabelling _)\n}\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 moveRight (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inr (i, PUnit.unit)) \u2261r\n    moveRight (mk xl xr xL xR)\n      (\u2191((Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)).trans (Equiv.prodPUnit xr)) (Sum.inr (i, PUnit.unit)))\n[PROOFSTEP]\ntry intro i\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 moveRight (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inr (i, PUnit.unit)) \u2261r\n    moveRight (mk xl xr xL xR)\n      (\u2191((Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)).trans (Equiv.prodPUnit xr)) (Sum.inr (i, PUnit.unit)))\n[PROOFSTEP]\nintro i\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 moveRight (mk xl xr xL xR * mk PUnit PEmpty (fun x => 0) PEmpty.elim) (Sum.inr (i, PUnit.unit)) \u2261r\n    moveRight (mk xl xr xL xR)\n      (\u2191((Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)).trans (Equiv.prodPUnit xr)) (Sum.inr (i, PUnit.unit)))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 xR i * mk PUnit PEmpty (fun x => 0) PEmpty.elim + mk xl xr xL xR * 0 - xR i * 0 \u2261r\n    xR (\u2191(Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)) (Sum.inr (i, PUnit.unit))).fst\n[PROOFSTEP]\napply (Relabelling.subCongr (Relabelling.refl _) (mulZeroRelabelling _)).trans\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 xR i * mk PUnit PEmpty (fun x => 0) PEmpty.elim + mk xl xr xL xR * 0 - 0 \u2261r\n    xR (\u2191(Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)) (Sum.inr (i, PUnit.unit))).fst\n[PROOFSTEP]\nrw [sub_zero]\n[GOAL]\ncase refine_2.inr.mk.unit\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ni : xr\n\u22a2 xR i * mk PUnit PEmpty (fun x => 0) PEmpty.elim + mk xl xr xL xR * 0 + 0 \u2261r\n    xR (\u2191(Equiv.emptySum (xl \u00d7 PEmpty) (xr \u00d7 PUnit)) (Sum.inr (i, PUnit.unit))).fst\n[PROOFSTEP]\nexact\n  (addZeroRelabelling _).trans <|\n    (((mulOneRelabelling _).addCongr (mulZeroRelabelling _)).trans <| addZeroRelabelling _)\n[GOAL]\nx y z : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) =\n    Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nlet x := mk xl xr xL xR\n[GOAL]\nx\u271d y z : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) =\n    Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nlet y := mk yl yr yL yR\n[GOAL]\nx\u271d y\u271d z : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) =\n    Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nlet z := mk zl zr zL zR\n[GOAL]\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) =\n    Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nrefine' quot_eq_of_mk'_quot_eq _ _ _ _\n[GOAL]\ncase refine'_1\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) \u2243\n    LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase refine'_1.toFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) \u2192\n    LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9)\n[GOAL]\ncase refine'_1.toFun.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xl\nsnd\u271d : yl\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.toFun.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xr\nsnd\u271d : yr\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.toFun.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xl\nsnd\u271d : yr\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.toFun.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xr\nsnd\u271d : yl\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)) \u2192\n    LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nrintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9)\n[GOAL]\ncase refine'_1.invFun.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yl\nsnd\u271d : zl\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yr\nsnd\u271d : zr\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yl\nsnd\u271d : zr\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.invFun.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yr\nsnd\u271d : zl\n\u22a2 LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_1.left_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n        fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd)) fun val =>\n            Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>\n            Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd))) fun val =>\n          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9)\n[GOAL]\ncase refine'_1.left_inv.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xl\nsnd\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inl (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inl (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.left_inv.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xr\nsnd\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inl (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inl (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.left_inv.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xl\nsnd\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inr (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inr (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.left_inv.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xr\nsnd\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inr (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inr (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 Function.RightInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n        fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd)) fun val =>\n            Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>\n            Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd))) fun val =>\n          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd))\n[PROOFSTEP]\nrintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9)\n[GOAL]\ncase refine'_1.right_inv.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yl\nsnd\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inl (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inl (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yr\nsnd\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inl (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inl (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yl\nsnd\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inr (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inr (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1.right_inv.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yr\nsnd\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inr (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inr (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) \u2243\n    RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase refine'_2.toFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) \u2192\n    RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9)\n[GOAL]\ncase refine'_2.toFun.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xl\nsnd\u271d : yl\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.toFun.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xr\nsnd\u271d : yr\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.toFun.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xl\nsnd\u271d : yr\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.toFun.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xr\nsnd\u271d : yl\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)) \u2192\n    RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nrintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9)\n[GOAL]\ncase refine'_2.invFun.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yl\nsnd\u271d : zr\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yr\nsnd\u271d : zl\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yl\nsnd\u271d : zl\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.invFun.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yr\nsnd\u271d : zr\n\u22a2 RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)\n[PROOFSTEP]\nsolve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]\n[GOAL]\ncase refine'_2.left_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 LeftInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n        fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd)) fun val =>\n            Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>\n            Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd))) fun val =>\n          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd))\n[PROOFSTEP]\nrintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9)\n[GOAL]\ncase refine'_2.left_inv.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xl\nsnd\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inl (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inl (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.left_inv.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zr\nfst\u271d : xr\nsnd\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inl (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inl (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.left_inv.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xl\nsnd\u271d : yr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inr (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inr (Sum.inl (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.left_inv.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nsnd\u271d\u00b9 : zl\nfst\u271d : xr\nsnd\u271d : yl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n        (Sum.inr (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9))) =\n    Sum.inr (Sum.inr (fst\u271d, snd\u271d), snd\u271d\u00b9)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.right_inv\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 Function.RightInverse\n    (fun a =>\n      Sum.casesOn a\n        (fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n              fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n        fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd)) fun val =>\n            Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n    fun a =>\n    Sum.casesOn a\n      (fun val =>\n        Prod.casesOn val fun fst snd =>\n          Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>\n            Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n      fun val =>\n      Prod.casesOn val fun fst snd =>\n        Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd))) fun val =>\n          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd))\n[PROOFSTEP]\nrintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9)\n[GOAL]\ncase refine'_2.right_inv.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yl\nsnd\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inl (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inl (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_mul_assoc reference below, because otherwise the decreasing_by block\n      -- failed. Each branch previously ended with: `simp [quot_mul_assoc]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_2.right_inv.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xl\nfst\u271d : yr\nsnd\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inl (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inl (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_mul_assoc reference below, because otherwise the decreasing_by block\n      -- failed. Each branch previously ended with: `simp [quot_mul_assoc]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_2.right_inv.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yl\nsnd\u271d : zl\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inr (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d)))) =\n    Sum.inr (fst\u271d\u00b9, Sum.inl (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_mul_assoc reference below, because otherwise the decreasing_by block\n      -- failed. Each branch previously ended with: `simp [quot_mul_assoc]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_2.right_inv.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nfst\u271d\u00b9 : xr\nfst\u271d : yr\nsnd\u271d : zr\n\u22a2 (fun a =>\n        Sum.casesOn a\n          (fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n          fun val =>\n          Prod.casesOn val fun fst snd =>\n            Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n              fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n      ((fun a =>\n          Sum.casesOn a\n            (fun val =>\n              Prod.casesOn val fun fst snd =>\n                Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n            fun val =>\n            Prod.casesOn val fun fst snd =>\n              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n        (Sum.inr (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d)))) =\n    Sum.inr (fst\u271d\u00b9, Sum.inr (fst\u271d, snd\u271d))\n[PROOFSTEP]\nrfl\n  -- Porting note: explicitly wrote out arguments to each recursive\n      -- quot_mul_assoc reference below, because otherwise the decreasing_by block\n      -- failed. Each branch previously ended with: `simp [quot_mul_assoc]; abel`\n      -- See https://github.com/leanprover/lean4/issues/2288\n[GOAL]\ncase refine'_3\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 \u2200 (i : LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n    Quotient.mk setoid (moveLeft (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) i) =\n      Quotient.mk setoid\n        (moveLeft (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n          (\u2191{\n                toFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun fst snd =>\n                        Sum.casesOn fst\n                          (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>\n                          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                    fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n                invFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun fst snd =>\n                        Sum.casesOn snd\n                          (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd)) fun val =>\n                          Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                    fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n                left_inv :=\n                  (_ :\n                    \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Prod.casesOn val fun fst snd =>\n                                    Sum.casesOn fst\n                                      (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                                fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                            x) =\n                        x),\n                right_inv :=\n                  (_ :\n                    \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Prod.casesOn val fun fst snd =>\n                                    Sum.casesOn snd\n                                      (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                                fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                            x) =\n                        x) }\n            i))\n[PROOFSTEP]\nrintro (\u27e8\u27e8i, j\u27e9 | \u27e8i, j\u27e9, k\u27e9 | \u27e8\u27e8i, j\u27e9 | \u27e8i, j\u27e9, k\u27e9)\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inl (Sum.inl (i, j), k))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inl (Sum.inl (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xL i * y + x * yL j - xL i * yL j) * z + x * y * zL k - (xL i * y + x * yL j - xL i * yL j) * zL k\u27e7 =\n    \u27e6xL i * (y * z) + x * (yL j * z + y * zL k - yL j * zL k) - xL i * (yL j * z + y * zL k - yL j * zL k)\u27e7\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid\n      ((xL i * y + x * yL j - xL i * yL j) * z + x * y * zL k - (xL i * y + x * yL j - xL i * yL j) * zL k) =\n    Quotient.mk setoid\n      (xL i * (y * z) + x * (yL j * z + y * zL k - yL j * zL k) - xL i * (yL j * z + y * zL k - yL j * zL k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k)) -\n        Quotient.mk setoid (xL i * yL j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yL j) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yL j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inl (Sum.inr (i, j), k))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inl (Sum.inr (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xR i * y + x * yR j - xR i * yR j) * z + x * y * zL k - (xR i * y + x * yR j - xR i * yR j) * zL k\u27e7 =\n    \u27e6xR i * (y * z) + x * (yR j * z + y * zL k - yR j * zL k) - xR i * (yR j * z + y * zL k - yR j * zL k)\u27e7\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid\n      ((xR i * y + x * yR j - xR i * yR j) * z + x * y * zL k - (xR i * y + x * yR j - xR i * yR j) * zL k) =\n    Quotient.mk setoid\n      (xR i * (y * z) + x * (yR j * z + y * zL k - yR j * zL k) - xR i * (yR j * z + y * zL k - yR j * zL k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k)) -\n        Quotient.mk setoid (xR i * yR j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yR j) (zL k)]\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yR j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inr (Sum.inl (i, j), k))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inr (Sum.inl (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xL i * y + x * yR j - xL i * yR j) * z + x * y * zR k - (xL i * y + x * yR j - xL i * yR j) * zR k\u27e7 =\n    \u27e6xL i * (y * z) + x * (yR j * z + y * zR k - yR j * zR k) - xL i * (yR j * z + y * zR k - yR j * zR k)\u27e7\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid\n      ((xL i * y + x * yR j - xL i * yR j) * z + x * y * zR k - (xL i * y + x * yR j - xL i * yR j) * zR k) =\n    Quotient.mk setoid\n      (xL i * (y * z) + x * (yR j * z + y * zR k - yR j * zR k) - xL i * (yR j * z + y * zR k - yR j * zR k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k)) -\n        Quotient.mk setoid (xL i * yR j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yR j) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yR j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (moveLeft (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inr (Sum.inr (i, j), k))) =\n    Quotient.mk setoid\n      (moveLeft (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : LeftMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inr (Sum.inr (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xR i * y + x * yL j - xR i * yL j) * z + x * y * zR k - (xR i * y + x * yL j - xR i * yL j) * zR k\u27e7 =\n    \u27e6xR i * (y * z) + x * (yL j * z + y * zR k - yL j * zR k) - xR i * (yL j * z + y * zR k - yL j * zR k)\u27e7\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid\n      ((xR i * y + x * yL j - xR i * yL j) * z + x * y * zR k - (xR i * y + x * yL j - xR i * yL j) * zR k) =\n    Quotient.mk setoid\n      (xR i * (y * z) + x * (yL j * z + y * zR k - yL j * zR k) - xR i * (yL j * z + y * zR k - yL j * zR k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k)) -\n        Quotient.mk setoid (xR i * yL j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yL j) (zR k)]\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_3.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yL j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\n\u22a2 \u2200 (j : RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n    Quotient.mk setoid (moveRight (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) j) =\n      Quotient.mk setoid\n        (moveRight (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n          (\u2191{\n                toFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun fst snd =>\n                        Sum.casesOn fst\n                          (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>\n                          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                    fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n                invFun := fun a =>\n                  Sum.casesOn a\n                    (fun val =>\n                      Prod.casesOn val fun fst snd =>\n                        Sum.casesOn snd\n                          (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd)) fun val =>\n                          Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                    fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n                left_inv :=\n                  (_ :\n                    \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Prod.casesOn val fun fst snd =>\n                                    Sum.casesOn fst\n                                      (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                                fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                            x) =\n                        x),\n                right_inv :=\n                  (_ :\n                    \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                      (fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          ((fun a =>\n                              Sum.casesOn a\n                                (fun val =>\n                                  Prod.casesOn val fun fst snd =>\n                                    Sum.casesOn snd\n                                      (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                                fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                            x) =\n                        x) }\n            j))\n[PROOFSTEP]\nrintro (\u27e8\u27e8i, j\u27e9 | \u27e8i, j\u27e9, k\u27e9 | \u27e8\u27e8i, j\u27e9 | \u27e8i, j\u27e9, k\u27e9)\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inl (Sum.inl (i, j), k))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inl (Sum.inl (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xL i * y + x * yL j - xL i * yL j) * z + x * y * zR k - (xL i * y + x * yL j - xL i * yL j) * zR k\u27e7 =\n    \u27e6xL i * (y * z) + x * (yL j * z + y * zR k - yL j * zR k) - xL i * (yL j * z + y * zR k - yL j * zR k)\u27e7\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid\n      ((xL i * y + x * yL j - xL i * yL j) * z + x * y * zR k - (xL i * y + x * yL j - xL i * yL j) * zR k) =\n    Quotient.mk setoid\n      (xL i * (y * z) + x * (yL j * z + y * zR k - yL j * zR k) - xL i * (yL j * z + y * zR k - yL j * zR k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * yL j * zR k) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k)) -\n        Quotient.mk setoid (xL i * yL j * zR k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yL j) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inl.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xl\nj : yl\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zR k))) -\n      (Quotient.mk setoid (xL i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xL i * (yL j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inl (Sum.inr (i, j), k))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inl (Sum.inr (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xR i * y + x * yR j - xR i * yR j) * z + x * y * zR k - (xR i * y + x * yR j - xR i * yR j) * zR k\u27e7 =\n    \u27e6xR i * (y * z) + x * (yR j * z + y * zR k - yR j * zR k) - xR i * (yR j * z + y * zR k - yR j * zR k)\u27e7\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid\n      ((xR i * y + x * yR j - xR i * yR j) * z + x * y * zR k - (xR i * y + x * yR j - xR i * yR j) * zR k) =\n    Quotient.mk setoid\n      (xR i * (y * z) + x * (yR j * z + y * zR k - yR j * zR k) - xR i * (yR j * z + y * zR k - yR j * zR k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zR k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zR k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * yR j * zR k) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k)) -\n        Quotient.mk setoid (xR i * yR j * zR k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yR j) (zR k)]\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inl.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zr\ni : xr\nj : yr\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zR k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zR k))) -\n      (Quotient.mk setoid (xR i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zR k)) -\n        Quotient.mk setoid (xR i * (yR j * zR k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inr (Sum.inl (i, j), k))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inr (Sum.inl (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xL i * y + x * yR j - xL i * yR j) * z + x * y * zL k - (xL i * y + x * yR j - xL i * yR j) * zL k\u27e7 =\n    \u27e6xL i * (y * z) + x * (yR j * z + y * zL k - yR j * zL k) - xL i * (yR j * z + y * zL k - yR j * zL k)\u27e7\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid\n      ((xL i * y + x * yR j - xL i * yR j) * z + x * y * zL k - (xL i * y + x * yR j - xL i * yR j) * zL k) =\n    Quotient.mk setoid\n      (xL i * (y * z) + x * (yR j * z + y * zL k - yR j * zL k) - xL i * (yR j * z + y * zL k - yR j * zL k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yR j * mk zl zr zL zR) -\n          Quotient.mk setoid (xL i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * yR j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yR j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * yR j * zL k) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yR j) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k)) -\n        Quotient.mk setoid (xL i * yR j * zL k)) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xL i) (yR j) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inl.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xl\nj : yr\n\u22a2 Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k))) =\n    Quotient.mk setoid (xL i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yR j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yR j * zL k))) -\n      (Quotient.mk setoid (xL i * (yR j * mk zl zr zL zR)) + Quotient.mk setoid (xL i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xL i * (yR j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (moveRight (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR) (Sum.inr (Sum.inr (i, j), k))) =\n    Quotient.mk setoid\n      (moveRight (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))\n        (\u2191{\n              toFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                        fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                      fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)),\n              invFun := fun a =>\n                Sum.casesOn a\n                  (fun val =>\n                    Prod.casesOn val fun fst snd =>\n                      Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                        fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                  fun val =>\n                  Prod.casesOn val fun fst snd =>\n                    Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                      fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd),\n              left_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn snd\n                                (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn fst\n                                    (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                    fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                          x) =\n                      x),\n              right_inv :=\n                (_ :\n                  \u2200 (x : RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))),\n                    (fun a =>\n                          Sum.casesOn a\n                            (fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn fst\n                                  (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))\n                                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))\n                            fun val =>\n                            Prod.casesOn val fun fst snd =>\n                              Sum.casesOn fst\n                                (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))\n                                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))\n                        ((fun a =>\n                            Sum.casesOn a\n                              (fun val =>\n                                Prod.casesOn val fun fst snd =>\n                                  Sum.casesOn snd\n                                    (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))\n                                    fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))\n                              fun val =>\n                              Prod.casesOn val fun fst snd =>\n                                Sum.casesOn snd\n                                  (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))\n                                  fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))\n                          x) =\n                      x) }\n          (Sum.inr (Sum.inr (i, j), k))))\n[PROOFSTEP]\nchange\n  \u27e6(xR i * y + x * yL j - xR i * yL j) * z + x * y * zL k - (xR i * y + x * yL j - xR i * yL j) * zL k\u27e7 =\n    \u27e6xR i * (y * z) + x * (yL j * z + y * zL k - yL j * zL k) - xR i * (yL j * z + y * zL k - yL j * zL k)\u27e7\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid\n      ((xR i * y + x * yL j - xR i * yL j) * z + x * y * zL k - (xR i * y + x * yL j - xR i * yL j) * zL k) =\n    Quotient.mk setoid\n      (xR i * (y * z) + x * (yL j * z + y * zL k - yL j * zL k) - xR i * (yL j * z + y * zL k - yL j * zL k))\n[PROOFSTEP]\nsimp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib, quot_left_distrib_sub, quot_left_distrib]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * mk yl yr yL yR * mk zl zr zL zR) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * yL j * mk zl zr zL zR) -\n          Quotient.mk setoid (xR i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * yL j * mk zl zr zL zR) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yL j) (mk zl zr zL zR)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * mk yl yr yL yR * zL k) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * mk yl yr yL yR * zL k) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (mk yl yr yL yR) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * yL j * zL k) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (mk xl xr xL xR) (yL j) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k)) -\n        Quotient.mk setoid (xR i * yL j * zL k)) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nrw [quot_mul_assoc (xR i) (yL j) (zL k)]\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_4.inr.mk.inr.mk\nx\u271d y\u271d z\u271d : PGame\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx : PGame := mk xl xr xL xR\ny : PGame := mk yl yr yL yR\nz : PGame := mk zl zr zL zR\nk : zl\ni : xr\nj : yl\n\u22a2 Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) -\n          Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) +\n        Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n      (Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) + Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k))) =\n    Quotient.mk setoid (xR i * (mk yl yr yL yR * mk zl zr zL zR)) +\n        (Quotient.mk setoid (mk xl xr xL xR * (yL j * mk zl zr zL zR)) +\n            Quotient.mk setoid (mk xl xr xL xR * (mk yl yr yL yR * zL k)) -\n          Quotient.mk setoid (mk xl xr xL xR * (yL j * zL k))) -\n      (Quotient.mk setoid (xR i * (yL j * mk zl zr zL zR)) + Quotient.mk setoid (xR i * (mk yl yr yL yR * zL k)) -\n        Quotient.mk setoid (xR i * (yL j * zL k)))\n[PROOFSTEP]\nabel\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yL j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yR j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yR j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yL j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yL j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zr\ni : xr\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yR j, snd := zR k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yR j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yR j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xl\nj : yr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xL i, snd := { fst := yR j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yL j, snd := mk zl zr zL zR } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xr\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := mk yl yr yL yR, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := mk xl xr xL xR, snd := { fst := yL j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u_1\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nzl zr : Type u_1\nzL : zl \u2192 PGame\nzR : zr \u2192 PGame\nx\u271d :\n  \u2200 (y : (_ : PGame) \u00d7' (_ : PGame) \u00d7' PGame),\n    (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n            Prod.instWellFoundedRelationProd).1\n        y { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } } \u2192\n      Quotient.mk setoid (y.1 * y.2.1 * y.2.2) = Quotient.mk setoid (y.1 * (y.2.1 * y.2.2))\nk : zl\ni : xr\nj : yl\n\u22a2 (invImage (fun a => PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => (x, y, snd))\n        Prod.instWellFoundedRelationProd).1\n    { fst := xR i, snd := { fst := yL j, snd := zL k } }\n    { fst := mk xl xr xL xR, snd := { fst := mk yl yr yL yR, snd := mk zl zr zL zR } }\n[PROOFSTEP]\npgame_wf_tac\n[GOAL]\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\n\u22a2 InvTy l r true \u2192 False\n[PROOFSTEP]\nrintro (_ | _ | _ | a | a)\n[GOAL]\ncase right\u2081\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : l\na\u271d : InvTy l r false\n\u22a2 False\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\ncase right\u2082\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : r\na\u271d : InvTy l r true\n\u22a2 False\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\n\u22a2 \u2200 (a : InvTy l r false), a = default\n[PROOFSTEP]\nrintro (a | a | a)\n[GOAL]\ncase zero\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\n\u22a2 InvTy.zero = default\ncase left\u2081\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\na : r\na\u271d : InvTy l r false\n\u22a2 InvTy.left\u2081 a a\u271d = default\ncase left\u2082\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\na : l\na\u271d : InvTy l r true\n\u22a2 InvTy.left\u2082 a a\u271d = default\n[PROOFSTEP]\nrfl\n[GOAL]\ncase left\u2081\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\na : r\na\u271d : InvTy l r false\n\u22a2 InvTy.left\u2081 a a\u271d = default\ncase left\u2082\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\na : l\na\u271d : InvTy l r true\n\u22a2 InvTy.left\u2082 a a\u271d = default\n[PROOFSTEP]\nall_goals exact isEmptyElim a\n[GOAL]\ncase left\u2081\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\na : r\na\u271d : InvTy l r false\n\u22a2 InvTy.left\u2081 a a\u271d = default\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\ncase left\u2082\nl r : Type u\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\nsrc\u271d : Inhabited (InvTy l r false) := InvTy.instInhabited l r\na : l\na\u271d : InvTy l r true\n\u22a2 InvTy.left\u2082 a a\u271d = default\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\nl r : Type u\nb : Bool\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ni : InvTy l r b\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\n\u22a2 invVal L R IHl IHr i = 0\n[PROOFSTEP]\ncases' i with a _ a _ a _ a\n[GOAL]\ncase zero\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\n\u22a2 invVal L R IHl IHr InvTy.zero = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase left\u2081\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : r\na\u271d : InvTy l r false\n\u22a2 invVal L R IHl IHr (InvTy.left\u2081 a a\u271d) = 0\ncase left\u2082\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : l\na\u271d : InvTy l r true\n\u22a2 invVal L R IHl IHr (InvTy.left\u2082 a a\u271d) = 0\ncase right\u2081\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : l\na\u271d : InvTy l r false\n\u22a2 invVal L R IHl IHr (InvTy.right\u2081 a a\u271d) = 0\ncase right\u2082\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : r\na\u271d : InvTy l r true\n\u22a2 invVal L R IHl IHr (InvTy.right\u2082 a a\u271d) = 0\n[PROOFSTEP]\nall_goals exact isEmptyElim a\n[GOAL]\ncase left\u2081\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : r\na\u271d : InvTy l r false\n\u22a2 invVal L R IHl IHr (InvTy.left\u2081 a a\u271d) = 0\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\ncase left\u2082\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : l\na\u271d : InvTy l r true\n\u22a2 invVal L R IHl IHr (InvTy.left\u2082 a a\u271d) = 0\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\ncase right\u2081\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : l\na\u271d : InvTy l r false\n\u22a2 invVal L R IHl IHr (InvTy.right\u2081 a a\u271d) = 0\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\ncase right\u2082\nl r : Type u\nL : l \u2192 PGame\nR : r \u2192 PGame\nIHl : l \u2192 PGame\nIHr : r \u2192 PGame\ninst\u271d\u00b9 : IsEmpty l\ninst\u271d : IsEmpty r\na : r\na\u271d : InvTy l r true\n\u22a2 invVal L R IHl IHr (InvTy.right\u2082 a a\u271d) = 0\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 0 \u29cf inv' (mk xl xr xL xR)\n[PROOFSTEP]\nconvert lf_mk _ _ InvTy.zero\n[GOAL]\ncase h.e'_1\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\n\u22a2 0 = invVal (fun i => xL \u2191i) xR (fun i => inv' (xL \u2191i)) (fun i => inv' (xR i)) InvTy.zero\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 inv' 0 \u2261r 1\n[PROOFSTEP]\nchange mk _ _ _ _ \u2261r 1\n[GOAL]\n\u22a2 mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n      (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i)) \u2261r\n    1\n[PROOFSTEP]\nrefine' \u27e8_, _, fun i => _, IsEmpty.elim _\u27e9\n[GOAL]\ncase refine'_1\n\u22a2 LeftMoves\n      (mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))) \u2243\n    LeftMoves 1\n[PROOFSTEP]\napply Equiv.equivPUnit (InvTy _ _ _)\n[GOAL]\ncase refine'_2\n\u22a2 RightMoves\n      (mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))) \u2243\n    RightMoves 1\n[PROOFSTEP]\napply Equiv.equivPEmpty (InvTy _ _ _)\n[GOAL]\ncase refine'_3\ni :\n  LeftMoves\n    (mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n      (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i)))\n\u22a2 moveLeft\n      (mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i)))\n      i \u2261r\n    moveLeft 1 (\u2191(Equiv.equivPUnit (InvTy { i // 0 < PEmpty.elim i } PEmpty false)) i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\ni :\n  LeftMoves\n    (mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n      (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i)))\n\u22a2 0 \u2261r 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\n\u22a2 IsEmpty\n    (RightMoves\n      (mk (InvTy { i // 0 < PEmpty.elim i } PEmpty false) (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => PEmpty.elim \u2191i) PEmpty.elim (fun i => inv' (PEmpty.elim \u2191i)) fun i => inv' (PEmpty.elim i))))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_4\n\u22a2 IsEmpty (InvTy { i // 0 < PEmpty.elim i } PEmpty true)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 inv' 1 \u2261r 1\n[PROOFSTEP]\nchange Relabelling (mk _ _ _ _) 1\n[GOAL]\n\u22a2 mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i)) \u2261r\n    1\n[PROOFSTEP]\nhave : IsEmpty { _i : PUnit.{u + 1} // (0 : PGame.{u}) < 0 } :=\n  by\n  rw [lt_self_iff_false]\n  infer_instance\n[GOAL]\n\u22a2 IsEmpty { _i // 0 < 0 }\n[PROOFSTEP]\nrw [lt_self_iff_false]\n[GOAL]\n\u22a2 IsEmpty { _i // False }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i)) \u2261r\n    1\n[PROOFSTEP]\nrefine' \u27e8_, _, fun i => _, IsEmpty.elim _\u27e9\n[GOAL]\ncase refine'_1\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 LeftMoves\n      (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i =>\n          inv' (PEmpty.elim i))) \u2243\n    LeftMoves 1\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_2\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 RightMoves\n      (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i =>\n          inv' (PEmpty.elim i))) \u2243\n    RightMoves 1\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_3\nthis : IsEmpty { _i // 0 < 0 }\ni :\n  LeftMoves\n    (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i)))\n\u22a2 moveLeft\n      (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i)))\n      i \u2261r\n    moveLeft 1 (\u2191(id ?refine'_1) i)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_4\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 IsEmpty\n    (RightMoves\n      (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n        (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i =>\n          inv' (PEmpty.elim i))))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 InvTy { i // 0 < 0 } PEmpty false \u2243 PUnit\n[PROOFSTEP]\napply Equiv.equivPUnit\n[GOAL]\ncase refine'_2\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 InvTy { i // 0 < 0 } PEmpty true \u2243 PEmpty\n[PROOFSTEP]\napply Equiv.equivOfIsEmpty\n[GOAL]\ncase refine'_3\nthis : IsEmpty { _i // 0 < 0 }\ni :\n  LeftMoves\n    (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i)))\n\u22a2 invVal (fun i => 0) PEmpty.elim (fun i => inv' 0) (fun i => inv' (PEmpty.elim i)) i \u2261r 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\nthis : IsEmpty { _i // 0 < 0 }\ni :\n  LeftMoves\n    (mk (InvTy { i // 0 < (fun x => 0) i } PEmpty false) (InvTy { i // 0 < (fun x => 0) i } PEmpty true)\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i))\n      (invVal (fun i => (fun x => 0) \u2191i) PEmpty.elim (fun i => inv' ((fun x => 0) \u2191i)) fun i => inv' (PEmpty.elim i)))\n\u22a2 0 \u2261r 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4\nthis : IsEmpty { _i // 0 < 0 }\n\u22a2 IsEmpty (InvTy { i // 0 < 0 } PEmpty true)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 PGame \u2192 PGame\n[PROOFSTEP]\nclassical exact fun x => if x \u2248 0 then 0 else if 0 < x then inv' x else -inv' (-x)\n[GOAL]\n\u22a2 PGame \u2192 PGame\n[PROOFSTEP]\nexact fun x => if x \u2248 0 then 0 else if 0 < x then inv' x else -inv' (-x)\n[GOAL]\nx : PGame\nh : x \u2248 0\n\u22a2 x\u207b\u00b9 = 0\n[PROOFSTEP]\nclassical exact if_pos h\n[GOAL]\nx : PGame\nh : x \u2248 0\n\u22a2 x\u207b\u00b9 = 0\n[PROOFSTEP]\nexact if_pos h\n[GOAL]\nx : PGame\nh : 0 < x\n\u22a2 x\u207b\u00b9 = inv' x\n[PROOFSTEP]\nclassical exact (if_neg h.lf.not_equiv').trans (if_pos h)\n[GOAL]\nx : PGame\nh : 0 < x\n\u22a2 x\u207b\u00b9 = inv' x\n[PROOFSTEP]\nexact (if_neg h.lf.not_equiv').trans (if_pos h)\n[GOAL]\nx : PGame\nh : x \u29cf 0\n\u22a2 x\u207b\u00b9 = -inv' (-x)\n[PROOFSTEP]\nclassical exact (if_neg h.not_equiv).trans (if_neg h.not_gt)\n[GOAL]\nx : PGame\nh : x \u29cf 0\n\u22a2 x\u207b\u00b9 = -inv' (-x)\n[PROOFSTEP]\nexact (if_neg h.not_equiv).trans (if_neg h.not_gt)\n[GOAL]\n\u22a2 1\u207b\u00b9 \u2261r 1\n[PROOFSTEP]\nrw [inv_eq_of_pos PGame.zero_lt_one]\n[GOAL]\n\u22a2 inv' 1 \u2261r 1\n[PROOFSTEP]\nexact inv'One\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Game.Basic", "llama_tokens": 264377, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711908591638, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.588725725851367}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\nf : Option \u03b1 \u2192 M\ns : Finset \u03b1\n\u22a2 \u220f x in \u2191insertNone s, f x = f none * \u220f x in s, f (some x)\n[PROOFSTEP]\nsimp [insertNone]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\nf : \u03b1 \u2192 M\ns : Finset (Option \u03b1)\n\u22a2 \u220f x in \u2191eraseNone s, f x = \u220f x in s, Option.elim' 1 f x\n[PROOFSTEP]\nclassical calc\n  \u220f x in eraseNone s, f x = \u220f x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n    (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n  _ = \u220f x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n  _ = \u220f x in s, Option.elim' 1 f x := prod_erase _ rfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\nf : \u03b1 \u2192 M\ns : Finset (Option \u03b1)\n\u22a2 \u220f x in \u2191eraseNone s, f x = \u220f x in s, Option.elim' 1 f x\n[PROOFSTEP]\ncalc\n  \u220f x in eraseNone s, f x = \u220f x in (eraseNone s).map Embedding.some, Option.elim' 1 f x :=\n    (prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm\n  _ = \u220f x in s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]\n  _ = \u220f x in s, Option.elim' 1 f x := prod_erase _ rfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u_2\ninst\u271d : CommMonoid M\nf : \u03b1 \u2192 M\ns : Finset (Option \u03b1)\n\u22a2 \u220f x in map Embedding.some (\u2191eraseNone s), Option.elim' 1 f x = \u220f x in erase s none, Option.elim' 1 f x\n[PROOFSTEP]\nrw [map_some_eraseNone]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.Option", "llama_tokens": 665, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711908591638, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.588725725851367}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\nS : Type ?u.27\ninst\u271d\u00b2 : Mul R\ninst\u271d\u00b9 : Add R\ninst\u271d : Distrib S\nf : R \u2192 S\nhf : Injective f\nadd : \u2200 (x y : R), f (x + y) = f x + f y\nmul : \u2200 (x y : R), f (x * y) = f x * f y\nx y z : R\n\u22a2 f (x * (y + z)) = f (x * y + x * z)\n[PROOFSTEP]\nsimp only [*, left_distrib]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\nS : Type ?u.27\ninst\u271d\u00b2 : Mul R\ninst\u271d\u00b9 : Add R\ninst\u271d : Distrib S\nf : R \u2192 S\nhf : Injective f\nadd : \u2200 (x y : R), f (x + y) = f x + f y\nmul : \u2200 (x y : R), f (x * y) = f x * f y\nx y z : R\n\u22a2 f ((x + y) * z) = f (x * z + y * z)\n[PROOFSTEP]\nsimp only [*, right_distrib]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\nS : Type ?u.1511\ninst\u271d\u00b2 : Distrib R\ninst\u271d\u00b9 : Add S\ninst\u271d : Mul S\nf : R \u2192 S\nhf : Surjective f\nadd : \u2200 (x y : R), f (x + y) = f x + f y\nmul : \u2200 (x y : R), f (x * y) = f x * f y\nx y z : R\n\u22a2 f x * (f y + f z) = f x * f y + f x * f z\n[PROOFSTEP]\nsimp only [\u2190 add, \u2190 mul, left_distrib]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\nS : Type ?u.1511\ninst\u271d\u00b2 : Distrib R\ninst\u271d\u00b9 : Add S\ninst\u271d : Mul S\nf : R \u2192 S\nhf : Surjective f\nadd : \u2200 (x y : R), f (x + y) = f x + f y\nmul : \u2200 (x y : R), f (x * y) = f x * f y\nx y z : R\n\u22a2 (f x + f y) * f z = f x * f z + f y * f z\n[PROOFSTEP]\nsimp only [\u2190 add, \u2190 mul, right_distrib]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : HasDistribNeg \u03b1\ninst\u271d\u00b9 : Neg \u03b2\ninst\u271d : Mul \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\nneg : \u2200 (a : \u03b2), f (-a) = -f a\nmul : \u2200 (a b : \u03b2), f (a * b) = f a * f b\nsrc\u271d\u00b9 : InvolutiveNeg \u03b2 := Injective.involutiveNeg f hf neg\nsrc\u271d : Mul \u03b2 := inst\u271d\nx y : \u03b2\n\u22a2 f (-x * y) = f (-(x * y))\n[PROOFSTEP]\nerw [neg, mul, neg, neg_mul, mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : HasDistribNeg \u03b1\ninst\u271d\u00b9 : Neg \u03b2\ninst\u271d : Mul \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : Injective f\nneg : \u2200 (a : \u03b2), f (-a) = -f a\nmul : \u2200 (a b : \u03b2), f (a * b) = f a * f b\nsrc\u271d\u00b9 : InvolutiveNeg \u03b2 := Injective.involutiveNeg f hf neg\nsrc\u271d : Mul \u03b2 := inst\u271d\nx y : \u03b2\n\u22a2 f (x * -y) = f (-(x * y))\n[PROOFSTEP]\nerw [neg, mul, neg, mul_neg, mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : HasDistribNeg \u03b1\ninst\u271d\u00b9 : Neg \u03b2\ninst\u271d : Mul \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nneg : \u2200 (a : \u03b1), f (-a) = -f a\nmul : \u2200 (a b : \u03b1), f (a * b) = f a * f b\nsrc\u271d\u00b9 : InvolutiveNeg \u03b2 := Surjective.involutiveNeg f hf neg\nsrc\u271d : Mul \u03b2 := inst\u271d\nx y : \u03b1\n\u22a2 -f x * f y = -(f x * f y)\n[PROOFSTEP]\nerw [\u2190 neg, \u2190 mul, neg_mul, neg, mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : HasDistribNeg \u03b1\ninst\u271d\u00b9 : Neg \u03b2\ninst\u271d : Mul \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nneg : \u2200 (a : \u03b1), f (-a) = -f a\nmul : \u2200 (a b : \u03b1), f (a * b) = f a * f b\nsrc\u271d\u00b9 : InvolutiveNeg \u03b2 := Surjective.involutiveNeg f hf neg\nsrc\u271d : Mul \u03b2 := inst\u271d\nx y : \u03b1\n\u22a2 f x * -f y = -(f x * f y)\n[PROOFSTEP]\nerw [\u2190 neg, \u2190 mul, mul_neg, neg, mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.InjSurj", "llama_tokens": 1517, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256591565729, "lm_q2_score": 0.6992544273261176, "lm_q1q2_score": 0.5886503192019609}}
{"text": "[GOAL]\nu : \u2124\u02e3\n\u22a2 natAbs \u2191u * natAbs \u2191u\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 natAbs_mul, Units.mul_inv]\n[GOAL]\nu : \u2124\u02e3\n\u22a2 natAbs 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nu : \u2124\u02e3\n\u22a2 natAbs \u2191u\u207b\u00b9 * natAbs \u2191u = 1\n[PROOFSTEP]\nrw [\u2190 natAbs_mul, Units.inv_mul]\n[GOAL]\nu : \u2124\u02e3\n\u22a2 natAbs 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nu : \u2124\u02e3\n\u22a2 u = 1 \u2228 u = -1\n[PROOFSTEP]\nsimpa only [Units.ext_iff, units_natAbs] using natAbs_eq u\n[GOAL]\na : \u2124\n\u22a2 IsUnit a \u2194 a = 1 \u2228 a = -1\n[PROOFSTEP]\nrefine' \u27e8fun h => isUnit_eq_one_or h, fun h => _\u27e9\n[GOAL]\na : \u2124\nh : a = 1 \u2228 a = -1\n\u22a2 IsUnit a\n[PROOFSTEP]\nrcases h with (rfl | rfl)\n[GOAL]\ncase inl\n\u22a2 IsUnit 1\n[PROOFSTEP]\nexact isUnit_one\n[GOAL]\ncase inr\n\u22a2 IsUnit (-1)\n[PROOFSTEP]\nexact isUnit_one.neg\n[GOAL]\na b : \u2124\nha : IsUnit a\nhb : IsUnit b\n\u22a2 a = b \u2228 a = -b\n[PROOFSTEP]\nrcases isUnit_eq_one_or hb with (rfl | rfl)\n[GOAL]\ncase inl\na : \u2124\nha : IsUnit a\nhb : IsUnit 1\n\u22a2 a = 1 \u2228 a = -1\n[PROOFSTEP]\nexact isUnit_eq_one_or ha\n[GOAL]\ncase inr\na : \u2124\nha : IsUnit a\nhb : IsUnit (-1)\n\u22a2 a = -1 \u2228 a = - -1\n[PROOFSTEP]\nrwa [or_comm, neg_neg, \u2190 isUnit_iff]\n[GOAL]\nz w : \u2124\nh : z * w = 1\n\u22a2 z = 1 \u2227 w = 1 \u2228 z = -1 \u2227 w = -1\n[PROOFSTEP]\nhave h' : w * z = 1 := mul_comm z w \u25b8 h\n[GOAL]\nz w : \u2124\nh : z * w = 1\nh' : w * z = 1\n\u22a2 z = 1 \u2227 w = 1 \u2228 z = -1 \u2227 w = -1\n[PROOFSTEP]\nrcases eq_one_or_neg_one_of_mul_eq_one h with (rfl | rfl)\n[GOAL]\ncase inl\nw : \u2124\nh : 1 * w = 1\nh' : w * 1 = 1\n\u22a2 1 = 1 \u2227 w = 1 \u2228 1 = -1 \u2227 w = -1\n[PROOFSTEP]\nrcases eq_one_or_neg_one_of_mul_eq_one h' with (rfl | rfl)\n[GOAL]\ncase inr\nw : \u2124\nh : -1 * w = 1\nh' : w * -1 = 1\n\u22a2 -1 = 1 \u2227 w = 1 \u2228 -1 = -1 \u2227 w = -1\n[PROOFSTEP]\nrcases eq_one_or_neg_one_of_mul_eq_one h' with (rfl | rfl)\n[GOAL]\ncase inl.inl\nh h' : 1 * 1 = 1\n\u22a2 1 = 1 \u2227 1 = 1 \u2228 1 = -1 \u2227 1 = -1\n[PROOFSTEP]\ntauto\n[GOAL]\ncase inl.inr\nh : 1 * -1 = 1\nh' : -1 * 1 = 1\n\u22a2 1 = 1 \u2227 -1 = 1 \u2228 1 = -1 \u2227 -1 = -1\n[PROOFSTEP]\ntauto\n[GOAL]\ncase inr.inl\nh : -1 * 1 = 1\nh' : 1 * -1 = 1\n\u22a2 -1 = 1 \u2227 1 = 1 \u2228 -1 = -1 \u2227 1 = -1\n[PROOFSTEP]\ntauto\n[GOAL]\ncase inr.inr\nh h' : -1 * -1 = 1\n\u22a2 -1 = 1 \u2227 -1 = 1 \u2228 -1 = -1 \u2227 -1 = -1\n[PROOFSTEP]\ntauto\n[GOAL]\nz w : \u2124\n\u22a2 z * w = 1 \u2194 z = 1 \u2227 w = 1 \u2228 z = -1 \u2227 w = -1\n[PROOFSTEP]\nrefine' \u27e8eq_one_or_neg_one_of_mul_eq_one', fun h => Or.elim h (fun H => _) fun H => _\u27e9\n[GOAL]\ncase refine'_1\nz w : \u2124\nh : z = 1 \u2227 w = 1 \u2228 z = -1 \u2227 w = -1\nH : z = 1 \u2227 w = 1\n\u22a2 z * w = 1\n[PROOFSTEP]\nrcases H with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_2\nz w : \u2124\nh : z = 1 \u2227 w = 1 \u2228 z = -1 \u2227 w = -1\nH : z = -1 \u2227 w = -1\n\u22a2 z * w = 1\n[PROOFSTEP]\nrcases H with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\nh : 1 = 1 \u2227 1 = 1 \u2228 1 = -1 \u2227 1 = -1\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.intro\nh : -1 = 1 \u2227 -1 = 1 \u2228 -1 = -1 \u2227 -1 = -1\n\u22a2 -1 * -1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nz w : \u2124\nh : z * w = -1\n\u22a2 z = 1 \u2227 w = -1 \u2228 z = -1 \u2227 w = 1\n[PROOFSTEP]\nrcases isUnit_eq_one_or (IsUnit.mul_iff.mp (Int.isUnit_iff.mpr (Or.inr h))).1 with (rfl | rfl)\n[GOAL]\ncase inl\nw : \u2124\nh : 1 * w = -1\n\u22a2 1 = 1 \u2227 w = -1 \u2228 1 = -1 \u2227 w = 1\n[PROOFSTEP]\nexact Or.inl \u27e8rfl, one_mul w \u25b8 h\u27e9\n[GOAL]\ncase inr\nw : \u2124\nh : -1 * w = -1\n\u22a2 -1 = 1 \u2227 w = -1 \u2228 -1 = -1 \u2227 w = 1\n[PROOFSTEP]\nexact Or.inr \u27e8rfl, neg_inj.mp (neg_one_mul w \u25b8 h)\u27e9\n[GOAL]\nz w : \u2124\n\u22a2 z * w = -1 \u2194 z = 1 \u2227 w = -1 \u2228 z = -1 \u2227 w = 1\n[PROOFSTEP]\nrefine' \u27e8eq_one_or_neg_one_of_mul_eq_neg_one', fun h => Or.elim h (fun H => _) fun H => _\u27e9\n[GOAL]\ncase refine'_1\nz w : \u2124\nh : z = 1 \u2227 w = -1 \u2228 z = -1 \u2227 w = 1\nH : z = 1 \u2227 w = -1\n\u22a2 z * w = -1\n[PROOFSTEP]\nrcases H with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_2\nz w : \u2124\nh : z = 1 \u2227 w = -1 \u2228 z = -1 \u2227 w = 1\nH : z = -1 \u2227 w = 1\n\u22a2 z * w = -1\n[PROOFSTEP]\nrcases H with \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\nh : 1 = 1 \u2227 -1 = -1 \u2228 1 = -1 \u2227 -1 = 1\n\u22a2 1 * -1 = -1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.intro\nh : -1 = 1 \u2227 1 = -1 \u2228 -1 = -1 \u2227 1 = 1\n\u22a2 -1 * 1 = -1\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2124\n\u22a2 IsUnit n \u2194 natAbs n = 1\n[PROOFSTEP]\nsimp [natAbs_eq_iff, isUnit_iff, Nat.cast_zero]\n[GOAL]\nn : \u2115\n\u22a2 IsUnit \u2191n \u2194 IsUnit n\n[PROOFSTEP]\nsimp [isUnit_iff_natAbs_eq]\n[GOAL]\na b c d : \u2124\nha : IsUnit a\nhb : IsUnit b\nhc : IsUnit c\nhd : IsUnit d\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nrw [isUnit_iff] at ha hb hc hd \n[GOAL]\na b c d : \u2124\nha : a = 1 \u2228 a = -1\nhb : b = 1 \u2228 b = -1\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases ha\n[GOAL]\ncase inl\na b c d : \u2124\nhb : b = 1 \u2228 b = -1\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\nh\u271d : a = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hb\n[GOAL]\ncase inr\na b c d : \u2124\nhb : b = 1 \u2228 b = -1\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\nh\u271d : a = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hb\n[GOAL]\ncase inl.inl\na b c d : \u2124\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b9 : a = 1\nh\u271d : b = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hc\n[GOAL]\ncase inl.inr\na b c d : \u2124\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b9 : a = 1\nh\u271d : b = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hc\n[GOAL]\ncase inr.inl\na b c d : \u2124\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b9 : a = -1\nh\u271d : b = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hc\n[GOAL]\ncase inr.inr\na b c d : \u2124\nhc : c = 1 \u2228 c = -1\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b9 : a = -1\nh\u271d : b = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hc\n[GOAL]\ncase inl.inl.inl\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = 1\nh\u271d\u00b9 : b = 1\nh\u271d : c = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inl.inl.inr\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = 1\nh\u271d\u00b9 : b = 1\nh\u271d : c = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inl.inr.inl\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = 1\nh\u271d\u00b9 : b = -1\nh\u271d : c = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inl.inr.inr\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = 1\nh\u271d\u00b9 : b = -1\nh\u271d : c = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inr.inl.inl\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = -1\nh\u271d\u00b9 : b = 1\nh\u271d : c = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inr.inl.inr\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = -1\nh\u271d\u00b9 : b = 1\nh\u271d : c = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inr.inr.inl\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = -1\nh\u271d\u00b9 : b = -1\nh\u271d : c = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inr.inr.inr\na b c d : \u2124\nhd : d = 1 \u2228 d = -1\nh\u271d\u00b2 : a = -1\nh\u271d\u00b9 : b = -1\nh\u271d : c = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase inl.inl.inl.inl\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inl.inl.inr\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inl.inr.inl\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inl.inr.inr\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inr.inl.inl\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inr.inl.inr\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inr.inr.inl\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inr.inr.inr\na b c d : \u2124\nh\u271d\u00b3 : a = 1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inl.inl.inl\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inl.inl.inr\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inl.inr.inl\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inl.inr.inr\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inr.inl.inl\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inr.inl.inr\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inr.inr.inl\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inr.inr.inr.inr\na b c d : \u2124\nh\u271d\u00b3 : a = -1\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase inl.inl.inl.inl\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inl.inl.inr\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inl.inr.inl\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inl.inr.inr\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inr.inl.inl\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inr.inl.inr\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inr.inr.inl\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inr.inr.inr\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 1 + b = c + d \u2194 1 = c \u2227 b = d \u2228 1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inl.inl.inl\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inl.inl.inr\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inl.inr.inl\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inl.inr.inr\nb c d : \u2124\nh\u271d\u00b2 : b = 1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inr.inl.inl\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inr.inl.inr\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inr.inr.inl\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inr.inr.inr.inr\nb c d : \u2124\nh\u271d\u00b2 : b = -1\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 -1 + b = c + d \u2194 -1 = c \u2227 b = d \u2228 -1 = d \u2227 b = c\n[PROOFSTEP]\nsubst b\n[GOAL]\ncase inl.inl.inl.inl\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 1 + 1 = c + d \u2194 1 = c \u2227 1 = d \u2228 1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inl.inl.inr\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 1 + 1 = c + d \u2194 1 = c \u2227 1 = d \u2228 1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inl.inr.inl\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 1 + 1 = c + d \u2194 1 = c \u2227 1 = d \u2228 1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inl.inr.inr\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 1 + 1 = c + d \u2194 1 = c \u2227 1 = d \u2228 1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inr.inl.inl\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 1 + -1 = c + d \u2194 1 = c \u2227 -1 = d \u2228 1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inr.inl.inr\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 1 + -1 = c + d \u2194 1 = c \u2227 -1 = d \u2228 1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inr.inr.inl\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 1 + -1 = c + d \u2194 1 = c \u2227 -1 = d \u2228 1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inr.inr.inr\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 1 + -1 = c + d \u2194 1 = c \u2227 -1 = d \u2228 1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inl.inl.inl\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 -1 + 1 = c + d \u2194 -1 = c \u2227 1 = d \u2228 -1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inl.inl.inr\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 -1 + 1 = c + d \u2194 -1 = c \u2227 1 = d \u2228 -1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inl.inr.inl\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 -1 + 1 = c + d \u2194 -1 = c \u2227 1 = d \u2228 -1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inl.inr.inr\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 -1 + 1 = c + d \u2194 -1 = c \u2227 1 = d \u2228 -1 = d \u2227 1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inr.inl.inl\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = 1\n\u22a2 -1 + -1 = c + d \u2194 -1 = c \u2227 -1 = d \u2228 -1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inr.inl.inr\nc d : \u2124\nh\u271d\u00b9 : c = 1\nh\u271d : d = -1\n\u22a2 -1 + -1 = c + d \u2194 -1 = c \u2227 -1 = d \u2228 -1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inr.inr.inl\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = 1\n\u22a2 -1 + -1 = c + d \u2194 -1 = c \u2227 -1 = d \u2228 -1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inr.inr.inr.inr\nc d : \u2124\nh\u271d\u00b9 : c = -1\nh\u271d : d = -1\n\u22a2 -1 + -1 = c + d \u2194 -1 = c \u2227 -1 = d \u2228 -1 = d \u2227 -1 = c\n[PROOFSTEP]\nsubst c\n[GOAL]\ncase inl.inl.inl.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 1 + 1 = 1 + d \u2194 1 = 1 \u2227 1 = d \u2228 1 = d \u2227 1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inl.inl.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 1 + 1 = 1 + d \u2194 1 = 1 \u2227 1 = d \u2228 1 = d \u2227 1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inl.inr.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 1 + 1 = -1 + d \u2194 1 = -1 \u2227 1 = d \u2228 1 = d \u2227 1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inl.inr.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 1 + 1 = -1 + d \u2194 1 = -1 \u2227 1 = d \u2228 1 = d \u2227 1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inr.inl.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 1 + -1 = 1 + d \u2194 1 = 1 \u2227 -1 = d \u2228 1 = d \u2227 -1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inr.inl.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 1 + -1 = 1 + d \u2194 1 = 1 \u2227 -1 = d \u2228 1 = d \u2227 -1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inr.inr.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 1 + -1 = -1 + d \u2194 1 = -1 \u2227 -1 = d \u2228 1 = d \u2227 -1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inr.inr.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 1 + -1 = -1 + d \u2194 1 = -1 \u2227 -1 = d \u2228 1 = d \u2227 -1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inl.inl.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 -1 + 1 = 1 + d \u2194 -1 = 1 \u2227 1 = d \u2228 -1 = d \u2227 1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inl.inl.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 -1 + 1 = 1 + d \u2194 -1 = 1 \u2227 1 = d \u2228 -1 = d \u2227 1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inl.inr.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 -1 + 1 = -1 + d \u2194 -1 = -1 \u2227 1 = d \u2228 -1 = d \u2227 1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inl.inr.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 -1 + 1 = -1 + d \u2194 -1 = -1 \u2227 1 = d \u2228 -1 = d \u2227 1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inr.inl.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 -1 + -1 = 1 + d \u2194 -1 = 1 \u2227 -1 = d \u2228 -1 = d \u2227 -1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inr.inl.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 -1 + -1 = 1 + d \u2194 -1 = 1 \u2227 -1 = d \u2228 -1 = d \u2227 -1 = 1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inr.inr.inl\nd : \u2124\nh\u271d : d = 1\n\u22a2 -1 + -1 = -1 + d \u2194 -1 = -1 \u2227 -1 = d \u2228 -1 = d \u2227 -1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inr.inr.inr.inr\nd : \u2124\nh\u271d : d = -1\n\u22a2 -1 + -1 = -1 + d \u2194 -1 = -1 \u2227 -1 = d \u2228 -1 = d \u2227 -1 = -1\n[PROOFSTEP]\nsubst d\n[GOAL]\ncase inl.inl.inl.inl\n\u22a2 1 + 1 = 1 + 1 \u2194 1 = 1 \u2227 1 = 1 \u2228 1 = 1 \u2227 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inl.inl.inr\n\u22a2 1 + 1 = 1 + -1 \u2194 1 = 1 \u2227 1 = -1 \u2228 1 = -1 \u2227 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inl.inr.inl\n\u22a2 1 + 1 = -1 + 1 \u2194 1 = -1 \u2227 1 = 1 \u2228 1 = 1 \u2227 1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inl.inr.inr\n\u22a2 1 + 1 = -1 + -1 \u2194 1 = -1 \u2227 1 = -1 \u2228 1 = -1 \u2227 1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr.inl.inl\n\u22a2 1 + -1 = 1 + 1 \u2194 1 = 1 \u2227 -1 = 1 \u2228 1 = 1 \u2227 -1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr.inl.inr\n\u22a2 1 + -1 = 1 + -1 \u2194 1 = 1 \u2227 -1 = -1 \u2228 1 = -1 \u2227 -1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr.inr.inl\n\u22a2 1 + -1 = -1 + 1 \u2194 1 = -1 \u2227 -1 = 1 \u2228 1 = 1 \u2227 -1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr.inr.inr\n\u22a2 1 + -1 = -1 + -1 \u2194 1 = -1 \u2227 -1 = -1 \u2228 1 = -1 \u2227 -1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl.inl.inl\n\u22a2 -1 + 1 = 1 + 1 \u2194 -1 = 1 \u2227 1 = 1 \u2228 -1 = 1 \u2227 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl.inl.inr\n\u22a2 -1 + 1 = 1 + -1 \u2194 -1 = 1 \u2227 1 = -1 \u2228 -1 = -1 \u2227 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl.inr.inl\n\u22a2 -1 + 1 = -1 + 1 \u2194 -1 = -1 \u2227 1 = 1 \u2228 -1 = 1 \u2227 1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl.inr.inr\n\u22a2 -1 + 1 = -1 + -1 \u2194 -1 = -1 \u2227 1 = -1 \u2228 -1 = -1 \u2227 1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.inl.inl\n\u22a2 -1 + -1 = 1 + 1 \u2194 -1 = 1 \u2227 -1 = 1 \u2228 -1 = 1 \u2227 -1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.inl.inr\n\u22a2 -1 + -1 = 1 + -1 \u2194 -1 = 1 \u2227 -1 = -1 \u2228 -1 = -1 \u2227 -1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.inr.inl\n\u22a2 -1 + -1 = -1 + 1 \u2194 -1 = -1 \u2227 -1 = 1 \u2228 -1 = 1 \u2227 -1 = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.inr.inr\n\u22a2 -1 + -1 = -1 + -1 \u2194 -1 = -1 \u2227 -1 = -1 \u2228 -1 = -1 \u2227 -1 = -1\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Units", "llama_tokens": 11185, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.808067204308405, "lm_q2_score": 0.727975460709318, "lm_q1q2_score": 0.5882530953405016}}
{"text": "[GOAL]\nz w : \u2102\n\u22a2 z < w \u2194 z \u2264 w \u2227 \u00acw \u2264 z\n[PROOFSTEP]\ndsimp\n[GOAL]\nz w : \u2102\n\u22a2 z.re < w.re \u2227 z.im = w.im \u2194 (z.re \u2264 w.re \u2227 z.im = w.im) \u2227 \u00ac(w.re \u2264 z.re \u2227 w.im = z.im)\n[PROOFSTEP]\nrw [lt_iff_le_not_le]\n[GOAL]\nz w : \u2102\n\u22a2 (z.re \u2264 w.re \u2227 \u00acw.re \u2264 z.re) \u2227 z.im = w.im \u2194 (z.re \u2264 w.re \u2227 z.im = w.im) \u2227 \u00ac(w.re \u2264 z.re \u2227 w.im = z.im)\n[PROOFSTEP]\ntauto\n[GOAL]\nx y : \u211d\n\u22a2 \u2191x \u2264 \u2191y \u2194 x \u2264 y\n[PROOFSTEP]\nsimp [le_def, ofReal']\n[GOAL]\nx y : \u211d\n\u22a2 \u2191x < \u2191y \u2194 x < y\n[PROOFSTEP]\nsimp [lt_def, ofReal']\n[GOAL]\nz w : \u2102\n\u22a2 \u00acz \u2264 w \u2194 w.re < z.re \u2228 z.im \u2260 w.im\n[PROOFSTEP]\nrw [le_def, not_and_or, not_le]\n[GOAL]\nz w : \u2102\n\u22a2 \u00acz < w \u2194 w.re \u2264 z.re \u2228 z.im \u2260 w.im\n[PROOFSTEP]\nrw [lt_def, not_and_or, not_lt]\n[GOAL]\nr : \u211d\nz : \u2102\nhz : \u2191r \u2264 z\n\u22a2 z = \u2191z.re\n[PROOFSTEP]\next\n[GOAL]\ncase a\nr : \u211d\nz : \u2102\nhz : \u2191r \u2264 z\n\u22a2 z.re = (\u2191z.re).re\ncase a r : \u211d z : \u2102 hz : \u2191r \u2264 z \u22a2 z.im = (\u2191z.re).im\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nr : \u211d\nz : \u2102\nhz : \u2191r \u2264 z\n\u22a2 z.im = (\u2191z.re).im\n[PROOFSTEP]\nsimp only [\u2190 (Complex.le_def.1 hz).2, Complex.zero_im, Complex.ofReal_im]\n", "meta": {"mathlib_filename": "Mathlib.Data.Complex.Order", "llama_tokens": 648, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289387914176259, "lm_q2_score": 0.7090191276365462, "lm_q1q2_score": 0.587733458755018}}
{"text": "[GOAL]\nB : Type u_1\nF : Type u_2\nZ : Type u_3\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalSpace Z\nproj : Z \u2192 B\ninst\u271d : Nonempty F\nh : IsHomeomorphicTrivialFiberBundle F proj\n\u22a2 Function.Surjective proj\n[PROOFSTEP]\nobtain \u27e8e, rfl\u27e9 := h.proj_eq\n[GOAL]\ncase intro\nB : Type u_1\nF : Type u_2\nZ : Type u_3\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : Nonempty F\ne : Z \u2243\u209c B \u00d7 F\nh : IsHomeomorphicTrivialFiberBundle F (Prod.fst \u2218 \u2191e)\n\u22a2 Function.Surjective (Prod.fst \u2218 \u2191e)\n[PROOFSTEP]\nexact Prod.fst_surjective.comp e.surjective\n[GOAL]\nB : Type u_1\nF : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalSpace Z\nproj : Z \u2192 B\nh : IsHomeomorphicTrivialFiberBundle F proj\n\u22a2 Continuous proj\n[PROOFSTEP]\nobtain \u27e8e, rfl\u27e9 := h.proj_eq\n[GOAL]\ncase intro\nB : Type u_1\nF : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalSpace Z\ne : Z \u2243\u209c B \u00d7 F\nh : IsHomeomorphicTrivialFiberBundle F (Prod.fst \u2218 \u2191e)\n\u22a2 Continuous (Prod.fst \u2218 \u2191e)\n[PROOFSTEP]\nexact continuous_fst.comp e.continuous\n[GOAL]\nB : Type u_1\nF : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalSpace Z\nproj : Z \u2192 B\nh : IsHomeomorphicTrivialFiberBundle F proj\n\u22a2 IsOpenMap proj\n[PROOFSTEP]\nobtain \u27e8e, rfl\u27e9 := h.proj_eq\n[GOAL]\ncase intro\nB : Type u_1\nF : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : TopologicalSpace Z\ne : Z \u2243\u209c B \u00d7 F\nh : IsHomeomorphicTrivialFiberBundle F (Prod.fst \u2218 \u2191e)\n\u22a2 IsOpenMap (Prod.fst \u2218 \u2191e)\n[PROOFSTEP]\nexact isOpenMap_fst.comp e.isOpenMap\n", "meta": {"mathlib_filename": "Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle", "llama_tokens": 755, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677660619633, "lm_q2_score": 0.6926419767901476, "lm_q1q2_score": 0.5873380697394838}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\n\u22a2 \u2191x = 0 \u2194 p \u2223 x\n[PROOFSTEP]\nrw [\u2190 CharP.cast_eq_zero_iff R p x]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\n\u22a2 \u2191x = 0 \u2194 \u2191x = 0\n[PROOFSTEP]\nchange algebraMap \u2115 A x = 0 \u2194 algebraMap \u2115 R x = 0\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\n\u22a2 \u2191(algebraMap \u2115 A) x = 0 \u2194 \u2191(algebraMap \u2115 R) x = 0\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_apply \u2115 R A x]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\n\u22a2 \u2191(algebraMap R A) (\u2191(algebraMap \u2115 R) x) = 0 \u2194 \u2191(algebraMap \u2115 R) x = 0\n[PROOFSTEP]\nrefine' Iff.trans _ h.eq_iff\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\n\u22a2 \u2191(algebraMap R A) (\u2191(algebraMap \u2115 R) x) = 0 \u2194 \u2191(algebraMap R A) (\u2191(algebraMap \u2115 R) x) = \u2191(algebraMap R A) 0\n[PROOFSTEP]\nrw [RingHom.map_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\ninst\u271d : CharZero R\nx y : \u2115\nhxy : \u2191x = \u2191y\n\u22a2 x = y\n[PROOFSTEP]\nchange algebraMap \u2115 A x = algebraMap \u2115 A y at hxy \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\ninst\u271d : CharZero R\nx y : \u2115\nhxy : \u2191(algebraMap \u2115 A) x = \u2191(algebraMap \u2115 A) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_apply \u2115 R A x] at hxy \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\ninst\u271d : CharZero R\nx y : \u2115\nhxy : \u2191(algebraMap R A) (\u2191(algebraMap \u2115 R) x) = \u2191(algebraMap \u2115 A) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_apply \u2115 R A y] at hxy \n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\ninst\u271d : CharZero R\nx y : \u2115\nhxy : \u2191(algebraMap R A) (\u2191(algebraMap \u2115 R) x) = \u2191(algebraMap R A) (\u2191(algebraMap \u2115 R) y)\n\u22a2 x = y\n[PROOFSTEP]\nexact CharZero.cast_injective (h hxy)\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : CommSemiring L\ninst\u271d\u00b9 : Nontrivial L\ninst\u271d : Algebra K L\n\u22a2 ringChar K = ringChar L\n[PROOFSTEP]\nrw [ringChar.eq_iff, Algebra.charP_iff K L]\n[GOAL]\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : CommSemiring L\ninst\u271d\u00b9 : Nontrivial L\ninst\u271d : Algebra K L\n\u22a2 CharP L (ringChar L)\n[PROOFSTEP]\napply ringChar.charP\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.Algebra", "llama_tokens": 1449, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951025545426, "lm_q2_score": 0.6959583124210896, "lm_q1q2_score": 0.5873158114342818}}
{"text": "[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\n\u22a2 \u2045g\u2081, g\u2082\u2046 = 1 \u2194 g\u2081 * g\u2082 = g\u2082 * g\u2081\n[PROOFSTEP]\nrw [commutatorElement_def, mul_inv_eq_one, mul_inv_eq_iff_eq_mul]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\n\u22a2 \u2045g\u2081, g\u2082\u2046\u207b\u00b9 = \u2045g\u2082, g\u2081\u2046\n[PROOFSTEP]\nsimp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\n\u22a2 \u2191f \u2045g\u2081, g\u2082\u2046 = \u2045\u2191f g\u2081, \u2191f g\u2082\u2046\n[PROOFSTEP]\nsimp_rw [commutatorElement_def, map_mul f, map_inv f]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u22a2 \u2045H\u2081, H\u2082\u2046 = \u22a5 \u2194 H\u2081 \u2264 centralizer \u2191H\u2082\n[PROOFSTEP]\nrw [eq_bot_iff, commutator_le]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u22a2 (\u2200 (g\u2081 : G), g\u2081 \u2208 H\u2081 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2082 \u2192 \u2045g\u2081, g\u2082\u2046 \u2208 \u22a5) \u2194 H\u2081 \u2264 centralizer \u2191H\u2082\n[PROOFSTEP]\nrefine' forall_congr' fun p => forall_congr' fun _hp => forall_congr' fun q => forall_congr' fun hq => _\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\np : G\n_hp : p \u2208 H\u2081\nq : G\nhq : q \u2208 H\u2082\n\u22a2 \u2045p, q\u2046 \u2208 \u22a5 \u2194 q * p = p * q\n[PROOFSTEP]\nrw [mem_bot, commutatorElement_eq_one_iff_mul_comm, eq_comm]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh1 : \u2045\u2045H\u2082, H\u2083\u2046, H\u2081\u2046 = \u22a5\nh2 : \u2045\u2045H\u2083, H\u2081\u2046, H\u2082\u2046 = \u22a5\n\u22a2 \u2045\u2045H\u2081, H\u2082\u2046, H\u2083\u2046 = \u22a5\n[PROOFSTEP]\nsimp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le, mem_centralizer_iff_commutator_eq_one, \u2190\n  commutatorElement_def] at h1 h2 \u22a2\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh1 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2082 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2083 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2081 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nh2 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2083 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2081 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2082 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\n\u22a2 \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2081 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2082 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2083 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\n[PROOFSTEP]\nintro x hx y hy z hz\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh1 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2082 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2083 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2081 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nh2 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2083 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2081 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2082 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nx : G\nhx : x \u2208 H\u2081\ny : G\nhy : y \u2208 H\u2082\nz : G\nhz : z \u2208 \u2191H\u2083\n\u22a2 \u2045z, \u2045x, y\u2046\u2046 = 1\n[PROOFSTEP]\ntrans x * z * \u2045y, \u2045z\u207b\u00b9, x\u207b\u00b9\u2046\u2046\u207b\u00b9 * z\u207b\u00b9 * y * \u2045x\u207b\u00b9, \u2045y\u207b\u00b9, z\u2046\u2046\u207b\u00b9 * y\u207b\u00b9 * x\u207b\u00b9\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh1 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2082 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2083 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2081 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nh2 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2083 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2081 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2082 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nx : G\nhx : x \u2208 H\u2081\ny : G\nhy : y \u2208 H\u2082\nz : G\nhz : z \u2208 \u2191H\u2083\n\u22a2 \u2045z, \u2045x, y\u2046\u2046 = x * z * \u2045y, \u2045z\u207b\u00b9, x\u207b\u00b9\u2046\u2046\u207b\u00b9 * z\u207b\u00b9 * y * \u2045x\u207b\u00b9, \u2045y\u207b\u00b9, z\u2046\u2046\u207b\u00b9 * y\u207b\u00b9 * x\u207b\u00b9\n[PROOFSTEP]\ngroup\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh1 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2082 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2083 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2081 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nh2 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2083 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2081 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2082 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nx : G\nhx : x \u2208 H\u2081\ny : G\nhy : y \u2208 H\u2082\nz : G\nhz : z \u2208 \u2191H\u2083\n\u22a2 x * z * \u2045y, \u2045z\u207b\u00b9, x\u207b\u00b9\u2046\u2046\u207b\u00b9 * z\u207b\u00b9 * y * \u2045x\u207b\u00b9, \u2045y\u207b\u00b9, z\u2046\u2046\u207b\u00b9 * y\u207b\u00b9 * x\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [h1 _ (H\u2082.inv_mem hy) _ hz _ (H\u2081.inv_mem hx), h2 _ (H\u2083.inv_mem hz) _ (H\u2081.inv_mem hx) _ hy]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh1 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2082 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2083 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2081 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nh2 : \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2083 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2081 \u2192 \u2200 (h : G), h \u2208 \u2191H\u2082 \u2192 \u2045h, \u2045g\u2081, g\u2082\u2046\u2046 = 1\nx : G\nhx : x \u2208 H\u2081\ny : G\nhy : y \u2208 H\u2082\nz : G\nhz : z \u2208 \u2191H\u2083\n\u22a2 x * z * 1\u207b\u00b9 * z\u207b\u00b9 * y * 1\u207b\u00b9 * y\u207b\u00b9 * x\u207b\u00b9 = 1\n[PROOFSTEP]\ngroup\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\n\u22a2 Normal \u2045H\u2081, H\u2082\u2046\n[PROOFSTEP]\nlet base : Set G := {x | \u2203 g\u2081 \u2208 H\u2081, \u2203 g\u2082 \u2208 H\u2082, \u2045g\u2081, g\u2082\u2046 = x}\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\n\u22a2 Normal \u2045H\u2081, H\u2082\u2046\n[PROOFSTEP]\nchange (closure base).Normal\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\n\u22a2 Normal (closure base)\n[PROOFSTEP]\nsuffices h_base : base = Group.conjugatesOfSet base\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\nh_base : base = Group.conjugatesOfSet base\n\u22a2 Normal (closure base)\n[PROOFSTEP]\nrw [h_base]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\nh_base : base = Group.conjugatesOfSet base\n\u22a2 Normal (closure (Group.conjugatesOfSet base))\n[PROOFSTEP]\nexact Subgroup.normalClosure_normal\n[GOAL]\ncase h_base\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\n\u22a2 base = Group.conjugatesOfSet base\n[PROOFSTEP]\nrefine' Set.Subset.antisymm Group.subset_conjugatesOfSet fun a h => _\n[GOAL]\ncase h_base\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\na : G\nh : a \u2208 Group.conjugatesOfSet base\n\u22a2 a \u2208 base\n[PROOFSTEP]\nsimp_rw [Group.mem_conjugatesOfSet_iff, isConj_iff] at h \n[GOAL]\ncase h_base\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\na : G\nh : \u2203 a_1, a_1 \u2208 {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x} \u2227 \u2203 c, c * a_1 * c\u207b\u00b9 = a\n\u22a2 a \u2208 base\n[PROOFSTEP]\nrcases h with \u27e8b, \u27e8c, hc, e, he, rfl\u27e9, d, rfl\u27e9\n[GOAL]\ncase h_base.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nh\u2081 : Normal H\u2081\nh\u2082 : Normal H\u2082\nbase : Set G := {x | \u2203 g\u2081, g\u2081 \u2208 H\u2081 \u2227 \u2203 g\u2082, g\u2082 \u2208 H\u2082 \u2227 \u2045g\u2081, g\u2082\u2046 = x}\nc : G\nhc : c \u2208 H\u2081\ne : G\nhe : e \u2208 H\u2082\nd : G\n\u22a2 d * \u2045c, e\u2046 * d\u207b\u00b9 \u2208 base\n[PROOFSTEP]\nexact \u27e8_, h\u2081.conj_mem c hc d, _, h\u2082.conj_mem e he d, (conjugate_commutatorElement c e d).symm\u27e9\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\n\u22a2 map f \u2045H\u2081, H\u2082\u2046 = \u2045map f H\u2081, map f H\u2082\u2046\n[PROOFSTEP]\nsimp_rw [le_antisymm_iff, map_le_iff_le_comap, commutator_le, mem_comap, map_commutatorElement]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\n\u22a2 (\u2200 (g\u2081 : G), g\u2081 \u2208 H\u2081 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2082 \u2192 \u2045\u2191f g\u2081, \u2191f g\u2082\u2046 \u2208 \u2045map f H\u2081, map f H\u2082\u2046) \u2227\n    \u2200 (g\u2081 : G'), g\u2081 \u2208 map f H\u2081 \u2192 \u2200 (g\u2082 : G'), g\u2082 \u2208 map f H\u2082 \u2192 \u2045g\u2081, g\u2082\u2046 \u2208 map f \u2045H\u2081, H\u2082\u2046\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\n\u22a2 \u2200 (g\u2081 : G), g\u2081 \u2208 H\u2081 \u2192 \u2200 (g\u2082 : G), g\u2082 \u2208 H\u2082 \u2192 \u2045\u2191f g\u2081, \u2191f g\u2082\u2046 \u2208 \u2045map f H\u2081, map f H\u2082\u2046\n[PROOFSTEP]\nintro p hp q hq\n[GOAL]\ncase left\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\np : G\nhp : p \u2208 H\u2081\nq : G\nhq : q \u2208 H\u2082\n\u22a2 \u2045\u2191f p, \u2191f q\u2046 \u2208 \u2045map f H\u2081, map f H\u2082\u2046\n[PROOFSTEP]\nexact commutator_mem_commutator (mem_map_of_mem _ hp) (mem_map_of_mem _ hq)\n[GOAL]\ncase right\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\n\u22a2 \u2200 (g\u2081 : G'), g\u2081 \u2208 map f H\u2081 \u2192 \u2200 (g\u2082 : G'), g\u2082 \u2208 map f H\u2082 \u2192 \u2045g\u2081, g\u2082\u2046 \u2208 map f \u2045H\u2081, H\u2082\u2046\n[PROOFSTEP]\nrintro _ \u27e8p, hp, rfl\u27e9 _ \u27e8q, hq, rfl\u27e9\n[GOAL]\ncase right.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\np : G\nhp : p \u2208 \u2191H\u2081\nq : G\nhq : q \u2208 \u2191H\u2082\n\u22a2 \u2045\u2191f p, \u2191f q\u2046 \u2208 map f \u2045H\u2081, H\u2082\u2046\n[PROOFSTEP]\nrw [\u2190 map_commutatorElement]\n[GOAL]\ncase right.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf\u271d : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\nf : G \u2192* G'\np : G\nhp : p \u2208 \u2191H\u2081\nq : G\nhq : q \u2208 \u2191H\u2082\n\u22a2 \u2191f \u2045p, q\u2046 \u2208 map f \u2045H\u2081, H\u2082\u2046\n[PROOFSTEP]\nexact mem_map_of_mem _ (commutator_mem_commutator hp hq)\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046 = prod \u2045H\u2081, H\u2082\u2046 \u2045K\u2081, K\u2082\u2046\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046 \u2264 prod \u2045H\u2081, H\u2082\u2046 \u2045K\u2081, K\u2082\u2046\n[PROOFSTEP]\nrw [commutator_le]\n[GOAL]\ncase a\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 \u2200 (g\u2081 : G \u00d7 G'), g\u2081 \u2208 prod H\u2081 K\u2081 \u2192 \u2200 (g\u2082 : G \u00d7 G'), g\u2082 \u2208 prod H\u2082 K\u2082 \u2192 \u2045g\u2081, g\u2082\u2046 \u2208 prod \u2045H\u2081, H\u2082\u2046 \u2045K\u2081, K\u2082\u2046\n[PROOFSTEP]\nrintro \u27e8p\u2081, p\u2082\u27e9 \u27e8hp\u2081, hp\u2082\u27e9 \u27e8q\u2081, q\u2082\u27e9 \u27e8hq\u2081, hq\u2082\u27e9\n[GOAL]\ncase a.mk.intro.mk.intro\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\np\u2081 : G\np\u2082 : G'\nhp\u2081 : (p\u2081, p\u2082).fst \u2208 \u2191H\u2081.toSubmonoid\nhp\u2082 : (p\u2081, p\u2082).snd \u2208 \u2191K\u2081.toSubmonoid\nq\u2081 : G\nq\u2082 : G'\nhq\u2081 : (q\u2081, q\u2082).fst \u2208 \u2191H\u2082.toSubmonoid\nhq\u2082 : (q\u2081, q\u2082).snd \u2208 \u2191K\u2082.toSubmonoid\n\u22a2 \u2045(p\u2081, p\u2082), (q\u2081, q\u2082)\u2046 \u2208 prod \u2045H\u2081, H\u2082\u2046 \u2045K\u2081, K\u2082\u2046\n[PROOFSTEP]\nexact \u27e8commutator_mem_commutator hp\u2081 hq\u2081, commutator_mem_commutator hp\u2082 hq\u2082\u27e9\n[GOAL]\ncase a\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 prod \u2045H\u2081, H\u2082\u2046 \u2045K\u2081, K\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046\n[PROOFSTEP]\nrw [prod_le_iff]\n[GOAL]\ncase a\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inl G G') \u2045H\u2081, H\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046 \u2227\n    map (MonoidHom.inr G G') \u2045K\u2081, K\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.left\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inl G G') \u2045H\u2081, H\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046\n[PROOFSTEP]\nrw [map_commutator]\n[GOAL]\ncase a.left\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 \u2045map (MonoidHom.inl G G') H\u2081, map (MonoidHom.inl G G') H\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046\n[PROOFSTEP]\napply commutator_mono\n[GOAL]\ncase a.left.h\u2081\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inl G G') H\u2081 \u2264 prod H\u2081 K\u2081\n[PROOFSTEP]\nsimp [le_prod_iff, map_map, MonoidHom.fst_comp_inl, MonoidHom.snd_comp_inl, MonoidHom.fst_comp_inr,\n  MonoidHom.snd_comp_inr]\n[GOAL]\ncase a.left.h\u2082\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inl G G') H\u2082 \u2264 prod H\u2082 K\u2082\n[PROOFSTEP]\nsimp [le_prod_iff, map_map, MonoidHom.fst_comp_inl, MonoidHom.snd_comp_inl, MonoidHom.fst_comp_inr,\n  MonoidHom.snd_comp_inr]\n[GOAL]\ncase a.right\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inr G G') \u2045K\u2081, K\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046\n[PROOFSTEP]\nrw [map_commutator]\n[GOAL]\ncase a.right\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 \u2045map (MonoidHom.inr G G') K\u2081, map (MonoidHom.inr G G') K\u2082\u2046 \u2264 \u2045prod H\u2081 K\u2081, prod H\u2082 K\u2082\u2046\n[PROOFSTEP]\napply commutator_mono\n[GOAL]\ncase a.right.h\u2081\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inr G G') K\u2081 \u2264 prod H\u2081 K\u2081\n[PROOFSTEP]\nsimp [le_prod_iff, map_map, MonoidHom.fst_comp_inl, MonoidHom.snd_comp_inl, MonoidHom.fst_comp_inr,\n  MonoidHom.snd_comp_inr]\n[GOAL]\ncase a.right.h\u2082\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Group G'\ninst\u271d : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081\u271d K\u2082\u271d : Subgroup G\nK\u2081 K\u2082 : Subgroup G'\n\u22a2 map (MonoidHom.inr G G') K\u2082 \u2264 prod H\u2082 K\u2082\n[PROOFSTEP]\nsimp [le_prod_iff, map_map, MonoidHom.fst_comp_inl, MonoidHom.snd_comp_inl, MonoidHom.fst_comp_inr,\n  MonoidHom.snd_comp_inr]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\n\u22a2 \u2045pi Set.univ H, pi Set.univ K\u2046 = pi Set.univ fun i => \u2045H i, K i\u2046\n[PROOFSTEP]\nclassical\napply le_antisymm (commutator_pi_pi_le H K)\n\u00b7 rw [pi_le_iff]\n  intro i hi\n  rw [map_commutator]\n  apply commutator_mono <;>\n    \u00b7 rw [le_pi_iff]\n      intro j _hj\n      rintro _ \u27e8_, \u27e8x, hx, rfl\u27e9, rfl\u27e9\n      by_cases h : j = i\n      \u00b7 subst h\n        simpa using hx\n      \u00b7 simp [h, one_mem]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\n\u22a2 \u2045pi Set.univ H, pi Set.univ K\u2046 = pi Set.univ fun i => \u2045H i, K i\u2046\n[PROOFSTEP]\napply le_antisymm (commutator_pi_pi_le H K)\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\n\u22a2 (pi Set.univ fun i => \u2045H i, K i\u2046) \u2264 \u2045pi Set.univ H, pi Set.univ K\u2046\n[PROOFSTEP]\nrw [pi_le_iff]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\n\u22a2 \u2200 (i : \u03b7), map (MonoidHom.single (fun i => Gs i) i) \u2045H i, K i\u2046 \u2264 \u2045pi Set.univ H, pi Set.univ K\u2046\n[PROOFSTEP]\nintro i hi\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\n\u22a2 hi \u2208 map (MonoidHom.single (fun i => Gs i) i) \u2045H i, K i\u2046 \u2192 hi \u2208 \u2045pi Set.univ H, pi Set.univ K\u2046\n[PROOFSTEP]\nrw [map_commutator]\n[GOAL]\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\n\u22a2 hi \u2208 \u2045map (MonoidHom.single (fun i => Gs i) i) (H i), map (MonoidHom.single (fun i => Gs i) i) (K i)\u2046 \u2192\n    hi \u2208 \u2045pi Set.univ H, pi Set.univ K\u2046\n[PROOFSTEP]\napply commutator_mono\n[GOAL]\ncase h\u2081\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\n\u22a2 map (MonoidHom.single (fun i => Gs i) i) (H i) \u2264 pi Set.univ H\n[PROOFSTEP]\nrw [le_pi_iff]\n[GOAL]\ncase h\u2081\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\n\u22a2 \u2200 (i_1 : \u03b7),\n    i_1 \u2208 Set.univ \u2192 map (Pi.evalMonoidHom (fun i => Gs i) i_1) (map (MonoidHom.single (fun i => Gs i) i) (H i)) \u2264 H i_1\n[PROOFSTEP]\nintro j _hj\n[GOAL]\ncase h\u2081\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\n\u22a2 map (Pi.evalMonoidHom (fun i => Gs i) j) (map (MonoidHom.single (fun i => Gs i) i) (H i)) \u2264 H j\n[PROOFSTEP]\nrintro _ \u27e8_, \u27e8x, hx, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h\u2081.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) i) x) \u2208 H j\n[PROOFSTEP]\nby_cases h : j = i\n[GOAL]\ncase pos\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs i\nhx : x \u2208 \u2191(H i)\nh : j = i\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) i) x) \u2208 H j\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs j\nhx : x \u2208 \u2191(H j)\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) j) x) \u2208 H j\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase neg\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs i\nhx : x \u2208 \u2191(H i)\nh : \u00acj = i\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) i) x) \u2208 H j\n[PROOFSTEP]\nsimp [h, one_mem]\n[GOAL]\ncase h\u2082\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\n\u22a2 map (MonoidHom.single (fun i => Gs i) i) (K i) \u2264 pi Set.univ K\n[PROOFSTEP]\nrw [le_pi_iff]\n[GOAL]\ncase h\u2082\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\n\u22a2 \u2200 (i_1 : \u03b7),\n    i_1 \u2208 Set.univ \u2192 map (Pi.evalMonoidHom (fun i => Gs i) i_1) (map (MonoidHom.single (fun i => Gs i) i) (K i)) \u2264 K i_1\n[PROOFSTEP]\nintro j _hj\n[GOAL]\ncase h\u2082\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\n\u22a2 map (Pi.evalMonoidHom (fun i => Gs i) j) (map (MonoidHom.single (fun i => Gs i) i) (K i)) \u2264 K j\n[PROOFSTEP]\nrintro _ \u27e8_, \u27e8x, hx, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h\u2082.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs i\nhx : x \u2208 \u2191(K i)\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) i) x) \u2208 K j\n[PROOFSTEP]\nby_cases h : j = i\n[GOAL]\ncase pos\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs i\nhx : x \u2208 \u2191(K i)\nh : j = i\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) i) x) \u2208 K j\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs j\nhx : x \u2208 \u2191(K j)\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) j) x) \u2208 K j\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase neg\nG : Type u_1\nG' : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : MonoidHomClass F G G'\nf : F\ng\u2081 g\u2082 g\u2083 g : G\nH\u2081 H\u2082 H\u2083 K\u2081 K\u2082 : Subgroup G\n\u03b7 : Type u_4\ninst\u271d\u00b9 : Finite \u03b7\nGs : \u03b7 \u2192 Type u_5\ninst\u271d : (i : \u03b7) \u2192 Group (Gs i)\nH K : (i : \u03b7) \u2192 Subgroup (Gs i)\ni : \u03b7\nhi : (i : \u03b7) \u2192 Gs i\nj : \u03b7\n_hj : j \u2208 Set.univ\nx : Gs i\nhx : x \u2208 \u2191(K i)\nh : \u00acj = i\n\u22a2 \u2191(Pi.evalMonoidHom (fun i => Gs i) j) (\u2191(MonoidHom.single (fun i => Gs i) i) x) \u2208 K j\n[PROOFSTEP]\nsimp [h, one_mem]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Commutator", "llama_tokens": 14379, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.6893056167854461, "lm_q1q2_score": 0.5872558568149774}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : LinearOrderedField \ud835\udd5c\nG H : SimpleGraph \u03b1\n\u03b5 \u03b4 : \ud835\udd5c\nn : \u2115\ns : Finset \u03b1\nh\u03b5 : FarFromTriangleFree G \u03b5\nh : \u03b4 \u2264 \u03b5\n\u22a2 \u03b4 * \u2191(Fintype.card \u03b1 ^ 2) \u2264 \u03b5 * \u2191(Fintype.card \u03b1 ^ 2)\n[PROOFSTEP]\ngcongr\n[GOAL]\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nG H : SimpleGraph \u03b1\n\u03b5 \u03b4 : \ud835\udd5c\nn : \u2115\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nh\u2080 : FarFromTriangleFree G \u03b5\nh\u2081 : CliqueFree G 3\n\u22a2 \u03b5 \u2264 0\n[PROOFSTEP]\nhave := h\u2080 (empty_subset _)\n[GOAL]\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nG H : SimpleGraph \u03b1\n\u03b5 \u03b4 : \ud835\udd5c\nn : \u2115\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nh\u2080 : FarFromTriangleFree G \u03b5\nh\u2081 : CliqueFree G 3\nthis : (fun H => CliqueFree H 3) (deleteEdges G \u2191\u2205) \u2192 \u03b5 * \u2191(Fintype.card \u03b1 ^ 2) \u2264 \u2191(Finset.card \u2205)\n\u22a2 \u03b5 \u2264 0\n[PROOFSTEP]\nrw [coe_empty, Finset.card_empty, cast_zero, deleteEdges_empty_eq] at this \n[GOAL]\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nG H : SimpleGraph \u03b1\n\u03b5 \u03b4 : \ud835\udd5c\nn : \u2115\ns : Finset \u03b1\ninst\u271d : Nonempty \u03b1\nh\u2080 : FarFromTriangleFree G \u03b5\nh\u2081 : CliqueFree G 3\nthis : (fun H => CliqueFree H 3) G \u2192 \u03b5 * \u2191(Fintype.card \u03b1 ^ 2) \u2264 0\n\u22a2 \u03b5 \u2264 0\n[PROOFSTEP]\nexact nonpos_of_mul_nonpos_left (this h\u2081) (cast_pos.2 <| sq_pos_of_pos Fintype.card_pos)\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Triangle.Basic", "llama_tokens": 647, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619220634456, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.5872083105371021}}
{"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : MulZeroClass \u03b1\nx\u271d : ULift \u03b1\n\u22a2 \u2191Equiv.ulift (0 * x\u271d) = \u2191Equiv.ulift 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d : MulZeroClass \u03b1\nx\u271d : ULift \u03b1\n\u22a2 \u2191Equiv.ulift (x\u271d * 0) = \u2191Equiv.ulift 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Distrib \u03b1\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : ULift \u03b1\n\u22a2 \u2191Equiv.ulift (x\u271d\u00b2 * (x\u271d\u00b9 + x\u271d)) = \u2191Equiv.ulift (x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d)\n[PROOFSTEP]\nsimp [left_distrib]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Distrib \u03b1\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : ULift \u03b1\n\u22a2 \u2191Equiv.ulift ((x\u271d\u00b2 + x\u271d\u00b9) * x\u271d) = \u2191Equiv.ulift (x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d)\n[PROOFSTEP]\nsimp [right_distrib]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.ULift", "llama_tokens": 360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339837155239, "lm_q2_score": 0.6992544210587586, "lm_q1q2_score": 0.5871877006263636}}
{"text": "[GOAL]\nlo hi a : \u2115\nh : Nat.succ lo \u2264 a \u2227 a \u2264 hi\n\u22a2 Nat.succ (Nat.pred a) \u2264 hi\n[PROOFSTEP]\nrw [Nat.succ_pred_eq_of_pos]\n[GOAL]\nlo hi a : \u2115\nh : Nat.succ lo \u2264 a \u2227 a \u2264 hi\n\u22a2 a \u2264 hi\nlo hi a : \u2115 h : Nat.succ lo \u2264 a \u2227 a \u2264 hi \u22a2 0 < a\n[PROOFSTEP]\nexact h.right\n[GOAL]\nlo hi a : \u2115\nh : Nat.succ lo \u2264 a \u2227 a \u2264 hi\n\u22a2 0 < a\n[PROOFSTEP]\nexact lt_of_le_of_lt (Nat.zero_le lo) h.left\n", "meta": {"mathlib_filename": "Mathlib.Testing.SlimCheck.Gen", "llama_tokens": 213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.880797071719777, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.5860916774798524}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : MulSemiringAction M R\nm : M\n\u22a2 HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m)\n[PROOFSTEP]\nsuffices DistribMulAction.toAddMonoidHom R[X] m = (mapRingHom (MulSemiringAction.toRingHom M R m)).toAddMonoidHom\n  by\n  ext1 r\n  exact FunLike.congr_fun this r\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : MulSemiringAction M R\nm : M\nthis : DistribMulAction.toAddMonoidHom R[X] m = RingHom.toAddMonoidHom (mapRingHom (MulSemiringAction.toRingHom M R m))\n\u22a2 HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m)\n[PROOFSTEP]\next1 r\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : MulSemiringAction M R\nm : M\nthis : DistribMulAction.toAddMonoidHom R[X] m = RingHom.toAddMonoidHom (mapRingHom (MulSemiringAction.toRingHom M R m))\nr : R[X]\n\u22a2 m \u2022 r = map (MulSemiringAction.toRingHom M R m) r\n[PROOFSTEP]\nexact FunLike.congr_fun this r\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : MulSemiringAction M R\nm : M\n\u22a2 DistribMulAction.toAddMonoidHom R[X] m = RingHom.toAddMonoidHom (mapRingHom (MulSemiringAction.toRingHom M R m))\n[PROOFSTEP]\next n r : 2\n[GOAL]\ncase h.h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : MulSemiringAction M R\nm : M\nn : \u2115\nr : R\n\u22a2 \u2191(AddMonoidHom.comp (DistribMulAction.toAddMonoidHom R[X] m) (LinearMap.toAddMonoidHom (monomial n))) r =\n    \u2191(AddMonoidHom.comp (RingHom.toAddMonoidHom (mapRingHom (MulSemiringAction.toRingHom M R m)))\n          (LinearMap.toAddMonoidHom (monomial n)))\n      r\n[PROOFSTEP]\nchange m \u2022 monomial n r = map (MulSemiringAction.toRingHom M R m) (monomial n r)\n[GOAL]\ncase h.h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : MulSemiringAction M R\nm : M\nn : \u2115\nr : R\n\u22a2 m \u2022 \u2191(monomial n) r = map (MulSemiringAction.toRingHom M R m) (\u2191(monomial n) r)\n[PROOFSTEP]\nrw [Polynomial.map_monomial, Polynomial.smul_monomial, MulSemiringAction.toRingHom_apply]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nR : Type u_2\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : MulSemiringAction M R\nS : Type u_3\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : MulSemiringAction M S\nm : M\nf : S[X]\nx r : S\n\u22a2 eval (m \u2022 x) (m \u2022 \u2191C r) = m \u2022 eval x (\u2191C r)\n[PROOFSTEP]\nrw [smul_C, eval_C, eval_C]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nR : Type u_2\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : MulSemiringAction M R\nS : Type u_3\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : MulSemiringAction M S\nm : M\nf\u271d : S[X]\nx : S\nf g : S[X]\nihf : eval (m \u2022 x) (m \u2022 f) = m \u2022 eval x f\nihg : eval (m \u2022 x) (m \u2022 g) = m \u2022 eval x g\n\u22a2 eval (m \u2022 x) (m \u2022 (f + g)) = m \u2022 eval x (f + g)\n[PROOFSTEP]\nrw [smul_add, eval_add, ihf, ihg, eval_add, smul_add]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nR : Type u_2\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : MulSemiringAction M R\nS : Type u_3\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : MulSemiringAction M S\nm : M\nf : S[X]\nx : S\nn : \u2115\nr : S\nx\u271d : eval (m \u2022 x) (m \u2022 (\u2191C r * X ^ n)) = m \u2022 eval x (\u2191C r * X ^ n)\n\u22a2 eval (m \u2022 x) (m \u2022 (\u2191C r * X ^ (n + 1))) = m \u2022 eval x (\u2191C r * X ^ (n + 1))\n[PROOFSTEP]\nrw [smul_mul', smul_pow', smul_C, smul_X, eval_mul, eval_C, eval_pow, eval_X, eval_mul, eval_C, eval_pow, eval_X,\n  smul_mul', smul_pow']\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : Monoid M\nR : Type u_2\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : MulSemiringAction M R\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : MulSemiringAction M S\nG : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : MulSemiringAction G S\ng : G\nf : S[X]\nx : S\n\u22a2 eval (g \u2022 x) f = g \u2022 eval x (g\u207b\u00b9 \u2022 f)\n[PROOFSTEP]\nrw [\u2190 smul_eval_smul, smul_inv_smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2076 : Monoid M\nR : Type u_2\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : MulSemiringAction M R\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : MulSemiringAction M S\nG : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : MulSemiringAction G S\ng : G\nf : S[X]\nx : S\n\u22a2 eval x (g \u2022 f) = g \u2022 eval (g\u207b\u00b9 \u2022 x) f\n[PROOFSTEP]\nrw [\u2190 smul_eval_smul, smul_inv_smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Fintype G\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : MulSemiringAction G R\nx : R\n\u22a2 \u2191\u2191\u2191(aeval x) (X - \u2191C (ofQuotientStabilizer G x \u21911)) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Fintype G\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : MulSemiringAction G R\nx : R\ng : G\ng' : G \u29f8 stabilizer G x\n\u22a2 g \u2022 (X - \u2191C (ofQuotientStabilizer G x g')) = X - \u2191C (ofQuotientStabilizer G x ((fun x_1 x_2 => x_1 \u2022 x_2) g g'))\n[PROOFSTEP]\nrw [ofQuotientStabilizer_smul, smul_sub, Polynomial.smul_X, Polynomial.smul_C]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nG : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Fintype G\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : MulSemiringAction G R\nx : R\ng : G\nn : \u2115\n\u22a2 g \u2022 Polynomial.coeff (prodXSubSmul G R x) n = Polynomial.coeff (prodXSubSmul G R x) n\n[PROOFSTEP]\nrw [\u2190 Polynomial.coeff_smul, prodXSubSmul.smul]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nP : Type u_2\ninst\u271d\u00b3 : CommSemiring P\ninst\u271d\u00b2 : MulSemiringAction M P\nQ : Type u_3\ninst\u271d\u00b9 : CommSemiring Q\ninst\u271d : MulSemiringAction M Q\ng : P \u2192+*[M] Q\nm : M\np : P[X]\nb : P\n\u22a2 map (\u2191g) (m \u2022 \u2191C b) = m \u2022 map (\u2191g) (\u2191C b)\n[PROOFSTEP]\nrw [smul_C, map_C, coe_fn_coe, g.map_smul, map_C, coe_fn_coe, smul_C]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nP : Type u_2\ninst\u271d\u00b3 : CommSemiring P\ninst\u271d\u00b2 : MulSemiringAction M P\nQ : Type u_3\ninst\u271d\u00b9 : CommSemiring Q\ninst\u271d : MulSemiringAction M Q\ng : P \u2192+*[M] Q\nm : M\np\u271d p q : P[X]\nihp : map (\u2191g) (m \u2022 p) = m \u2022 map (\u2191g) p\nihq : map (\u2191g) (m \u2022 q) = m \u2022 map (\u2191g) q\n\u22a2 map (\u2191g) (m \u2022 (p + q)) = m \u2022 map (\u2191g) (p + q)\n[PROOFSTEP]\nrw [smul_add, Polynomial.map_add, ihp, ihq, Polynomial.map_add, smul_add]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Monoid M\nP : Type u_2\ninst\u271d\u00b3 : CommSemiring P\ninst\u271d\u00b2 : MulSemiringAction M P\nQ : Type u_3\ninst\u271d\u00b9 : CommSemiring Q\ninst\u271d : MulSemiringAction M Q\ng : P \u2192+*[M] Q\nm : M\np : P[X]\nn : \u2115\nb : P\nx\u271d : map (\u2191g) (m \u2022 (\u2191C b * X ^ n)) = m \u2022 map (\u2191g) (\u2191C b * X ^ n)\n\u22a2 map (\u2191g) (m \u2022 (\u2191C b * X ^ (n + 1))) = m \u2022 map (\u2191g) (\u2191C b * X ^ (n + 1))\n[PROOFSTEP]\nrw [smul_mul', smul_C, smul_pow', smul_X, Polynomial.map_mul, map_C, Polynomial.map_pow, map_X, coe_fn_coe, g.map_smul,\n  Polynomial.map_mul, map_C, Polynomial.map_pow, map_X, smul_mul', smul_C, smul_pow', smul_X, coe_fn_coe]\n  -- porting note: added `.toRingHom`\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Polynomial.GroupRingAction", "llama_tokens": 3130, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711680567799, "lm_q2_score": 0.6893056231680121, "lm_q1q2_score": 0.5858899056722218}}
{"text": "[GOAL]\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\n\u22a2 WellFounded (Upto.GT p)\n[PROOFSTEP]\nsuffices Upto.GT p = InvImage (\u00b7 < \u00b7) fun y : Nat.Upto p => x - y.val\n  by\n  rw [this]\n  exact (measure _).wf\n[GOAL]\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\nthis : Upto.GT p = InvImage (fun x x_1 => x < x_1) fun y => x - \u2191y\n\u22a2 WellFounded (Upto.GT p)\n[PROOFSTEP]\nrw [this]\n[GOAL]\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\nthis : Upto.GT p = InvImage (fun x x_1 => x < x_1) fun y => x - \u2191y\n\u22a2 WellFounded (InvImage (fun x x_1 => x < x_1) fun y => x - \u2191y)\n[PROOFSTEP]\nexact (measure _).wf\n[GOAL]\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\n\u22a2 Upto.GT p = InvImage (fun x x_1 => x < x_1) fun y => x - \u2191y\n[PROOFSTEP]\next \u27e8a, ha\u27e9 \u27e8b, _\u27e9\n[GOAL]\ncase h.mk.h.mk.a\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\na : \u2115\nha : \u2200 (j : \u2115), j < a \u2192 \u00acp j\nb : \u2115\nproperty\u271d : \u2200 (j : \u2115), j < b \u2192 \u00acp j\n\u22a2 Upto.GT p { val := a, property := ha } { val := b, property := property\u271d } \u2194\n    InvImage (fun x x_1 => x < x_1) (fun y => x - \u2191y) { val := a, property := ha } { val := b, property := property\u271d }\n[PROOFSTEP]\ndsimp [InvImage, Upto.GT]\n[GOAL]\ncase h.mk.h.mk.a\np : \u2115 \u2192 Prop\nx : \u2115\nh : p x\na : \u2115\nha : \u2200 (j : \u2115), j < a \u2192 \u00acp j\nb : \u2115\nproperty\u271d : \u2200 (j : \u2115), j < b \u2192 \u00acp j\n\u22a2 a > b \u2194 x - a < x - b\n[PROOFSTEP]\nrw [tsub_lt_tsub_iff_left_of_le (le_of_not_lt fun h' => ha _ h' h)]\n[GOAL]\np : \u2115 \u2192 Prop\nx : Upto p\nh : \u00acp \u2191x\nj : \u2115\nh' : j < Nat.succ \u2191x\n\u22a2 \u00acp j\n[PROOFSTEP]\nrcases Nat.lt_succ_iff_lt_or_eq.1 h' with (h' | rfl) <;> [exact x.2 _ h'; exact h]\n[GOAL]\np : \u2115 \u2192 Prop\nx : Upto p\nh : \u00acp \u2191x\nj : \u2115\nh' : j < Nat.succ \u2191x\n\u22a2 \u00acp j\n[PROOFSTEP]\nrcases Nat.lt_succ_iff_lt_or_eq.1 h' with (h' | rfl)\n[GOAL]\ncase inl\np : \u2115 \u2192 Prop\nx : Upto p\nh : \u00acp \u2191x\nj : \u2115\nh'\u271d : j < Nat.succ \u2191x\nh' : j < \u2191x\n\u22a2 \u00acp j\n[PROOFSTEP]\nexact x.2 _ h'\n[GOAL]\ncase inr\np : \u2115 \u2192 Prop\nx : Upto p\nh : \u00acp \u2191x\nh' : \u2191x < Nat.succ \u2191x\n\u22a2 \u00acp \u2191x\n[PROOFSTEP]\nexact h\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Upto", "llama_tokens": 1002, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.837619947119304, "lm_q2_score": 0.6992544085240401, "lm_q1q2_score": 0.5857094406908466}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\ni : \u03b9\na : A i\nn : \u2115\n\u22a2 mk i a ^ n = mk (n \u2022 i) (GMonoid.gnpow n a)\n[PROOFSTEP]\nmatch n with\n| 0 =>\n  rw [pow_zero]\n  exact (GMonoid.gnpow_zero' \u27e8_, a\u27e9).symm\n| n + 1 =>\n  rw [pow_succ, mk_pow a n, mk_mul_mk]\n  exact (GMonoid.gnpow_succ' n \u27e8_, a\u27e9).symm\n[GOAL]\n\u03b9 : Type u_1\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\ni : \u03b9\na : A i\nn : \u2115\n\u22a2 mk i a ^ 0 = mk (0 \u2022 i) (GMonoid.gnpow 0 a)\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\n\u03b9 : Type u_1\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\ni : \u03b9\na : A i\nn : \u2115\n\u22a2 1 = mk (0 \u2022 i) (GMonoid.gnpow 0 a)\n[PROOFSTEP]\nexact (GMonoid.gnpow_zero' \u27e8_, a\u27e9).symm\n[GOAL]\n\u03b9 : Type u_1\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\ni : \u03b9\na : A i\nn\u271d n : \u2115\n\u22a2 mk i a ^ (n + 1) = mk ((n + 1) \u2022 i) (GMonoid.gnpow (n + 1) a)\n[PROOFSTEP]\nrw [pow_succ, mk_pow a n, mk_mul_mk]\n[GOAL]\n\u03b9 : Type u_1\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\ni : \u03b9\na : A i\nn\u271d n : \u2115\n\u22a2 mk (i + n \u2022 i) (GMul.mul a (GMonoid.gnpow n a)) = mk ((n + 1) \u2022 i) (GMonoid.gnpow (n + 1) a)\n[PROOFSTEP]\nexact (GMonoid.gnpow_succ' n \u27e8_, a\u27e9).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedMonoid.GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\n\u22a2 dProdIndex l f\u03b9 = sum (map f\u03b9 l)\n[PROOFSTEP]\nmatch l with\n| [] => simp\n| head :: tail => simp [List.dProdIndex_eq_map_sum tail f\u03b9]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedMonoid.GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\n\u22a2 dProdIndex [] f\u03b9 = sum (map f\u03b9 [])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GradedMonoid.GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nhead : \u03b1\ntail : List \u03b1\n\u22a2 dProdIndex (head :: tail) f\u03b9 = sum (map f\u03b9 (head :: tail))\n[PROOFSTEP]\nsimp [List.dProdIndex_eq_map_sum tail f\u03b9]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 A (f\u03b9 a)\n\u22a2 mk (List.dProdIndex l f\u03b9) (List.dProd l f\u03b9 fA) = List.prod (List.map (fun a => mk (f\u03b9 a) (fA a)) l)\n[PROOFSTEP]\nmatch l with\n| [] => simp; rfl\n| head :: tail => simp [\u2190 GradedMonoid.mk_list_dProd tail _ _, GradedMonoid.mk_mul_mk, List.prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 A (f\u03b9 a)\n\u22a2 mk (List.dProdIndex [] f\u03b9) (List.dProd [] f\u03b9 fA) = List.prod (List.map (fun a => mk (f\u03b9 a) (fA a)) [])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 A (f\u03b9 a)\n\u22a2 mk 0 GOne.one = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 A (f\u03b9 a)\nhead : \u03b1\ntail : List \u03b1\n\u22a2 mk (List.dProdIndex (head :: tail) f\u03b9) (List.dProd (head :: tail) f\u03b9 fA) =\n    List.prod (List.map (fun a => mk (f\u03b9 a) (fA a)) (head :: tail))\n[PROOFSTEP]\nsimp [\u2190 GradedMonoid.mk_list_dProd tail _ _, GradedMonoid.mk_mul_mk, List.prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nl : List \u03b1\nf : \u03b1 \u2192 GradedMonoid A\n\u22a2 List.prod (List.map f l) =\n    mk (List.dProdIndex l fun i => (f i).fst) (List.dProd l (fun i => (f i).fst) fun i => (f i).snd)\n[PROOFSTEP]\nrw [GradedMonoid.mk_list_dProd, GradedMonoid.mk]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nl : List \u03b1\nf : \u03b1 \u2192 GradedMonoid A\n\u22a2 List.prod (List.map f l) = List.prod (List.map (fun a => { fst := (f a).fst, snd := (f a).snd }) l)\n[PROOFSTEP]\nsimp_rw [Sigma.eta]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nA : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : GMonoid A\nn : \u2115\nf : Fin n \u2192 GradedMonoid A\n\u22a2 List.prod (List.ofFn f) =\n    mk (List.dProdIndex (List.finRange n) fun i => (f i).fst)\n      (List.dProd (List.finRange n) (fun i => (f i).fst) fun i => (f i).snd)\n[PROOFSTEP]\nrw [List.ofFn_eq_map, GradedMonoid.list_prod_map_eq_dProd]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Monoid R\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : \u03b1 \u2192 R\n\u22a2 dProd l f\u03b9 fA = prod (map fA l)\n[PROOFSTEP]\nmatch l with\n| [] =>\n  rw [List.dProd_nil, List.map_nil, List.prod_nil]\n  rfl\n| head :: tail =>\n  rw [List.dProd_cons, List.map_cons, List.prod_cons, List.dProd_monoid tail _ _]\n  rfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Monoid R\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : \u03b1 \u2192 R\n\u22a2 dProd [] f\u03b9 fA = prod (map fA [])\n[PROOFSTEP]\nrw [List.dProd_nil, List.map_nil, List.prod_nil]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Monoid R\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : \u03b1 \u2192 R\n\u22a2 GradedMonoid.GOne.one = 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Monoid R\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : \u03b1 \u2192 R\nhead : \u03b1\ntail : List \u03b1\n\u22a2 dProd (head :: tail) f\u03b9 fA = prod (map fA (head :: tail))\n[PROOFSTEP]\nrw [List.dProd_cons, List.map_cons, List.prod_cons, List.dProd_monoid tail _ _]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : AddMonoid \u03b9\ninst\u271d : Monoid R\nl : List \u03b1\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : \u03b1 \u2192 R\nhead : \u03b1\ntail : List \u03b1\n\u22a2 GradedMonoid.GMul.mul (fA head) (prod (map fA tail)) = fA head * prod (map fA tail)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn : \u2115\nr : R\ni : \u03b9\nh : r \u2208 A i\n\u22a2 r ^ n \u2208 A (n \u2022 i)\n[PROOFSTEP]\nmatch n with\n| 0 =>\n  rw [pow_zero, zero_nsmul]\n  exact one_mem_graded _\n| n + 1 =>\n  rw [pow_succ', succ_nsmul']\n  exact mul_mem_graded (pow_mem_graded n h) h\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn : \u2115\nr : R\ni : \u03b9\nh : r \u2208 A i\n\u22a2 r ^ 0 \u2208 A (0 \u2022 i)\n[PROOFSTEP]\nrw [pow_zero, zero_nsmul]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn : \u2115\nr : R\ni : \u03b9\nh : r \u2208 A i\n\u22a2 1 \u2208 A 0\n[PROOFSTEP]\nexact one_mem_graded _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn\u271d : \u2115\nr : R\ni : \u03b9\nh : r \u2208 A i\nn : \u2115\n\u22a2 r ^ (n + 1) \u2208 A ((n + 1) \u2022 i)\n[PROOFSTEP]\nrw [pow_succ', succ_nsmul']\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn\u271d : \u2115\nr : R\ni : \u03b9\nh : r \u2208 A i\nn : \u2115\n\u22a2 r ^ n * r \u2208 A (n \u2022 i + i)\n[PROOFSTEP]\nexact mul_mem_graded (pow_mem_graded n h) h\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\n\u03b9' : Type u_4\nl : List \u03b9'\ni : \u03b9' \u2192 \u03b9\nr : \u03b9' \u2192 R\nh : \u2200 (j : \u03b9'), j \u2208 l \u2192 r j \u2208 A (i j)\n\u22a2 List.prod (List.map r l) \u2208 A (List.sum (List.map i l))\n[PROOFSTEP]\nmatch l with\n| [] =>\n  rw [List.map_nil, List.map_nil, List.prod_nil, List.sum_nil]\n  exact one_mem_graded _\n| head :: tail =>\n  rw [List.map_cons, List.map_cons, List.prod_cons, List.sum_cons]\n  exact\n    mul_mem_graded (h _ <| List.mem_cons_self _ _)\n      (list_prod_map_mem_graded tail _ _ <| fun j hj => h _ <| List.mem_cons_of_mem _ hj)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\n\u03b9' : Type u_4\nl : List \u03b9'\ni : \u03b9' \u2192 \u03b9\nr : \u03b9' \u2192 R\nh : \u2200 (j : \u03b9'), j \u2208 [] \u2192 r j \u2208 A (i j)\n\u22a2 List.prod (List.map r []) \u2208 A (List.sum (List.map i []))\n[PROOFSTEP]\nrw [List.map_nil, List.map_nil, List.prod_nil, List.sum_nil]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\n\u03b9' : Type u_4\nl : List \u03b9'\ni : \u03b9' \u2192 \u03b9\nr : \u03b9' \u2192 R\nh : \u2200 (j : \u03b9'), j \u2208 [] \u2192 r j \u2208 A (i j)\n\u22a2 1 \u2208 A 0\n[PROOFSTEP]\nexact one_mem_graded _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\n\u03b9' : Type u_4\nl : List \u03b9'\ni : \u03b9' \u2192 \u03b9\nr : \u03b9' \u2192 R\nhead : \u03b9'\ntail : List \u03b9'\nh : \u2200 (j : \u03b9'), j \u2208 head :: tail \u2192 r j \u2208 A (i j)\n\u22a2 List.prod (List.map r (head :: tail)) \u2208 A (List.sum (List.map i (head :: tail)))\n[PROOFSTEP]\nrw [List.map_cons, List.map_cons, List.prod_cons, List.sum_cons]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\n\u03b9' : Type u_4\nl : List \u03b9'\ni : \u03b9' \u2192 \u03b9\nr : \u03b9' \u2192 R\nhead : \u03b9'\ntail : List \u03b9'\nh : \u2200 (j : \u03b9'), j \u2208 head :: tail \u2192 r j \u2208 A (i j)\n\u22a2 r head * List.prod (List.map r tail) \u2208 A (i head + List.sum (List.map i tail))\n[PROOFSTEP]\nexact\n  mul_mem_graded (h _ <| List.mem_cons_self _ _)\n    (list_prod_map_mem_graded tail _ _ <| fun j hj => h _ <| List.mem_cons_of_mem _ hj)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn : \u2115\ni : Fin n \u2192 \u03b9\nr : Fin n \u2192 R\nh : \u2200 (j : Fin n), r j \u2208 A (i j)\n\u22a2 List.prod (List.ofFn r) \u2208 A (List.sum (List.ofFn i))\n[PROOFSTEP]\nrw [List.ofFn_eq_map, List.ofFn_eq_map]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nn : \u2115\ni : Fin n \u2192 \u03b9\nr : Fin n \u2192 R\nh : \u2200 (j : Fin n), r j \u2208 A (i j)\n\u22a2 List.prod (List.map r (List.finRange n)) \u2208 A (List.sum (List.map i (List.finRange n)))\n[PROOFSTEP]\nexact list_prod_map_mem_graded _ _ _ fun _ _ => h _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 { x // x \u2208 A (f\u03b9 a) }\nl : List \u03b1\n\u22a2 \u2191(List.dProd l f\u03b9 fA) = List.prod (List.map (fun a => \u2191(fA a)) l)\n[PROOFSTEP]\nmatch l with\n| [] => rw [List.dProd_nil, coe_gOne, List.map_nil, List.prod_nil]\n| head :: tail => rw [List.dProd_cons, coe_gMul, List.map_cons, List.prod_cons, SetLike.coe_list_dProd _ _ _ tail]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 { x // x \u2208 A (f\u03b9 a) }\nl : List \u03b1\n\u22a2 \u2191(List.dProd [] f\u03b9 fA) = List.prod (List.map (fun a => \u2191(fA a)) [])\n[PROOFSTEP]\nrw [List.dProd_nil, coe_gOne, List.map_nil, List.prod_nil]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\n\u03b1 : Type u_3\nS : Type u_4\ninst\u271d\u00b3 : SetLike S R\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : AddMonoid \u03b9\nA : \u03b9 \u2192 S\ninst\u271d : GradedMonoid A\nf\u03b9 : \u03b1 \u2192 \u03b9\nfA : (a : \u03b1) \u2192 { x // x \u2208 A (f\u03b9 a) }\nl : List \u03b1\nhead : \u03b1\ntail : List \u03b1\n\u22a2 \u2191(List.dProd (head :: tail) f\u03b9 fA) = List.prod (List.map (fun a => \u2191(fA a)) (head :: tail))\n[PROOFSTEP]\nrw [List.dProd_cons, coe_gMul, List.map_cons, List.prod_cons, SetLike.coe_list_dProd _ _ _ tail]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GradedMonoid", "llama_tokens": 5744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577681122619883, "lm_q2_score": 0.682573740869499, "lm_q1q2_score": 0.5854899891852338}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d : Group G\nf : \u03b1 \u2192 G\nrels : Set (FreeGroup \u03b1)\nh : \u2200 (r : FreeGroup \u03b1), r \u2208 rels \u2192 \u2191(\u2191FreeGroup.lift f) r = 1\ng : PresentedGroup rels \u2192* G\nhg : \u2200 (x : \u03b1), \u2191g (PresentedGroup.of x) = f x\n\u22a2 \u2200 {x : PresentedGroup rels}, \u2191g x = \u2191(toGroup h) x\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d : Group G\nf : \u03b1 \u2192 G\nrels : Set (FreeGroup \u03b1)\nh : \u2200 (r : FreeGroup \u03b1), r \u2208 rels \u2192 \u2191(\u2191FreeGroup.lift f) r = 1\ng : PresentedGroup rels \u2192* G\nhg : \u2200 (x : \u03b1), \u2191g (PresentedGroup.of x) = f x\nx : PresentedGroup rels\n\u22a2 \u2191g x = \u2191(toGroup h) x\n[PROOFSTEP]\nrefine' QuotientGroup.induction_on x _\n[GOAL]\n\u03b1 : Type u_1\nG : Type u_2\ninst\u271d : Group G\nf : \u03b1 \u2192 G\nrels : Set (FreeGroup \u03b1)\nh : \u2200 (r : FreeGroup \u03b1), r \u2208 rels \u2192 \u2191(\u2191FreeGroup.lift f) r = 1\ng : PresentedGroup rels \u2192* G\nhg : \u2200 (x : \u03b1), \u2191g (PresentedGroup.of x) = f x\nx : PresentedGroup rels\n\u22a2 \u2200 (z : FreeGroup \u03b1), \u2191g \u2191z = \u2191(toGroup h) \u2191z\n[PROOFSTEP]\nexact fun _ \u21a6 FreeGroup.lift.unique (g.comp (QuotientGroup.mk' _)) hg\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.PresentedGroup", "llama_tokens": 492, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.86153820232079, "lm_q2_score": 0.679178692681616, "lm_q1q2_score": 0.5851383899475038}}
{"text": "[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u22a2 \u2200 (a b c : F \u27f6 G), a + b + c = a + (b + c)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\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 w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d b\u271d c\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (a\u271d + b\u271d + c\u271d) x\u271d = NatTrans.app (a\u271d + (b\u271d + c\u271d)) x\u271d\n[PROOFSTEP]\napply add_assoc\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u22a2 \u2200 (a : F \u27f6 G), 0 + a = a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d : F \u27f6 G\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (0 + a\u271d) x\u271d = NatTrans.app a\u271d x\u271d\n[PROOFSTEP]\napply zero_add\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u22a2 \u2200 (a : F \u27f6 G), a + 0 = a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d : F \u27f6 G\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (a\u271d + 0) x\u271d = NatTrans.app a\u271d x\u271d\n[PROOFSTEP]\napply add_zero\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u22a2 \u2200 (a b : F \u27f6 G), a - b = a + -b\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d b\u271d : F \u27f6 G\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d b\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (a\u271d - b\u271d) x\u271d = NatTrans.app (a\u271d + -b\u271d) x\u271d\n[PROOFSTEP]\napply sub_eq_add_neg\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u22a2 \u2200 (a : F \u27f6 G), -a + a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d : F \u27f6 G\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (-a\u271d + a\u271d) x\u271d = NatTrans.app 0 x\u271d\n[PROOFSTEP]\napply add_left_neg\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u22a2 \u2200 (a b : F \u27f6 G), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d b\u271d : F \u27f6 G\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\na\u271d b\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (a\u271d + b\u271d) x\u271d = NatTrans.app (b\u271d + a\u271d) x\u271d\n[PROOFSTEP]\napply add_comm\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\n\u22a2 \u2200 (P Q R : C \u2964 D) (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_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nP\u271d Q\u271d R\u271d : C \u2964 D\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 w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nP\u271d Q\u271d R\u271d : C \u2964 D\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\nx\u271d : C\n\u22a2 NatTrans.app ((f\u271d + f'\u271d) \u226b g\u271d) x\u271d = NatTrans.app (f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\napply add_comp\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\n\u22a2 \u2200 (P Q R : C \u2964 D) (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_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nP\u271d Q\u271d R\u271d : C \u2964 D\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 w.h\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.38, u_1} C\ninst\u271d\u00b9 : Category.{?u.42, u_2} D\ninst\u271d : Preadditive D\nP\u271d Q\u271d R\u271d : C \u2964 D\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\nx\u271d : C\n\u22a2 NatTrans.app (f\u271d \u226b (g\u271d + g'\u271d)) x\u271d = NatTrans.app (f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d) x\u271d\n[PROOFSTEP]\napply comp_add\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{u_5, u_1} C\ninst\u271d\u00b9 : Category.{u_4, u_2} D\ninst\u271d : Preadditive D\nF G : C \u2964 D\n\u03b9 : Type u_3\ns : Finset \u03b9\nX : C\n\u03b1 : \u03b9 \u2192 (F \u27f6 G)\n\u22a2 app (\u2211 i in s, \u03b1 i) X = \u2211 i in s, app (\u03b1 i) X\n[PROOFSTEP]\nsimp only [\u2190 appHom_apply, map_sum]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.FunctorCategory", "llama_tokens": 3531, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289387914176259, "lm_q2_score": 0.7057850154599562, "lm_q1q2_score": 0.5850525777160466}}
{"text": "[GOAL]\nn a b n' a' b' : \u2115\nH : IsFibAux n a b\nhn : 2 * n = n'\nh1 : a * (2 * b - a) = a'\nh2 : a * a + b * b = b'\n\u22a2 fib n' = a'\n[PROOFSTEP]\nrw [\u2190 hn, fib_two_mul, H.1, H.2, \u2190 h1]\n[GOAL]\nn a b n' a' b' : \u2115\nH : IsFibAux n a b\nhn : 2 * n = n'\nh1 : a * (2 * b - a) = a'\nh2 : a * a + b * b = b'\n\u22a2 fib (n' + 1) = b'\n[PROOFSTEP]\nrw [\u2190 hn, fib_two_mul_add_one, H.1, H.2, pow_two, pow_two, add_comm, h2]\n[GOAL]\nn a b n' a' b' : \u2115\nH : IsFibAux n a b\nhn : 2 * n + 1 = n'\nh1 : a * a + b * b = a'\nh2 : b * (2 * a + b) = b'\n\u22a2 fib n' = a'\n[PROOFSTEP]\nrw [\u2190 hn, fib_two_mul_add_one, H.1, H.2, pow_two, pow_two, add_comm, h1]\n[GOAL]\nn a b n' a' b' : \u2115\nH : IsFibAux n a b\nhn : 2 * n + 1 = n'\nh1 : a * a + b * b = a'\nh2 : b * (2 * a + b) = b'\n\u22a2 fib (n' + 1) = b'\n[PROOFSTEP]\nrw [\u2190 hn, fib_two_mul_add_two, H.1, H.2, h2]\n", "meta": {"mathlib_filename": "Mathlib.Tactic.NormNum.NatFib", "llama_tokens": 486, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.584419845614867}}
{"text": "[GOAL]\n\u22a2 \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n    x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n      (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 hxy\n[GOAL]\ncase mk\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) (x, y) \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, mem_Ioi, sqrt_pos, mem_Ioo, Complex.neg_pi_lt_arg, true_and_iff,\n  Complex.arg_lt_pi_iff]\n[GOAL]\ncase mk\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 0 < x ^ 2 + y ^ 2 \u2227 (0 \u2264 (\u2191Complex.equivRealProd.symm (x, y)).re \u2228 (\u2191Complex.equivRealProd.symm (x, y)).im \u2260 0)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.left\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 0 < x ^ 2 + y ^ 2\n[PROOFSTEP]\ncases' hxy with hxy hxy\n[GOAL]\ncase mk.left.inl\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst}\n\u22a2 0 < x ^ 2 + y ^ 2\n[PROOFSTEP]\ndsimp at hxy \n[GOAL]\ncase mk.left.inl\nx y : \u211d\nhxy : 0 < x\n\u22a2 0 < x ^ 2 + y ^ 2\n[PROOFSTEP]\nlinarith [sq_pos_of_ne_zero _ hxy.ne', sq_nonneg y]\n[GOAL]\ncase mk.left.inr\nx y : \u211d\nhxy : (x, y) \u2208 {q | q.snd \u2260 0}\n\u22a2 0 < x ^ 2 + y ^ 2\n[PROOFSTEP]\nlinarith [sq_nonneg x, sq_pos_of_ne_zero _ hxy]\n[GOAL]\ncase mk.right\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 0 \u2264 (\u2191Complex.equivRealProd.symm (x, y)).re \u2228 (\u2191Complex.equivRealProd.symm (x, y)).im \u2260 0\n[PROOFSTEP]\ncases' hxy with hxy hxy\n[GOAL]\ncase mk.right.inl\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst}\n\u22a2 0 \u2264 (\u2191Complex.equivRealProd.symm (x, y)).re \u2228 (\u2191Complex.equivRealProd.symm (x, y)).im \u2260 0\n[PROOFSTEP]\nexact Or.inl (le_of_lt hxy)\n[GOAL]\ncase mk.right.inr\nx y : \u211d\nhxy : (x, y) \u2208 {q | q.snd \u2260 0}\n\u22a2 0 \u2264 (\u2191Complex.equivRealProd.symm (x, y)).re \u2228 (\u2191Complex.equivRealProd.symm (x, y)).im \u2260 0\n[PROOFSTEP]\nexact Or.inr hxy\n[GOAL]\n\u22a2 \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n    x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n[PROOFSTEP]\nrintro \u27e8r, \u03b8\u27e9 \u27e8hr, h\u03b8\u27e9\n[GOAL]\ncase mk.intro\nr \u03b8 : \u211d\nhr : (r, \u03b8).fst \u2208 Ioi 0\nh\u03b8 : (r, \u03b8).snd \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n[PROOFSTEP]\ndsimp at hr h\u03b8 \n[GOAL]\ncase mk.intro\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n[PROOFSTEP]\nrcases eq_or_ne \u03b8 0 with (rfl | h'\u03b8)\n[GOAL]\ncase mk.intro.inl\nr : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : 0 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, 0) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n[PROOFSTEP]\nsimpa using hr\n[GOAL]\ncase mk.intro.inr\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\nh'\u03b8 : \u03b8 \u2260 0\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n[PROOFSTEP]\nright\n[GOAL]\ncase mk.intro.inr.h\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\nh'\u03b8 : \u03b8 \u2260 0\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8) \u2208 {q | q.snd \u2260 0}\n[PROOFSTEP]\nsimp at hr \n[GOAL]\ncase mk.intro.inr.h\nr \u03b8 : \u211d\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\nh'\u03b8 : \u03b8 \u2260 0\nhr : 0 < r\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8) \u2208 {q | q.snd \u2260 0}\n[PROOFSTEP]\nsimpa only [ne_of_gt hr, Ne.def, mem_setOf_eq, mul_eq_zero, false_or_iff, sin_eq_zero_iff_of_lt_of_lt h\u03b8.1 h\u03b8.2] using\n  h'\u03b8\n[GOAL]\n\u22a2 \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n    x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n      (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n          ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x) =\n        x\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 _\n[GOAL]\ncase mk\nx y : \u211d\na\u271d : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n      ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) (x, y)) =\n    (x, y)\n[PROOFSTEP]\nhave A : sqrt (x ^ 2 + y ^ 2) = Complex.abs (x + y * Complex.I) := by\n  simp [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re, Complex.ofReal_re,\n    Complex.mul_re, Complex.I_re, mul_zero, Complex.ofReal_im, Complex.I_im, sub_self, add_zero, Complex.add_im,\n    Complex.mul_im, mul_one, zero_add]\n[GOAL]\nx y : \u211d\na\u271d : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 sqrt (x ^ 2 + y ^ 2) = \u2191Complex.abs (\u2191x + \u2191y * Complex.I)\n[PROOFSTEP]\nsimp [Complex.abs_def, Complex.normSq, pow_two, MonoidWithZeroHom.coe_mk, Complex.add_re, Complex.ofReal_re,\n  Complex.mul_re, Complex.I_re, mul_zero, Complex.ofReal_im, Complex.I_im, sub_self, add_zero, Complex.add_im,\n  Complex.mul_im, mul_one, zero_add]\n[GOAL]\ncase mk\nx y : \u211d\na\u271d : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\nA : sqrt (x ^ 2 + y ^ 2) = \u2191Complex.abs (\u2191x + \u2191y * Complex.I)\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n      ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) (x, y)) =\n    (x, y)\n[PROOFSTEP]\nhave Z := Complex.abs_mul_cos_add_sin_mul_I (x + y * Complex.I)\n[GOAL]\ncase mk\nx y : \u211d\na\u271d : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\nA : sqrt (x ^ 2 + y ^ 2) = \u2191Complex.abs (\u2191x + \u2191y * Complex.I)\nZ :\n  \u2191(\u2191Complex.abs (\u2191x + \u2191y * Complex.I)) *\n      (Complex.cos \u2191(Complex.arg (\u2191x + \u2191y * Complex.I)) +\n        Complex.sin \u2191(Complex.arg (\u2191x + \u2191y * Complex.I)) * Complex.I) =\n    \u2191x + \u2191y * Complex.I\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n      ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) (x, y)) =\n    (x, y)\n[PROOFSTEP]\nsimp only [\u2190 Complex.ofReal_cos, \u2190 Complex.ofReal_sin, mul_add, \u2190 Complex.ofReal_mul, \u2190 mul_assoc] at Z \n[GOAL]\ncase mk\nx y : \u211d\na\u271d : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\nA : sqrt (x ^ 2 + y ^ 2) = \u2191Complex.abs (\u2191x + \u2191y * Complex.I)\nZ :\n  \u2191(\u2191Complex.abs (\u2191x + \u2191y * Complex.I) * cos (Complex.arg (\u2191x + \u2191y * Complex.I))) +\n      \u2191(\u2191Complex.abs (\u2191x + \u2191y * Complex.I) * sin (Complex.arg (\u2191x + \u2191y * Complex.I))) * Complex.I =\n    \u2191x + \u2191y * Complex.I\n\u22a2 (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n      ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) (x, y)) =\n    (x, y)\n[PROOFSTEP]\nsimpa [A, -Complex.ofReal_cos, -Complex.ofReal_sin] using Complex.ext_iff.1 Z\n[GOAL]\n\u22a2 \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n    x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n      (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n          ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x) =\n        x\n[PROOFSTEP]\nrintro \u27e8r, \u03b8\u27e9 \u27e8hr, h\u03b8\u27e9\n[GOAL]\ncase mk.intro\nr \u03b8 : \u211d\nhr : (r, \u03b8).fst \u2208 Ioi 0\nh\u03b8 : (r, \u03b8).snd \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n      ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8)) =\n    (r, \u03b8)\n[PROOFSTEP]\ndsimp at hr h\u03b8 \n[GOAL]\ncase mk.intro\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n      ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) (r, \u03b8)) =\n    (r, \u03b8)\n[PROOFSTEP]\nsimp only [Prod.mk.inj_iff]\n[GOAL]\ncase mk.intro\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 sqrt ((r * cos \u03b8) ^ 2 + (r * sin \u03b8) ^ 2) = r \u2227 Complex.arg (\u2191Complex.equivRealProd.symm (r * cos \u03b8, r * sin \u03b8)) = \u03b8\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.intro.left\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 sqrt ((r * cos \u03b8) ^ 2 + (r * sin \u03b8) ^ 2) = r\n[PROOFSTEP]\nconv_rhs => rw [\u2190 sqrt_sq (le_of_lt hr), \u2190 one_mul (r ^ 2), \u2190 sin_sq_add_cos_sq \u03b8]\n[GOAL]\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n| r\n[PROOFSTEP]\nrw [\u2190 sqrt_sq (le_of_lt hr), \u2190 one_mul (r ^ 2), \u2190 sin_sq_add_cos_sq \u03b8]\n[GOAL]\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n| r\n[PROOFSTEP]\nrw [\u2190 sqrt_sq (le_of_lt hr), \u2190 one_mul (r ^ 2), \u2190 sin_sq_add_cos_sq \u03b8]\n[GOAL]\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n| r\n[PROOFSTEP]\nrw [\u2190 sqrt_sq (le_of_lt hr), \u2190 one_mul (r ^ 2), \u2190 sin_sq_add_cos_sq \u03b8]\n[GOAL]\ncase mk.intro.left\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 sqrt ((r * cos \u03b8) ^ 2 + (r * sin \u03b8) ^ 2) = sqrt ((sin \u03b8 ^ 2 + cos \u03b8 ^ 2) * r ^ 2)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mk.intro.left.e_x\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 (r * cos \u03b8) ^ 2 + (r * sin \u03b8) ^ 2 = (sin \u03b8 ^ 2 + cos \u03b8 ^ 2) * r ^ 2\n[PROOFSTEP]\nring\n[GOAL]\ncase mk.intro.right\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 Complex.arg (\u2191Complex.equivRealProd.symm (r * cos \u03b8, r * sin \u03b8)) = \u03b8\n[PROOFSTEP]\nconvert Complex.arg_mul_cos_add_sin_mul_I hr \u27e8h\u03b8.1, h\u03b8.2.le\u27e9\n[GOAL]\ncase h.e'_2.h.e'_1\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 \u2191Complex.equivRealProd.symm (r * cos \u03b8, r * sin \u03b8) = \u2191r * (Complex.cos \u2191\u03b8 + Complex.sin \u2191\u03b8 * Complex.I)\n[PROOFSTEP]\nsimp only [Complex.equivRealProd_symm_apply, Complex.ofReal_mul, Complex.ofReal_cos, Complex.ofReal_sin]\n[GOAL]\ncase h.e'_2.h.e'_1\nr \u03b8 : \u211d\nhr : r \u2208 Ioi 0\nh\u03b8 : \u03b8 \u2208 Ioo (-\u03c0) \u03c0\n\u22a2 \u2191r * Complex.cos \u2191\u03b8 + \u2191r * Complex.sin \u2191\u03b8 * Complex.I = \u2191r * (Complex.cos \u2191\u03b8 + Complex.sin \u2191\u03b8 * Complex.I)\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 ContinuousOn\n    \u2191{ toFun := fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)),\n        invFun := fun p => (p.fst * cos p.snd, p.fst * sin p.snd), source := {q | 0 < q.fst} \u222a {q | q.snd \u2260 0},\n        target := Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0,\n        map_source' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x \u2208\n                  Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0),\n        map_target' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n                    ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n                    ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x) =\n                  x) }\n    { toFun := fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)),\n        invFun := fun p => (p.fst * cos p.snd, p.fst * sin p.snd), source := {q | 0 < q.fst} \u222a {q | q.snd \u2260 0},\n        target := Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0,\n        map_source' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x \u2208\n                  Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0),\n        map_target' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n                    ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n                    ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x) =\n                  x) }.source\n[PROOFSTEP]\napply ((continuous_fst.pow 2).add (continuous_snd.pow 2)).sqrt.continuousOn.prod\n[GOAL]\n\u22a2 ContinuousOn (fun x => Complex.arg (\u2191Complex.equivRealProd.symm x))\n    { toFun := fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)),\n        invFun := fun p => (p.fst * cos p.snd, p.fst * sin p.snd), source := {q | 0 < q.fst} \u222a {q | q.snd \u2260 0},\n        target := Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0,\n        map_source' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x \u2208\n                  Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0),\n        map_target' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n                    ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n                    ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x) =\n                  x) }.source\n[PROOFSTEP]\nhave A : MapsTo Complex.equivRealProd.symm ({q : \u211d \u00d7 \u211d | 0 < q.1} \u222a {q : \u211d \u00d7 \u211d | q.2 \u2260 0}) {z | 0 < z.re \u2228 z.im \u2260 0} :=\n  by rintro \u27e8x, y\u27e9 hxy; simpa only using hxy\n[GOAL]\n\u22a2 MapsTo (\u2191Complex.equivRealProd.symm) ({q | 0 < q.fst} \u222a {q | q.snd \u2260 0}) {z | 0 < z.re \u2228 z.im \u2260 0}\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 hxy\n[GOAL]\ncase mk\nx y : \u211d\nhxy : (x, y) \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}\n\u22a2 \u2191Complex.equivRealProd.symm (x, y) \u2208 {z | 0 < z.re \u2228 z.im \u2260 0}\n[PROOFSTEP]\nsimpa only using hxy\n[GOAL]\nA : MapsTo (\u2191Complex.equivRealProd.symm) ({q | 0 < q.fst} \u222a {q | q.snd \u2260 0}) {z | 0 < z.re \u2228 z.im \u2260 0}\n\u22a2 ContinuousOn (fun x => Complex.arg (\u2191Complex.equivRealProd.symm x))\n    { toFun := fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)),\n        invFun := fun p => (p.fst * cos p.snd, p.fst * sin p.snd), source := {q | 0 < q.fst} \u222a {q | q.snd \u2260 0},\n        target := Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0,\n        map_source' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x \u2208\n                  Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0),\n        map_target' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n                    ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n                    ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x) =\n                  x) }.source\n[PROOFSTEP]\nrefine' ContinuousOn.comp (f := Complex.equivRealProd.symm) (g := Complex.arg) (fun z hz => _) _ A\n[GOAL]\ncase refine'_1\nA : MapsTo (\u2191Complex.equivRealProd.symm) ({q | 0 < q.fst} \u222a {q | q.snd \u2260 0}) {z | 0 < z.re \u2228 z.im \u2260 0}\nz : \u2102\nhz : z \u2208 {z | 0 < z.re \u2228 z.im \u2260 0}\n\u22a2 ContinuousWithinAt Complex.arg {z | 0 < z.re \u2228 z.im \u2260 0} z\n[PROOFSTEP]\nexact (Complex.continuousAt_arg hz).continuousWithinAt\n[GOAL]\ncase refine'_2\nA : MapsTo (\u2191Complex.equivRealProd.symm) ({q | 0 < q.fst} \u222a {q | q.snd \u2260 0}) {z | 0 < z.re \u2228 z.im \u2260 0}\n\u22a2 ContinuousOn \u2191Complex.equivRealProd.symm\n    { toFun := fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)),\n        invFun := fun p => (p.fst * cos p.snd, p.fst * sin p.snd), source := {q | 0 < q.fst} \u222a {q | q.snd \u2260 0},\n        target := Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0,\n        map_source' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x \u2208\n                  Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0),\n        map_target' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0}),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 {q | 0 < q.fst} \u222a {q | q.snd \u2260 0} \u2192\n                (fun p => (p.fst * cos p.snd, p.fst * sin p.snd))\n                    ((fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q))) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : \u211d \u00d7 \u211d\u2984,\n              x \u2208 Ioi 0 \u00d7\u02e2 Ioo (-\u03c0) \u03c0 \u2192\n                (fun q => (sqrt (q.fst ^ 2 + q.snd ^ 2), Complex.arg (\u2191Complex.equivRealProd.symm q)))\n                    ((fun p => (p.fst * cos p.snd, p.fst * sin p.snd)) x) =\n                  x) }.source\n[PROOFSTEP]\nexact Complex.equivRealProdClm.symm.continuous.continuousOn\n[GOAL]\np : \u211d \u00d7 \u211d\n\u22a2 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord))\n    (\u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]])))\n    p\n[PROOFSTEP]\nrw [Matrix.toLin_finTwoProd_toContinuousLinearMap]\n[GOAL]\np : \u211d \u00d7 \u211d\n\u22a2 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord))\n    (ContinuousLinearMap.prod\n      (cos p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (-p.fst * sin p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d)\n      (sin p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (p.fst * cos p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d))\n    p\n[PROOFSTEP]\nconvert\n  HasFDerivAt.prod (\ud835\udd5c := \u211d) (hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))\n    (hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using\n  2\n[GOAL]\ncase h.e'_10.h.e'_15.h.h.h\np : \u211d \u00d7 \u211d\ne_4\u271d : instTopologicalSpaceProd = UniformSpace.toTopologicalSpace\ne_5\u271d : Prod.instAddCommMonoid = AddCommGroup.toAddCommMonoid\ne_8\u271d : NonUnitalNonAssocSemiring.toAddCommMonoid = AddCommGroup.toAddCommMonoid\nhe\u271d\u00b9 : Prod.instModule = NormedSpace.toModule\nhe\u271d : NormedSpace.toModule = NormedSpace.toModule\n\u22a2 cos p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (-p.fst * sin p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d =\n    p.fst \u2022 -sin p.snd \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d + (cos \u2218 Prod.snd) p \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d\n[PROOFSTEP]\nsimp [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]\n[GOAL]\ncase h.e'_10.h.e'_16.h.h.h\np : \u211d \u00d7 \u211d\ne_4\u271d : instTopologicalSpaceProd = UniformSpace.toTopologicalSpace\ne_5\u271d : Prod.instAddCommMonoid = AddCommGroup.toAddCommMonoid\ne_11\u271d : NonUnitalNonAssocSemiring.toAddCommMonoid = AddCommGroup.toAddCommMonoid\nhe\u271d\u00b9 : Prod.instModule = NormedSpace.toModule\nhe\u271d : NormedSpace.toModule = NormedSpace.toModule\n\u22a2 sin p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (p.fst * cos p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d =\n    p.fst \u2022 cos p.snd \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d + (sin \u2218 Prod.snd) p \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d\n[PROOFSTEP]\nsimp [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]\n[GOAL]\n\u22a2 polarCoord.source =\u1d50[volume] univ\n[PROOFSTEP]\nhave A : polarCoord.source\u1d9c \u2286 LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) :=\n  by\n  intro x hx\n  simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt, Classical.not_not] at\n    hx \n  exact hx.2\n[GOAL]\n\u22a2 polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n[PROOFSTEP]\nintro x hx\n[GOAL]\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.source\u1d9c\n\u22a2 x \u2208 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n[PROOFSTEP]\nsimp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt, Classical.not_not] at hx \n[GOAL]\nx : \u211d \u00d7 \u211d\nhx : x.fst \u2264 0 \u2227 x.snd = 0\n\u22a2 x \u2208 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n[PROOFSTEP]\nexact hx.2\n[GOAL]\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 polarCoord.source =\u1d50[volume] univ\n[PROOFSTEP]\nhave B : volume (LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) : Set (\u211d \u00d7 \u211d)) = 0 :=\n  by\n  apply Measure.addHaar_submodule\n  rw [Ne.def, LinearMap.ker_eq_top]\n  intro h\n  have : (LinearMap.snd \u211d \u211d \u211d) (0, 1) = (0 : \u211d \u00d7 \u211d \u2192\u2097[\u211d] \u211d) (0, 1) := by rw [h]\n  simp at this \n[GOAL]\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 \u2191\u2191volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n[PROOFSTEP]\napply Measure.addHaar_submodule\n[GOAL]\ncase hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) \u2260 \u22a4\n[PROOFSTEP]\nrw [Ne.def, LinearMap.ker_eq_top]\n[GOAL]\ncase hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 \u00acLinearMap.snd \u211d \u211d \u211d = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\n\u22a2 False\n[PROOFSTEP]\nhave : (LinearMap.snd \u211d \u211d \u211d) (0, 1) = (0 : \u211d \u00d7 \u211d \u2192\u2097[\u211d] \u211d) (0, 1) := by rw [h]\n[GOAL]\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\n\u22a2 \u2191(LinearMap.snd \u211d \u211d \u211d) (0, 1) = \u21910 (0, 1)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\nthis : \u2191(LinearMap.snd \u211d \u211d \u211d) (0, 1) = \u21910 (0, 1)\n\u22a2 False\n[PROOFSTEP]\nsimp at this \n[GOAL]\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nB : \u2191\u2191volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n\u22a2 polarCoord.source =\u1d50[volume] univ\n[PROOFSTEP]\nsimp only [ae_eq_univ]\n[GOAL]\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nB : \u2191\u2191volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n\u22a2 \u2191\u2191volume polarCoord.source\u1d9c = 0\n[PROOFSTEP]\nexact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p) = \u222b (p : \u211d \u00d7 \u211d), f p\n[PROOFSTEP]\nset B : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d := fun p =>\n  LinearMap.toContinuousLinearMap\n    (Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d) !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p) = \u222b (p : \u211d \u00d7 \u211d), f p\n[PROOFSTEP]\nhave A : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ => hasFDerivAt_polarCoord_symm p\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p) = \u222b (p : \u211d \u00d7 \u211d), f p\n[PROOFSTEP]\nhave B_det : \u2200 p, (B p).det = p.1 := by\n  intro p\n  conv_rhs => rw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]\n  simp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin, Matrix.det_fin_two_of, sub_neg_eq_add]\n  ring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\n\u22a2 \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n[PROOFSTEP]\nintro p\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 ContinuousLinearMap.det (B p) = p.fst\n[PROOFSTEP]\nconv_rhs => rw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\np : \u211d \u00d7 \u211d\n| p.fst\n[PROOFSTEP]\nrw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\np : \u211d \u00d7 \u211d\n| p.fst\n[PROOFSTEP]\nrw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\np : \u211d \u00d7 \u211d\n| p.fst\n[PROOFSTEP]\nrw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 ContinuousLinearMap.det (B p) = (cos p.snd ^ 2 + sin p.snd ^ 2) * p.fst\n[PROOFSTEP]\nsimp only [neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin, Matrix.det_fin_two_of, sub_neg_eq_add]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 cos p.snd * (p.fst * cos p.snd) + p.fst * sin p.snd * sin p.snd = (cos p.snd ^ 2 + sin p.snd ^ 2) * p.fst\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p) = \u222b (p : \u211d \u00d7 \u211d), f p\n[PROOFSTEP]\nsymm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 \u222b (p : \u211d \u00d7 \u211d), f p = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p)\n[PROOFSTEP]\ncalc\n  \u222b p, f p = \u222b p in polarCoord.source, f p := by\n    rw [\u2190 integral_univ]\n    apply set_integral_congr_set_ae\n    exact polarCoord_source_ae_eq_univ.symm\n  _ = \u222b p in polarCoord.target, abs (B p).det \u2022 f (polarCoord.symm p) := by\n    apply integral_target_eq_integral_abs_det_fderiv_smul volume A\n  _ = \u222b p in polarCoord.target, p.1 \u2022 f (polarCoord.symm p) :=\n    by\n    apply set_integral_congr polarCoord.open_target.measurableSet fun x hx => ?_\n    rw [B_det, abs_of_pos]\n    exact hx.1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 \u222b (p : \u211d \u00d7 \u211d), f p = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p\n[PROOFSTEP]\nrw [\u2190 integral_univ]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in univ, f x = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p\n[PROOFSTEP]\napply set_integral_congr_set_ae\n[GOAL]\ncase hst\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 univ =\u1d50[volume] polarCoord.source\n[PROOFSTEP]\nexact polarCoord_source_ae_eq_univ.symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p =\n    \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, |ContinuousLinearMap.det (B p)| \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p)\n[PROOFSTEP]\napply integral_target_eq_integral_abs_det_fderiv_smul volume A\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, |ContinuousLinearMap.det (B p)| \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p) =\n    \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) p)\n[PROOFSTEP]\napply set_integral_congr polarCoord.open_target.measurableSet fun x hx => ?_\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.target\n\u22a2 |ContinuousLinearMap.det (B x)| \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) x) =\n    x.fst \u2022 f (\u2191(LocalHomeomorph.symm polarCoord) x)\n[PROOFSTEP]\nrw [B_det, abs_of_pos]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    \u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]]))\nA :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 (LocalHomeomorph.symm polarCoord).toLocalEquiv.source \u2192 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord)) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), ContinuousLinearMap.det (B p) = p.fst\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.target\n\u22a2 0 < x.fst\n[PROOFSTEP]\nexact hx.1\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.PolarCoord", "llama_tokens": 17567, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511506439707, "lm_q2_score": 0.682573740869499, "lm_q1q2_score": 0.5841815215225201}}
{"text": "[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 (x + y) ^ 0 = x ^ 0 + \u21910 * x ^ (0 - 1) * y + 0 * y ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 (x + y) ^ 1 = x ^ 1 + \u21911 * x ^ (1 - 1) * y + 0 * y ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\n\u22a2 { k // (x + y) ^ (n + 2) = x ^ (n + 2) + \u2191(n + 2) * x ^ (n + 2 - 1) * y + k * y ^ 2 }\n[PROOFSTEP]\ncases' (powAddExpansion x y (n + 1)) with z hz\n[GOAL]\ncase mk\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\nz : R\nhz : (x + y) ^ (n + 1) = x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2\n\u22a2 { k // (x + y) ^ (n + 2) = x ^ (n + 2) + \u2191(n + 2) * x ^ (n + 2 - 1) * y + k * y ^ 2 }\n[PROOFSTEP]\nexists x * z + (n + 1) * x ^ n + z * y\n[GOAL]\ncase mk\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\nz : R\nhz : (x + y) ^ (n + 1) = x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2\n\u22a2 (x + y) ^ (n + 2) = x ^ (n + 2) + \u2191(n + 2) * x ^ (n + 2 - 1) * y + (x * z + (\u2191n + 1) * x ^ n + z * y) * y ^ 2\n[PROOFSTEP]\ncalc\n  (x + y) ^ (n + 2) = (x + y) * (x + y) ^ (n + 1) := by ring\n  _ = (x + y) * (x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) := by rw [hz]\n  _ = x ^ (n + 2) + \u2191(n + 2) * x ^ (n + 1) * y + (x * z + (n + 1) * x ^ n + z * y) * y ^ 2 :=\n    by\n    push_cast\n    ring!\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\nz : R\nhz : (x + y) ^ (n + 1) = x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2\n\u22a2 (x + y) ^ (n + 2) = (x + y) * (x + y) ^ (n + 1)\n[PROOFSTEP]\nring\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\nz : R\nhz : (x + y) ^ (n + 1) = x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2\n\u22a2 (x + y) * (x + y) ^ (n + 1) = (x + y) * (x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2)\n[PROOFSTEP]\nrw [hz]\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\nz : R\nhz : (x + y) ^ (n + 1) = x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2\n\u22a2 (x + y) * (x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) =\n    x ^ (n + 2) + \u2191(n + 2) * x ^ (n + 1) * y + (x * z + (\u2191n + 1) * x ^ n + z * y) * y ^ 2\n[PROOFSTEP]\npush_cast\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\u271d\nm n\u271d : \u2115\nR : Type u_1\ninst\u271d : CommSemiring R\nx y : R\nn : \u2115\nz : R\nhz : (x + y) ^ (n + 1) = x ^ (n + 1) + \u2191(n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2\n\u22a2 (x + y) * (x ^ (n + 1) + (\u2191n + 1) * x ^ (n + 1 - 1) * y + z * y ^ 2) =\n    x ^ (n + 2) + (\u2191n + 2) * x ^ (n + 1) * y + (x * z + (\u2191n + 1) * x ^ n + z * y) * y ^ 2\n[PROOFSTEP]\nring!\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na\u271d b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\ne : \u2115\na : R\n\u22a2 { k // a * (x + y) ^ e = a * (x ^ e + \u2191e * x ^ (e - 1) * y + k * y ^ 2) }\n[PROOFSTEP]\nexists (powAddExpansion x y e).val\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na\u271d b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\ne : \u2115\na : R\n\u22a2 a * (x + y) ^ e = a * (x ^ e + \u2191e * x ^ (e - 1) * y + \u2191(powAddExpansion x y e) * y ^ 2)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na\u271d b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\ne : \u2115\na : R\n\u22a2 (x + y) ^ e = x ^ e + \u2191e * x ^ (e - 1) * y + \u2191(powAddExpansion x y e) * y ^ 2\n[PROOFSTEP]\napply (powAddExpansion _ _ _).property\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 eval (x + y) f = sum f fun e a => a * (x ^ e + \u2191e * x ^ (e - 1) * y + \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2)\n[PROOFSTEP]\nunfold eval\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 eval\u2082 (RingHom.id R) (x + y) f =\n    sum f fun e a => a * (x ^ e + \u2191e * x ^ (e - 1) * y + \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2)\n[PROOFSTEP]\nrw [eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (sum f fun e a => \u2191(RingHom.id R) a * (x + y) ^ e) =\n    sum f fun e a => a * (x ^ e + \u2191e * x ^ (e - 1) * y + \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2)\n[PROOFSTEP]\ncongr with (n z)\n[GOAL]\ncase e_f.h.h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n\u271d : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\nn : \u2115\nz : R\n\u22a2 \u2191(RingHom.id R) z * (x + y) ^ n = z * (x ^ n + \u2191n * x ^ (n - 1) * y + \u2191(Polynomial.polyBinomAux1 x y n z) * y ^ 2)\n[PROOFSTEP]\napply (polyBinomAux1 x y _ _).property\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 eval (x + y) f =\n    ((sum f fun e a => a * x ^ e) + sum f fun e a => a * \u2191e * x ^ (e - 1) * y) +\n      sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2\n[PROOFSTEP]\nrw [poly_binom_aux2]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (sum f fun e a => a * (x ^ e + \u2191e * x ^ (e - 1) * y + \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2)) =\n    ((sum f fun e a => a * x ^ e) + sum f fun e a => a * \u2191e * x ^ (e - 1) * y) +\n      sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2\n[PROOFSTEP]\nsimp [left_distrib, sum_add, mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 { k // eval (x + y) f = eval x f + eval x (\u2191derivative f) * y + k * y ^ 2 }\n[PROOFSTEP]\nexists f.sum fun e a => a * (polyBinomAux1 x y e a).val\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 eval (x + y) f =\n    eval x f + eval x (\u2191derivative f) * y + (sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a)) * y ^ 2\n[PROOFSTEP]\nrw [poly_binom_aux3]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (((sum f fun e a => a * x ^ e) + sum f fun e a => a * \u2191e * x ^ (e - 1) * y) +\n      sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2) =\n    eval x f + eval x (\u2191derivative f) * y + (sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a)) * y ^ 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (sum f fun e a => a * x ^ e) = eval x f\n[PROOFSTEP]\nrw [\u2190 eval_eq_sum]\n[GOAL]\ncase e_a.e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (sum f fun e a => a * \u2191e * x ^ (e - 1) * y) = eval x (\u2191derivative f) * y\n[PROOFSTEP]\nrw [derivative_eval]\n[GOAL]\ncase e_a.e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (sum f fun e a => a * \u2191e * x ^ (e - 1) * y) = (sum f fun n a => a * \u2191n * x ^ (n - 1)) * y\n[PROOFSTEP]\nexact Finset.sum_mul.symm\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a) * y ^ 2) =\n    (sum f fun e a => a * \u2191(Polynomial.polyBinomAux1 x y e a)) * y ^ 2\n[PROOFSTEP]\nexact Finset.sum_mul.symm\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\n\u22a2 x ^ 0 - y ^ 0 = 0 * (x - y)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\n\u22a2 x ^ 1 - y ^ 1 = 1 * (x - y)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk\u271d : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\nk : \u2115\n\u22a2 { z // x ^ (k + 2) - y ^ (k + 2) = z * (x - y) }\n[PROOFSTEP]\ncases' @powSubPowFactor x y (k + 1) with z hz\n[GOAL]\ncase mk\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk\u271d : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\nk : \u2115\nz : R\nhz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)\n\u22a2 { z // x ^ (k + 2) - y ^ (k + 2) = z * (x - y) }\n[PROOFSTEP]\nexists z * x + y ^ (k + 1)\n[GOAL]\ncase mk\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk\u271d : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\nk : \u2115\nz : R\nhz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)\n\u22a2 x ^ (k + 2) - y ^ (k + 2) = (z * x + y ^ (k + 1)) * (x - y)\n[PROOFSTEP]\nlinear_combination (norm := ring) x * hz\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk\u271d : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nx y : R\nk : \u2115\nz : R\nhz : x ^ (k + 1) - y ^ (k + 1) = z * (x - y)\n\u22a2 x ^ (k + 2) - y ^ (k + 2) - (z * x + y ^ (k + 1)) * (x - y) - (x * (x ^ (k + 1) - y ^ (k + 1)) - x * (z * (x - y))) =\n    0\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 { z // eval x f - eval y f = z * (x - y) }\n[PROOFSTEP]\nrefine' \u27e8f.sum fun i r => r * (powSubPowFactor x y i).val, _\u27e9\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 eval x f - eval y f = (sum f fun i r => r * \u2191(powSubPowFactor x y i)) * (x - y)\n[PROOFSTEP]\ndelta eval\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 eval\u2082 (RingHom.id R) x f - eval\u2082 (RingHom.id R) y f = (sum f fun i r => r * \u2191(powSubPowFactor x y i)) * (x - y)\n[PROOFSTEP]\nrw [eval\u2082_eq_sum, eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 ((sum f fun e a => \u2191(RingHom.id R) a * x ^ e) - sum f fun e a => \u2191(RingHom.id R) a * y ^ e) =\n    (sum f fun i r => r * \u2191(powSubPowFactor x y i)) * (x - y)\n[PROOFSTEP]\nsimp only [sum, \u2190 Finset.sum_sub_distrib, Finset.sum_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (Finset.sum (support f) fun x_1 =>\n      \u2191(RingHom.id R) (coeff f x_1) * x ^ x_1 - \u2191(RingHom.id R) (coeff f x_1) * y ^ x_1) =\n    Finset.sum (support f) fun x_1 => coeff f x_1 * \u2191(powSubPowFactor x y x_1) * (x - y)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\n\u22a2 (Finset.sum (support f) fun x_1 => coeff f x_1 * x ^ x_1 - coeff f x_1 * y ^ x_1) =\n    Finset.sum (support f) fun x_1 => coeff f x_1 * \u2191(powSubPowFactor x y x_1) * (x - y)\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase e_f.h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type x\nk : Type y\nA : Type z\na b : R\nm n : \u2115\ninst\u271d : CommRing R\nf : R[X]\nx y : R\ni : \u2115\n\u22a2 coeff f i * x ^ i - coeff f i * y ^ i = coeff f i * \u2191(powSubPowFactor x y i) * (x - y)\n[PROOFSTEP]\nrw [mul_assoc, \u2190 (powSubPowFactor x y _).prop, mul_sub]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Identities", "llama_tokens": 6331, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8774767906859264, "lm_q2_score": 0.665410572017153, "lm_q1q2_score": 0.5838823332220979}}
{"text": "[GOAL]\np : \u2115\nh : 0 < p\n\u22a2 0 < mersenne p\n[PROOFSTEP]\ndsimp [mersenne]\n[GOAL]\np : \u2115\nh : 0 < p\n\u22a2 0 < 2 ^ p - 1\n[PROOFSTEP]\ncalc\n  0 < 2 ^ 1 - 1 := by norm_num\n  _ \u2264 2 ^ p - 1 := Nat.sub_le_sub_right (Nat.pow_le_pow_of_le_right (Nat.succ_pos 1) h) 1\n[GOAL]\np : \u2115\nh : 0 < p\n\u22a2 0 < 2 ^ 1 - 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\nhp : 1 < p\n\u22a2 1 + 1 = 2 ^ 1\n[PROOFSTEP]\nrw [one_add_one_eq_two, pow_one]\n[GOAL]\nk : \u2115\n\u22a2 mersenne k + 1 = 2 ^ k\n[PROOFSTEP]\nrw [mersenne, tsub_add_cancel_of_le]\n[GOAL]\nk : \u2115\n\u22a2 1 \u2264 2 ^ k\n[PROOFSTEP]\nexact one_le_pow_of_one_le (by norm_num) k\n[GOAL]\nk : \u2115\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\nhp : 0 < p\n\u22a2 1 < 2 ^ p\n[PROOFSTEP]\nexact_mod_cast Nat.one_lt_two_pow p hp\n[GOAL]\np : \u2115\nw : 0 < p\ni : \u2115\n\u22a2 0 \u2264 sMod p i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero\np : \u2115\nw : 0 < p\n\u22a2 0 \u2264 sMod p zero\n[PROOFSTEP]\ndsimp [sMod]\n[GOAL]\ncase succ\np : \u2115\nw : 0 < p\nn\u271d : \u2115\n\u22a2 0 \u2264 sMod p (succ n\u271d)\n[PROOFSTEP]\ndsimp [sMod]\n[GOAL]\ncase zero\np : \u2115\nw : 0 < p\n\u22a2 0 \u2264 4 % (2 ^ p - 1)\n[PROOFSTEP]\nexact sup_eq_right.mp rfl\n[GOAL]\ncase succ\np : \u2115\nw : 0 < p\nn\u271d : \u2115\n\u22a2 0 \u2264 (sMod p n\u271d ^ 2 - 2) % (2 ^ p - 1)\n[PROOFSTEP]\napply Int.emod_nonneg\n[GOAL]\ncase succ.a\np : \u2115\nw : 0 < p\nn\u271d : \u2115\n\u22a2 2 ^ p - 1 \u2260 0\n[PROOFSTEP]\nexact mersenne_int_ne_zero p w\n[GOAL]\np i : \u2115\n\u22a2 sMod p i % (2 ^ p - 1) = sMod p i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero\np : \u2115\n\u22a2 sMod p zero % (2 ^ p - 1) = sMod p zero\n[PROOFSTEP]\nsimp [sMod]\n[GOAL]\ncase succ\np n\u271d : \u2115\n\u22a2 sMod p (succ n\u271d) % (2 ^ p - 1) = sMod p (succ n\u271d)\n[PROOFSTEP]\nsimp [sMod]\n[GOAL]\np : \u2115\nw : 0 < p\ni : \u2115\n\u22a2 sMod p i < 2 ^ p - 1\n[PROOFSTEP]\nrw [\u2190 sMod_mod]\n[GOAL]\np : \u2115\nw : 0 < p\ni : \u2115\n\u22a2 sMod p i % (2 ^ p - 1) < 2 ^ p - 1\n[PROOFSTEP]\nrefine (Int.emod_lt _ (mersenne_int_ne_zero p w)).trans_eq ?_\n[GOAL]\np : \u2115\nw : 0 < p\ni : \u2115\n\u22a2 |2 ^ p - 1| = 2 ^ p - 1\n[PROOFSTEP]\nexact abs_of_nonneg (mersenne_int_pos w).le\n[GOAL]\np' i : \u2115\n\u22a2 sZMod (p' + 2) i = \u2191(s i)\n[PROOFSTEP]\ninduction' i with i ih\n[GOAL]\ncase zero\np' : \u2115\n\u22a2 sZMod (p' + 2) zero = \u2191(s zero)\n[PROOFSTEP]\ndsimp [s, sZMod]\n[GOAL]\ncase zero\np' : \u2115\n\u22a2 4 = \u21914\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ\np' i : \u2115\nih : sZMod (p' + 2) i = \u2191(s i)\n\u22a2 sZMod (p' + 2) (succ i) = \u2191(s (succ i))\n[PROOFSTEP]\npush_cast [s, sZMod, ih]\n[GOAL]\ncase succ\np' i : \u2115\nih : sZMod (p' + 2) i = \u2191(s i)\n\u22a2 \u2191(s i) ^ 2 - 2 = \u2191(s i) ^ 2 - 2\n[PROOFSTEP]\nrfl\n[GOAL]\nb p : \u2115\nw : 0 < b\n\u22a2 \u2191(b ^ p - 1) = \u2191b ^ p - 1\n[PROOFSTEP]\nhave : 1 \u2264 b ^ p := Nat.one_le_pow p b w\n[GOAL]\nb p : \u2115\nw : 0 < b\nthis : 1 \u2264 b ^ p\n\u22a2 \u2191(b ^ p - 1) = \u2191b ^ p - 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\np i : \u2115\n\u22a2 sZMod p i = \u2191(sMod p i)\n[PROOFSTEP]\ninduction i\n[GOAL]\ncase zero\np : \u2115\n\u22a2 sZMod p zero = \u2191(sMod p zero)\n[PROOFSTEP]\npush_cast [\u2190 Int.coe_nat_two_pow_pred p, sMod, sZMod, *]\n[GOAL]\ncase succ\np n\u271d : \u2115\nn_ih\u271d : sZMod p n\u271d = \u2191(sMod p n\u271d)\n\u22a2 sZMod p (succ n\u271d) = \u2191(sMod p (succ n\u271d))\n[PROOFSTEP]\npush_cast [\u2190 Int.coe_nat_two_pow_pred p, sMod, sZMod, *]\n[GOAL]\ncase zero\np : \u2115\n\u22a2 4 = 4\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\np n\u271d : \u2115\nn_ih\u271d : sZMod p n\u271d = \u2191(sMod p n\u271d)\n\u22a2 \u2191(sMod p n\u271d) ^ 2 - 2 = \u2191(sMod p n\u271d) ^ 2 - 2\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nw : 1 < p\n\u22a2 lucasLehmerResidue p = 0 \u2194 sMod p (p - 2) = 0\n[PROOFSTEP]\ndsimp [lucasLehmerResidue]\n[GOAL]\np : \u2115\nw : 1 < p\n\u22a2 sZMod p (p - 2) = 0 \u2194 sMod p (p - 2) = 0\n[PROOFSTEP]\nrw [sZMod_eq_sMod p]\n[GOAL]\np : \u2115\nw : 1 < p\n\u22a2 \u2191(sMod p (p - 2)) = 0 \u2194 sMod p (p - 2) = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np : \u2115\nw : 1 < p\n\u22a2 \u2191(sMod p (p - 2)) = 0 \u2192 sMod p (p - 2) = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\np : \u2115\nw : 1 < p\nh : \u2191(sMod p (p - 2)) = 0\n\u22a2 sMod p (p - 2) = 0\n[PROOFSTEP]\nsimp [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h \n[GOAL]\ncase mp\np : \u2115\nw : 1 < p\nh : 2 ^ p - 1 \u2223 sMod p (p - 2)\n\u22a2 sMod p (p - 2) = 0\n[PROOFSTEP]\napply Int.eq_zero_of_dvd_of_nonneg_of_lt _ _ h\n[GOAL]\np : \u2115\nw : 1 < p\nh : 2 ^ p - 1 \u2223 sMod p (p - 2)\n\u22a2 0 \u2264 sMod p (p - 2)\n[PROOFSTEP]\nclear h\n[GOAL]\np : \u2115\nw : 1 < p\nh : 2 ^ p - 1 \u2223 sMod p (p - 2)\n\u22a2 sMod p (p - 2) < 2 ^ p - 1\n[PROOFSTEP]\nclear h\n[GOAL]\np : \u2115\nw : 1 < p\n\u22a2 0 \u2264 sMod p (p - 2)\n[PROOFSTEP]\napply sMod_nonneg _ (Nat.lt_of_succ_lt w)\n[GOAL]\np : \u2115\nw : 1 < p\n\u22a2 sMod p (p - 2) < 2 ^ p - 1\n[PROOFSTEP]\nexact sMod_lt _ (Nat.lt_of_succ_lt w) (p - 2)\n[GOAL]\ncase mpr\np : \u2115\nw : 1 < p\n\u22a2 sMod p (p - 2) = 0 \u2192 \u2191(sMod p (p - 2)) = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\np : \u2115\nw : 1 < p\nh : sMod p (p - 2) = 0\n\u22a2 \u2191(sMod p (p - 2)) = 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\np : \u2115\nw : 1 < p\nh : sMod p (p - 2) = 0\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nx y : X q\nh\u2081 : x.fst = y.fst\nh\u2082 : x.snd = y.snd\n\u22a2 x = y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nq : \u2115+\ny : X q\nfst\u271d snd\u271d : ZMod \u2191q\nh\u2081 : (fst\u271d, snd\u271d).fst = y.fst\nh\u2082 : (fst\u271d, snd\u271d).snd = y.snd\n\u22a2 (fst\u271d, snd\u271d) = y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nq : \u2115+\nfst\u271d\u00b9 snd\u271d\u00b9 fst\u271d snd\u271d : ZMod \u2191q\nh\u2081 : (fst\u271d\u00b9, snd\u271d\u00b9).fst = (fst\u271d, snd\u271d).fst\nh\u2082 : (fst\u271d\u00b9, snd\u271d\u00b9).snd = (fst\u271d, snd\u271d).snd\n\u22a2 (fst\u271d\u00b9, snd\u271d\u00b9) = (fst\u271d, snd\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx y z : X q\n\u22a2 x * y * z = x * (y * z)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx y z : X q\n\u22a2 (x * y * z).fst = (x * (y * z)).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx y z : X q\n\u22a2 (x * y * z).snd = (x * (y * z)).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx y z : X q\n\u22a2 (x.fst * y.fst + 3 * x.snd * y.snd) * z.fst + 3 * (x.fst * y.snd + x.snd * y.fst) * z.snd =\n    x.fst * (y.fst * z.fst + 3 * y.snd * z.snd) + 3 * x.snd * (y.fst * z.snd + y.snd * z.fst)\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx y z : X q\n\u22a2 (x.fst * y.fst + 3 * x.snd * y.snd) * z.snd + (x.fst * y.snd + x.snd * y.fst) * z.fst =\n    x.fst * (y.fst * z.snd + y.snd * z.fst) + x.snd * (y.fst * z.fst + 3 * y.snd * z.snd)\n[PROOFSTEP]\nring\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx : X q\n\u22a2 1 * x = x\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx : X q\n\u22a2 (1 * x).fst = x.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx : X q\n\u22a2 (1 * x).snd = x.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx : X q\n\u22a2 x * 1 = x\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx : X q\n\u22a2 (x * 1).fst = x.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b9 : Mul (X q) := inferInstanceAs (Mul (X q))\nsrc\u271d : One (X q) := inferInstanceAs (One (X q))\nx : X q\n\u22a2 (x * 1).snd = x.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\n\u22a2 NatCast.natCast 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\n\u22a2 (NatCast.natCast 0).fst = 0.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\n\u22a2 (NatCast.natCast 0).snd = 0.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nx\u271d : \u2115\n\u22a2 NatCast.natCast (x\u271d + 1) = NatCast.natCast x\u271d + 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nx\u271d : \u2115\n\u22a2 (NatCast.natCast (x\u271d + 1)).fst = (NatCast.natCast x\u271d + 1).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nx\u271d : \u2115\n\u22a2 (NatCast.natCast (x\u271d + 1)).snd = (NatCast.natCast x\u271d + 1).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nn : \u2115\n\u22a2 IntCast.intCast \u2191n = \u2191n\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nn : \u2115\n\u22a2 (IntCast.intCast \u2191n).fst = (\u2191n).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nn : \u2115\n\u22a2 (IntCast.intCast \u2191n).snd = (\u2191n).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nn : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nn : \u2115\n\u22a2 (IntCast.intCast (Int.negSucc n)).fst = (-\u2191(n + 1)).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b2 : Monoid (X q) := inferInstanceAs (Monoid (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : NatCast (X q) := inferInstanceAs (NatCast (X q))\nn : \u2115\n\u22a2 (IntCast.intCast (Int.negSucc n)).snd = (-\u2191(n + 1)).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nx y z : X q\n\u22a2 x * (y + z) = x * y + x * z\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nx y z : X q\n\u22a2 (x * (y + z)).fst = (x * y + x * z).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nx y z : X q\n\u22a2 (x * (y + z)).snd = (x * y + x * z).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nq : \u2115+\nx y z : X q\n\u22a2 x.fst * (y.fst + z.fst) + 3 * x.snd * (y.snd + z.snd) =\n    x.fst * y.fst + 3 * x.snd * y.snd + (x.fst * z.fst + 3 * x.snd * z.snd)\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2082\nq : \u2115+\nx y z : X q\n\u22a2 x.fst * (y.snd + z.snd) + x.snd * (y.fst + z.fst) = x.fst * y.snd + x.snd * y.fst + (x.fst * z.snd + x.snd * z.fst)\n[PROOFSTEP]\nring\n[GOAL]\nq : \u2115+\nx y z : X q\n\u22a2 (x + y) * z = x * z + y * z\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nx y z : X q\n\u22a2 ((x + y) * z).fst = (x * z + y * z).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nx y z : X q\n\u22a2 ((x + y) * z).snd = (x * z + y * z).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nq : \u2115+\nx y z : X q\n\u22a2 (x.fst + y.fst) * z.fst + 3 * (x.snd + y.snd) * z.snd =\n    x.fst * z.fst + 3 * x.snd * z.snd + (y.fst * z.fst + 3 * y.snd * z.snd)\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2082\nq : \u2115+\nx y z : X q\n\u22a2 (x.fst + y.fst) * z.snd + (x.snd + y.snd) * z.fst = x.fst * z.snd + x.snd * z.fst + (y.fst * z.snd + y.snd * z.fst)\n[PROOFSTEP]\nring\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b2 : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : Monoid (X q) := inferInstanceAs (Monoid (X q))\nx\u271d : X q\n\u22a2 0 * x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b2 : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : Monoid (X q) := inferInstanceAs (Monoid (X q))\nx\u271d : X q\n\u22a2 (0 * x\u271d).fst = 0.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b2 : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : Monoid (X q) := inferInstanceAs (Monoid (X q))\nx\u271d : X q\n\u22a2 (0 * x\u271d).snd = 0.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d\u00b2 : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : Monoid (X q) := inferInstanceAs (Monoid (X q))\nx\u271d : X q\n\u22a2 x\u271d * 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d\u00b2 : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : Monoid (X q) := inferInstanceAs (Monoid (X q))\nx\u271d : X q\n\u22a2 (x\u271d * 0).fst = 0.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d\u00b2 : AddGroupWithOne (X q) := inferInstanceAs (AddGroupWithOne (X q))\nsrc\u271d\u00b9 : AddCommGroup (X q) := inferInstanceAs (AddCommGroup (X q))\nsrc\u271d : Monoid (X q) := inferInstanceAs (Monoid (X q))\nx\u271d : X q\n\u22a2 (x\u271d * 0).snd = 0.snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nsrc\u271d : Ring (X q) := inferInstanceAs (Ring (X q))\nx\u271d\u00b9 x\u271d : X q\n\u22a2 x\u271d\u00b9 * x\u271d = x\u271d * x\u271d\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d : Ring (X q) := inferInstanceAs (Ring (X q))\nx\u271d\u00b9 x\u271d : X q\n\u22a2 (x\u271d\u00b9 * x\u271d).fst = (x\u271d * x\u271d\u00b9).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d : Ring (X q) := inferInstanceAs (Ring (X q))\nx\u271d\u00b9 x\u271d : X q\n\u22a2 (x\u271d\u00b9 * x\u271d).snd = (x\u271d * x\u271d\u00b9).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\nq : \u2115+\nsrc\u271d : Ring (X q) := inferInstanceAs (Ring (X q))\nx\u271d\u00b9 x\u271d : X q\n\u22a2 x\u271d\u00b9.fst * x\u271d.fst + 3 * x\u271d\u00b9.snd * x\u271d.snd = x\u271d.fst * x\u271d\u00b9.fst + 3 * x\u271d.snd * x\u271d\u00b9.snd\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2082\nq : \u2115+\nsrc\u271d : Ring (X q) := inferInstanceAs (Ring (X q))\nx\u271d\u00b9 x\u271d : X q\n\u22a2 x\u271d\u00b9.fst * x\u271d.snd + x\u271d\u00b9.snd * x\u271d.fst = x\u271d.fst * x\u271d\u00b9.snd + x\u271d.snd * x\u271d\u00b9.fst\n[PROOFSTEP]\nring\n[GOAL]\nq : \u2115+\nn m : \u2124\n\u22a2 \u2191(n * m) = \u2191n * \u2191m\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nn m : \u2124\n\u22a2 (\u2191(n * m)).fst = (\u2191n * \u2191m).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nn m : \u2124\n\u22a2 (\u2191(n * m)).snd = (\u2191n * \u2191m).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\nn : \u2115\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq : \u2115+\nn : \u2115\n\u22a2 (\u2191\u2191n).fst = (\u2191n).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq : \u2115+\nn : \u2115\n\u22a2 (\u2191\u2191n).snd = (\u2191n).snd\n[PROOFSTEP]\nsimp\n[GOAL]\nq : \u2115+\n\u22a2 Fintype.card (X q) = \u2191q ^ 2\n[PROOFSTEP]\ndsimp [X]\n[GOAL]\nq : \u2115+\n\u22a2 Fintype.card (ZMod \u2191q \u00d7 ZMod \u2191q) = \u2191q ^ 2\n[PROOFSTEP]\nrw [Fintype.card_prod, ZMod.card q, sq]\n[GOAL]\nq : \u2115+\nw : 1 < q\n\u22a2 Fintype.card (X q)\u02e3 < \u2191q ^ 2\n[PROOFSTEP]\nhave : Fact (1 < (q : \u2115)) := \u27e8w\u27e9\n[GOAL]\nq : \u2115+\nw : 1 < q\nthis : Fact (1 < \u2191q)\n\u22a2 Fintype.card (X q)\u02e3 < \u2191q ^ 2\n[PROOFSTEP]\nconvert card_units_lt (X q)\n[GOAL]\ncase h.e'_4\nq : \u2115+\nw : 1 < q\nthis : Fact (1 < \u2191q)\n\u22a2 \u2191q ^ 2 = Fintype.card (X q)\n[PROOFSTEP]\nrw [card_eq]\n[GOAL]\nq\u271d q : \u2115+\n\u22a2 \u03c9 * \u03c9b = 1\n[PROOFSTEP]\ndsimp [\u03c9, \u03c9b]\n[GOAL]\nq\u271d q : \u2115+\n\u22a2 (2, 1) * (2, -1) = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nq\u271d q : \u2115+\n\u22a2 ((2, 1) * (2, -1)).fst = 1.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nq\u271d q : \u2115+\n\u22a2 ((2, 1) * (2, -1)).snd = 1.snd\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2081\nq\u271d q : \u2115+\n\u22a2 2 * 2 + -3 = 1\n[PROOFSTEP]\nring\n[GOAL]\nq\u271d q : \u2115+\n\u22a2 \u03c9b * \u03c9 = 1\n[PROOFSTEP]\nrw [mul_comm, \u03c9_mul_\u03c9b]\n[GOAL]\nq : \u2115+\ni : \u2115\n\u22a2 \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n[PROOFSTEP]\ninduction' i with i ih\n[GOAL]\ncase zero\nq : \u2115+\n\u22a2 \u2191(s zero) = \u03c9 ^ 2 ^ zero + \u03c9b ^ 2 ^ zero\n[PROOFSTEP]\ndsimp [s, \u03c9, \u03c9b]\n[GOAL]\ncase zero\nq : \u2115+\n\u22a2 \u21914 = (2, 1) ^ 1 + (2, -1) ^ 1\n[PROOFSTEP]\next\n[GOAL]\ncase zero.h\u2081\nq : \u2115+\n\u22a2 (\u21914).fst = ((2, 1) ^ 1 + (2, -1) ^ 1).fst\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase zero.h\u2082\nq : \u2115+\n\u22a2 (\u21914).snd = ((2, 1) ^ 1 + (2, -1) ^ 1).snd\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 \u2191(s (succ i)) = \u03c9 ^ 2 ^ succ i + \u03c9b ^ 2 ^ succ i\n[PROOFSTEP]\ncalc\n  (s (i + 1) : X q) = (s i ^ 2 - 2 : \u2124) := rfl\n  _ = (s i : X q) ^ 2 - 2 := by push_cast ; rfl\n  _ = (\u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i) ^ 2 - 2 := by rw [ih]\n  _ = (\u03c9 ^ 2 ^ i) ^ 2 + (\u03c9b ^ 2 ^ i) ^ 2 + 2 * (\u03c9b ^ 2 ^ i * \u03c9 ^ 2 ^ i) - 2 := by ring\n  _ = (\u03c9 ^ 2 ^ i) ^ 2 + (\u03c9b ^ 2 ^ i) ^ 2 := by rw [\u2190 mul_pow \u03c9b \u03c9, \u03c9b_mul_\u03c9, one_pow, mul_one, add_sub_cancel]\n  _ = \u03c9 ^ 2 ^ (i + 1) + \u03c9b ^ 2 ^ (i + 1) := by rw [\u2190 pow_mul, \u2190 pow_mul, _root_.pow_succ']\n[GOAL]\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 \u2191(s i ^ 2 - 2) = \u2191(s i) ^ 2 - 2\n[PROOFSTEP]\npush_cast\n[GOAL]\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 \u2191(s i) ^ 2 - 2 = \u2191(s i) ^ 2 - 2\n[PROOFSTEP]\nrfl\n[GOAL]\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 \u2191(s i) ^ 2 - 2 = (\u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i) ^ 2 - 2\n[PROOFSTEP]\nrw [ih]\n[GOAL]\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 (\u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i) ^ 2 - 2 = (\u03c9 ^ 2 ^ i) ^ 2 + (\u03c9b ^ 2 ^ i) ^ 2 + 2 * (\u03c9b ^ 2 ^ i * \u03c9 ^ 2 ^ i) - 2\n[PROOFSTEP]\nring\n[GOAL]\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 (\u03c9 ^ 2 ^ i) ^ 2 + (\u03c9b ^ 2 ^ i) ^ 2 + 2 * (\u03c9b ^ 2 ^ i * \u03c9 ^ 2 ^ i) - 2 = (\u03c9 ^ 2 ^ i) ^ 2 + (\u03c9b ^ 2 ^ i) ^ 2\n[PROOFSTEP]\nrw [\u2190 mul_pow \u03c9b \u03c9, \u03c9b_mul_\u03c9, one_pow, mul_one, add_sub_cancel]\n[GOAL]\nq : \u2115+\ni : \u2115\nih : \u2191(s i) = \u03c9 ^ 2 ^ i + \u03c9b ^ 2 ^ i\n\u22a2 (\u03c9 ^ 2 ^ i) ^ 2 + (\u03c9b ^ 2 ^ i) ^ 2 = \u03c9 ^ 2 ^ (i + 1) + \u03c9b ^ 2 ^ (i + 1)\n[PROOFSTEP]\nrw [\u2190 pow_mul, \u2190 pow_mul, _root_.pow_succ']\n[GOAL]\np' : \u2115\n\u22a2 2 < q (p' + 2)\n[PROOFSTEP]\nrefine (minFac_prime (one_lt_mersenne ?_).ne').two_le.lt_of_ne' ?_\n[GOAL]\ncase refine_1\np' : \u2115\n\u22a2 1 < p' + 2\n[PROOFSTEP]\nexact le_add_left _ _\n[GOAL]\ncase refine_2\np' : \u2115\n\u22a2 minFac (mersenne (p' + 2)) \u2260 2\n[PROOFSTEP]\nrw [Ne.def, minFac_eq_two_iff, mersenne, Nat.pow_succ']\n[GOAL]\ncase refine_2\np' : \u2115\n\u22a2 \u00ac2 \u2223 2 * 2 ^ (p' + 1) - 1\n[PROOFSTEP]\nexact Nat.two_not_dvd_two_mul_sub_one (Nat.one_le_two_pow _)\n[GOAL]\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\ndsimp [lucasLehmerResidue] at h \n[GOAL]\np' : \u2115\nh : sZMod (p' + 2) (p' + 2 - 2) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nrw [sZMod_eq_s p'] at h \n[GOAL]\np' : \u2115\nh : \u2191(s (p' + 2 - 2)) = 0\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nsimp [ZMod.int_cast_zmod_eq_zero_iff_dvd] at h \n[GOAL]\np' : \u2115\nh : 2 ^ (p' + 2) - 1 \u2223 s p'\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\ncases' h with k h\n[GOAL]\ncase intro\np' : \u2115\nk : \u2124\nh : s p' = (2 ^ (p' + 2) - 1) * k\n\u22a2 \u2203 k, \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : s p' = (2 ^ (p' + 2) - 1) * k\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nreplace h := congr_arg (fun n : \u2124 => (n : X (q (p' + 2)))) h\n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : (fun n => \u2191n) (s p') = (fun n => \u2191n) ((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u2191(s p') = \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nrw [closed_form] at h \n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p' = \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nreplace h := congr_arg (fun x => \u03c9 ^ 2 ^ p' * x) h\n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : (fun x => \u03c9 ^ 2 ^ p' * x) (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = (fun x => \u03c9 ^ 2 ^ p' * x) \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\ndsimp at h \n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nhave t : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1) := by ring\n[GOAL]\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\n\u22a2 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ p' * (\u03c9 ^ 2 ^ p' + \u03c9b ^ 2 ^ p') = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nrw [mul_add, \u2190 pow_add \u03c9, t, \u2190 mul_pow \u03c9 \u03c9b (2 ^ p'), \u03c9_mul_\u03c9b, one_pow] at h \n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u03c9 ^ 2 ^ p' * \u2191((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nrw [mul_comm, coe_mul] at h \n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u2191(2 ^ (p' + 2) - 1) * \u2191k * \u03c9 ^ 2 ^ p'\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nrw [mul_comm _ (k : X (q (p' + 2)))] at h \n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nh : \u03c9 ^ 2 ^ (p' + 1) + 1 = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p'\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nreplace h := eq_sub_of_add_eq h\n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nhave : 1 \u2264 2 ^ (p' + 2) := Nat.one_le_pow _ _ (by decide)\n[GOAL]\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncase h\np' : \u2115\nk : \u2124\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\nh : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(2 ^ (p' + 2) - 1) * \u03c9 ^ 2 ^ p' - 1\nthis : 1 \u2264 2 ^ (p' + 2)\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\np : \u2115\n\u22a2 \u2191(mersenne p) = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\np : \u2115\n\u22a2 (\u2191(mersenne p)).fst = 0.fst\n[PROOFSTEP]\nsimp [mersenne, q, ZMod.nat_cast_zmod_eq_zero_iff_dvd, -pow_pos]\n[GOAL]\ncase h\u2082\np : \u2115\n\u22a2 (\u2191(mersenne p)).snd = 0.snd\n[PROOFSTEP]\nsimp [mersenne, q, ZMod.nat_cast_zmod_eq_zero_iff_dvd, -pow_pos]\n[GOAL]\ncase h\u2081\np : \u2115\n\u22a2 minFac (2 ^ p - 1) \u2223 2 ^ p - 1\n[PROOFSTEP]\napply Nat.minFac_dvd\n[GOAL]\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = -1\n[PROOFSTEP]\ncases' \u03c9_pow_formula p' h with k w\n[GOAL]\ncase intro\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\nk : \u2124\nw : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * \u2191(mersenne (p' + 2)) * \u03c9 ^ 2 ^ p' - 1\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = -1\n[PROOFSTEP]\nrw [mersenne_coe_X] at w \n[GOAL]\ncase intro\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\nk : \u2124\nw : \u03c9 ^ 2 ^ (p' + 1) = \u2191k * 0 * \u03c9 ^ 2 ^ p' - 1\n\u22a2 \u03c9 ^ 2 ^ (p' + 1) = -1\n[PROOFSTEP]\nsimpa using w\n[GOAL]\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u03c9 ^ 2 ^ (p' + 2) = (\u03c9 ^ 2 ^ (p' + 1)) ^ 2\n[PROOFSTEP]\nrw [\u2190 pow_mul, \u2190 Nat.pow_succ]\n[GOAL]\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 (\u03c9 ^ 2 ^ (p' + 1)) ^ 2 = (-1) ^ 2\n[PROOFSTEP]\nrw [\u03c9_pow_eq_neg_one p' h]\n[GOAL]\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 (-1) ^ 2 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 orderOf (\u03c9Unit (p' + 2)) = 2 ^ (p' + 2)\n[PROOFSTEP]\napply Nat.eq_prime_pow_of_dvd_least_prime_pow\n[GOAL]\ncase pp\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 Nat.Prime 2\n[PROOFSTEP]\nexact Nat.prime_two\n[GOAL]\ncase h\u2081\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u00acorderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n[PROOFSTEP]\nintro o\n[GOAL]\ncase h\u2081\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\no : orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n\u22a2 False\n[PROOFSTEP]\nhave \u03c9_pow := orderOf_dvd_iff_pow_eq_one.1 o\n[GOAL]\ncase h\u2081\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\no : orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n\u03c9_pow : \u03c9Unit (p' + 2) ^ 2 ^ (p' + 1) = 1\n\u22a2 False\n[PROOFSTEP]\nreplace \u03c9_pow := congr_arg (Units.coeHom (X (q (p' + 2))) : Units (X (q (p' + 2))) \u2192 X (q (p' + 2))) \u03c9_pow\n[GOAL]\ncase h\u2081\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\no : orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n\u03c9_pow : \u2191(Units.coeHom (X (q (p' + 2)))) (\u03c9Unit (p' + 2) ^ 2 ^ (p' + 1)) = \u2191(Units.coeHom (X (q (p' + 2)))) 1\n\u22a2 False\n[PROOFSTEP]\nsimp at \u03c9_pow \n[GOAL]\ncase h\u2081\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\no : orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n\u03c9_pow : \u03c9 ^ 2 ^ (p' + 1) = 1\n\u22a2 False\n[PROOFSTEP]\nhave h : (1 : ZMod (q (p' + 2))) = -1 := congr_arg Prod.fst (\u03c9_pow.symm.trans (\u03c9_pow_eq_neg_one p' h))\n[GOAL]\ncase h\u2081\np' : \u2115\nh\u271d : lucasLehmerResidue (p' + 2) = 0\no : orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n\u03c9_pow : \u03c9 ^ 2 ^ (p' + 1) = 1\nh : 1 = -1\n\u22a2 False\n[PROOFSTEP]\nhaveI : Fact (2 < (q (p' + 2) : \u2115)) := \u27e8two_lt_q _\u27e9\n[GOAL]\ncase h\u2081\np' : \u2115\nh\u271d : lucasLehmerResidue (p' + 2) = 0\no : orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1)\n\u03c9_pow : \u03c9 ^ 2 ^ (p' + 1) = 1\nh : 1 = -1\nthis : Fact (2 < \u2191(q (p' + 2)))\n\u22a2 False\n[PROOFSTEP]\napply ZMod.neg_one_ne_one h.symm\n[GOAL]\ncase h\u2082\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 orderOf (\u03c9Unit (p' + 2)) \u2223 2 ^ (p' + 1 + 1)\n[PROOFSTEP]\napply orderOf_dvd_iff_pow_eq_one.2\n[GOAL]\ncase h\u2082\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u03c9Unit (p' + 2) ^ 2 ^ (p' + 1 + 1) = 1\n[PROOFSTEP]\napply Units.ext\n[GOAL]\ncase h\u2082.a\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u2191(\u03c9Unit (p' + 2) ^ 2 ^ (p' + 1 + 1)) = \u21911\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h\u2082.a\np' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u2191(\u03c9Unit (p' + 2)) ^ 2 ^ (p' + 1 + 1) = 1\n[PROOFSTEP]\nexact \u03c9_pow_eq_one p' h\n[GOAL]\np : \u2115\nw : 1 < p\n\u22a2 LucasLehmerTest p \u2192 Nat.Prime (mersenne p)\n[PROOFSTEP]\nlet p' := p - 2\n[GOAL]\np : \u2115\nw : 1 < p\np' : \u2115 := p - 2\n\u22a2 LucasLehmerTest p \u2192 Nat.Prime (mersenne p)\n[PROOFSTEP]\nhave z : p = p' + 2 := (tsub_eq_iff_eq_add_of_le w.nat_succ_le).mp rfl\n[GOAL]\np : \u2115\nw : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\n\u22a2 LucasLehmerTest p \u2192 Nat.Prime (mersenne p)\n[PROOFSTEP]\nhave w : 1 < p' + 2 := Nat.lt_of_sub_eq_succ rfl\n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\n\u22a2 LucasLehmerTest p \u2192 Nat.Prime (mersenne p)\n[PROOFSTEP]\ncontrapose\n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\n\u22a2 \u00acNat.Prime (mersenne p) \u2192 \u00acLucasLehmerTest p\n[PROOFSTEP]\nintro a t\n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : \u00acNat.Prime (mersenne p)\nt : LucasLehmerTest p\n\u22a2 False\n[PROOFSTEP]\nrw [z] at a \n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : \u00acNat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest p\n\u22a2 False\n[PROOFSTEP]\nrw [z] at t \n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : \u00acNat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\n\u22a2 False\n[PROOFSTEP]\nhave h\u2081 := order_ineq p' t\n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : \u00acNat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh\u2081 : 2 ^ (p' + 2) < \u2191(q (p' + 2)) ^ 2\n\u22a2 False\n[PROOFSTEP]\nhave h\u2082 := Nat.minFac_sq_le_self (mersenne_pos (Nat.lt_of_succ_lt w)) a\n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : \u00acNat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh\u2081 : 2 ^ (p' + 2) < \u2191(q (p' + 2)) ^ 2\nh\u2082 : Nat.minFac (mersenne (p' + 2)) ^ 2 \u2264 mersenne (p' + 2)\n\u22a2 False\n[PROOFSTEP]\nhave h := lt_of_lt_of_le h\u2081 h\u2082\n[GOAL]\np : \u2115\nw\u271d : 1 < p\np' : \u2115 := p - 2\nz : p = p' + 2\nw : 1 < p' + 2\na : \u00acNat.Prime (mersenne (p' + 2))\nt : LucasLehmerTest (p' + 2)\nh\u2081 : 2 ^ (p' + 2) < \u2191(q (p' + 2)) ^ 2\nh\u2082 : Nat.minFac (mersenne (p' + 2)) ^ 2 \u2264 mersenne (p' + 2)\nh : 2 ^ (p' + 2) < mersenne (p' + 2)\n\u22a2 False\n[PROOFSTEP]\nexact not_lt_of_ge (Nat.sub_le _ _) h\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\n\u22a2 \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n[PROOFSTEP]\nhave h1 :=\n  calc\n    4 = 2 ^ 2 := by norm_num\n    _ \u2264 2 ^ p := Nat.pow_le_pow_of_le_right (by norm_num) hp\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\n\u22a2 4 = 2 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\n\u22a2 2 > 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\n\u22a2 \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n[PROOFSTEP]\nhave h2 : 1 \u2264 2 ^ p := by linarith\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\n\u22a2 1 \u2264 2 ^ p\n[PROOFSTEP]\nlinarith\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\n\u22a2 \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n[PROOFSTEP]\ninduction k with\n| zero =>\n  rw [sMod', sMod, Int.ofNat_emod]\n  simp [h2]\n| succ k ih =>\n  rw [sMod', sMod, \u2190 ih]\n  have h3 : 2 \u2264 2 ^ p - 1 := by\n    zify [h2]\n    calc\n      (2 : Int) \u2264 4 - 1 := by norm_num\n      _ \u2264 2 ^ p - 1 := by zify at h1 ; exact Int.sub_le_sub_right h1 _\n  zify [h2, h3]\n  rw [\u2190 add_sub_assoc, sub_eq_add_neg, add_assoc, add_comm _ (-2), \u2190 add_assoc, Int.add_emod_self, \u2190 sub_eq_add_neg]\n[GOAL]\np k : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\n\u22a2 \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n[PROOFSTEP]\ninduction k with\n| zero =>\n  rw [sMod', sMod, Int.ofNat_emod]\n  simp [h2]\n| succ k ih =>\n  rw [sMod', sMod, \u2190 ih]\n  have h3 : 2 \u2264 2 ^ p - 1 := by\n    zify [h2]\n    calc\n      (2 : Int) \u2264 4 - 1 := by norm_num\n      _ \u2264 2 ^ p - 1 := by zify at h1 ; exact Int.sub_le_sub_right h1 _\n  zify [h2, h3]\n  rw [\u2190 add_sub_assoc, sub_eq_add_neg, add_assoc, add_comm _ (-2), \u2190 add_assoc, Int.add_emod_self, \u2190 sub_eq_add_neg]\n[GOAL]\ncase zero\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\n\u22a2 \u2191(sMod' (2 ^ p - 1) Nat.zero) = sMod p Nat.zero\n[PROOFSTEP]\n\n| zero =>\n  rw [sMod', sMod, Int.ofNat_emod]\n  simp [h2]\n[GOAL]\ncase zero\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\n\u22a2 \u2191(sMod' (2 ^ p - 1) Nat.zero) = sMod p Nat.zero\n[PROOFSTEP]\nrw [sMod', sMod, Int.ofNat_emod]\n[GOAL]\ncase zero\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\n\u22a2 \u21914 % \u2191(2 ^ p - 1) = 4 % (2 ^ p - 1)\n[PROOFSTEP]\nsimp [h2]\n[GOAL]\ncase succ\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 \u2191(sMod' (2 ^ p - 1) (Nat.succ k)) = sMod p (Nat.succ k)\n[PROOFSTEP]\n\n| succ k ih =>\n  rw [sMod', sMod, \u2190 ih]\n  have h3 : 2 \u2264 2 ^ p - 1 := by\n    zify [h2]\n    calc\n      (2 : Int) \u2264 4 - 1 := by norm_num\n      _ \u2264 2 ^ p - 1 := by zify at h1 ; exact Int.sub_le_sub_right h1 _\n  zify [h2, h3]\n  rw [\u2190 add_sub_assoc, sub_eq_add_neg, add_assoc, add_comm _ (-2), \u2190 add_assoc, Int.add_emod_self, \u2190 sub_eq_add_neg]\n[GOAL]\ncase succ\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 \u2191(sMod' (2 ^ p - 1) (Nat.succ k)) = sMod p (Nat.succ k)\n[PROOFSTEP]\nrw [sMod', sMod, \u2190 ih]\n[GOAL]\ncase succ\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 \u2191((sMod' (2 ^ p - 1) k ^ 2 + (2 ^ p - 1 - 2)) % (2 ^ p - 1)) = (\u2191(sMod' (2 ^ p - 1) k) ^ 2 - 2) % (2 ^ p - 1)\n[PROOFSTEP]\nhave h3 : 2 \u2264 2 ^ p - 1 := by\n  zify [h2]\n  calc\n    (2 : Int) \u2264 4 - 1 := by norm_num\n    _ \u2264 2 ^ p - 1 := by zify at h1 ; exact Int.sub_le_sub_right h1 _\n[GOAL]\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 2 \u2264 2 ^ p - 1\n[PROOFSTEP]\nzify [h2]\n[GOAL]\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 2 \u2264 2 ^ p - 1\n[PROOFSTEP]\ncalc\n  (2 : Int) \u2264 4 - 1 := by norm_num\n  _ \u2264 2 ^ p - 1 := by zify at h1 ; exact Int.sub_le_sub_right h1 _\n[GOAL]\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 2 \u2264 4 - 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\n\u22a2 4 - 1 \u2264 2 ^ p - 1\n[PROOFSTEP]\nzify at h1 \n[GOAL]\np : \u2115\nhp : 2 \u2264 p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\nh1 : 4 \u2264 2 ^ p\n\u22a2 4 - 1 \u2264 2 ^ p - 1\n[PROOFSTEP]\nexact Int.sub_le_sub_right h1 _\n[GOAL]\ncase succ\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\nh3 : 2 \u2264 2 ^ p - 1\n\u22a2 \u2191((sMod' (2 ^ p - 1) k ^ 2 + (2 ^ p - 1 - 2)) % (2 ^ p - 1)) = (\u2191(sMod' (2 ^ p - 1) k) ^ 2 - 2) % (2 ^ p - 1)\n[PROOFSTEP]\nzify [h2, h3]\n[GOAL]\ncase succ\np : \u2115\nhp : 2 \u2264 p\nh1 : 4 \u2264 2 ^ p\nh2 : 1 \u2264 2 ^ p\nk : \u2115\nih : \u2191(sMod' (2 ^ p - 1) k) = sMod p k\nh3 : 2 \u2264 2 ^ p - 1\n\u22a2 (\u2191(sMod' (2 ^ p - 1) k) ^ 2 + (2 ^ p - 1 - 2)) % (2 ^ p - 1) = (\u2191(sMod' (2 ^ p - 1) k) ^ 2 - 2) % (2 ^ p - 1)\n[PROOFSTEP]\nrw [\u2190 add_sub_assoc, sub_eq_add_neg, add_assoc, add_comm _ (-2), \u2190 add_assoc, Int.add_emod_self, \u2190 sub_eq_add_neg]\n[GOAL]\np : \u2115\nhp : Nat.blt 1 p = true\nh : sMod' (2 ^ p - 1) (p - 2) = 0\n\u22a2 LucasLehmerTest p\n[PROOFSTEP]\nrw [Nat.blt_eq] at hp \n[GOAL]\np : \u2115\nhp : 1 < p\nh : sMod' (2 ^ p - 1) (p - 2) = 0\n\u22a2 LucasLehmerTest p\n[PROOFSTEP]\nrw [LucasLehmerTest, LucasLehmer.residue_eq_zero_iff_sMod_eq_zero p hp, \u2190 sMod'_eq_sMod p _ hp, h]\n[GOAL]\np : \u2115\nhp : 1 < p\nh : sMod' (2 ^ p - 1) (p - 2) = 0\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nhp : Nat.blt 1 p = true\nh : Nat.ble 1 (sMod' (2 ^ p - 1) (p - 2)) = true\n\u22a2 \u00acLucasLehmerTest p\n[PROOFSTEP]\nrw [Nat.blt_eq] at hp \n[GOAL]\np : \u2115\nhp : 1 < p\nh : Nat.ble 1 (sMod' (2 ^ p - 1) (p - 2)) = true\n\u22a2 \u00acLucasLehmerTest p\n[PROOFSTEP]\nrw [Nat.ble_eq, Nat.succ_le, Nat.pos_iff_ne_zero] at h \n[GOAL]\np : \u2115\nhp : 1 < p\nh : sMod' (2 ^ p - 1) (p - 2) \u2260 0\n\u22a2 \u00acLucasLehmerTest p\n[PROOFSTEP]\nrw [LucasLehmerTest, LucasLehmer.residue_eq_zero_iff_sMod_eq_zero p hp, \u2190 sMod'_eq_sMod p _ hp]\n[GOAL]\np : \u2115\nhp : 1 < p\nh : sMod' (2 ^ p - 1) (p - 2) \u2260 0\n\u22a2 \u00ac\u2191(sMod' (2 ^ p - 1) (p - 2)) = 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nn k : \u2115\n\u22a2 1 * (k / 2 ^ n) + k % 2 ^ n = k / 2 ^ n + k % 2 ^ n\n[PROOFSTEP]\nrw [one_mul]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LucasLehmer", "llama_tokens": 19564, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8633916099737806, "lm_q2_score": 0.6757645879592641, "lm_q1q2_score": 0.5834494755614175}}
{"text": "[GOAL]\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\nn : \u2115\nf : Functions L n\nx : Fin n \u2192 M\n\u22a2 (funMap f fun i => Quotient.mk s (x i)) = Quotient.mk s (funMap f x)\n[PROOFSTEP]\nchange Quotient.map (@funMap L M ps.toStructure n f) Prestructure.fun_equiv (Quotient.finChoice _) = _\n[GOAL]\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\nn : \u2115\nf : Functions L n\nx : Fin n \u2192 M\n\u22a2 Quotient.map (funMap f) (_ : \u2200 (x y : Fin n \u2192 M), x \u2248 y \u2192 funMap f x \u2248 funMap f y)\n      (Quotient.finChoice fun i => Quotient.mk s (x i)) =\n    Quotient.mk s (funMap f x)\n[PROOFSTEP]\nrw [Quotient.finChoice_eq, Quotient.map_mk]\n[GOAL]\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\nn : \u2115\nr : Relations L n\nx : Fin n \u2192 M\n\u22a2 (RelMap r fun i => Quotient.mk s (x i)) \u2194 RelMap r x\n[PROOFSTEP]\nchange Quotient.lift (@RelMap L M ps.toStructure n r) Prestructure.rel_equiv (Quotient.finChoice _) \u2194 _\n[GOAL]\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\nn : \u2115\nr : Relations L n\nx : Fin n \u2192 M\n\u22a2 Quotient.lift (RelMap r) (_ : \u2200 (x y : Fin n \u2192 M), x \u2248 y \u2192 RelMap r x = RelMap r y)\n      (Quotient.finChoice fun i => Quotient.mk s (x i)) \u2194\n    RelMap r x\n[PROOFSTEP]\nrw [Quotient.finChoice_eq, Quotient.lift_mk]\n[GOAL]\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\n\u03b2 : Type u_2\nt : Term L \u03b2\nx : \u03b2 \u2192 M\n\u22a2 realize (fun i => Quotient.mk s (x i)) t = Quotient.mk s (realize x t)\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase var\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\n\u03b2 : Type u_2\nx : \u03b2 \u2192 M\n_a\u271d : \u03b2\n\u22a2 realize (fun i => Quotient.mk s (x i)) (var _a\u271d) = Quotient.mk s (realize x (var _a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nM : Type u_1\ns : Setoid M\nps : Prestructure L s\n\u03b2 : Type u_2\nx : \u03b2 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b2\nih : \u2200 (a : Fin l\u271d), realize (fun i => Quotient.mk s (x i)) (_ts\u271d a) = Quotient.mk s (realize x (_ts\u271d a))\n\u22a2 realize (fun i => Quotient.mk s (x i)) (func _f\u271d _ts\u271d) = Quotient.mk s (realize x (func _f\u271d _ts\u271d))\n[PROOFSTEP]\nsimp only [ih, funMap_quotient_mk', Term.realize]\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.Quotients", "llama_tokens": 986, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324893520001, "lm_q2_score": 0.7154239957834733, "lm_q1q2_score": 0.5832368850247157}}
{"text": "[GOAL]\nt : \u211d\n\u22a2 t \u2208 I \u2194 1 - t \u2208 I\n[PROOFSTEP]\nrw [mem_Icc, mem_Icc]\n[GOAL]\nt : \u211d\n\u22a2 0 \u2264 t \u2227 t \u2264 1 \u2194 0 \u2264 1 - t \u2227 1 - t \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nt : \u211d\n\u22a2 0 \u2264 t \u2227 t \u2264 1 \u2192 0 \u2264 1 - t \u2227 1 - t \u2264 1\n[PROOFSTEP]\nintro\n[GOAL]\ncase mpr\nt : \u211d\n\u22a2 0 \u2264 1 - t \u2227 1 - t \u2264 1 \u2192 0 \u2264 t \u2227 t \u2264 1\n[PROOFSTEP]\nintro\n[GOAL]\ncase mp\nt : \u211d\na\u271d : 0 \u2264 t \u2227 t \u2264 1\n\u22a2 0 \u2264 1 - t \u2227 1 - t \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr\nt : \u211d\na\u271d : 0 \u2264 1 - t \u2227 1 - t \u2264 1\n\u22a2 0 \u2264 t \u2227 t \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nt : \u211d\na\u271d : 0 \u2264 t \u2227 t \u2264 1\n\u22a2 0 \u2264 1 - t\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mp.right\nt : \u211d\na\u271d : 0 \u2264 t \u2227 t \u2264 1\n\u22a2 1 - t \u2264 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr.left\nt : \u211d\na\u271d : 0 \u2264 1 - t \u2227 1 - t \u2264 1\n\u22a2 0 \u2264 t\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr.right\nt : \u211d\na\u271d : 0 \u2264 1 - t \u2227 1 - t \u2264 1\n\u22a2 t \u2264 1\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u22a2 1 \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase right\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 \u2191(\u03c3 0) = \u21911\n[PROOFSTEP]\nsimp [symm]\n[GOAL]\n\u22a2 \u2191(\u03c3 1) = \u21910\n[PROOFSTEP]\nsimp [symm]\n[GOAL]\nx : \u2191I\n\u22a2 \u2191(\u03c3 (\u03c3 x)) = \u2191x\n[PROOFSTEP]\nsimp [symm]\n[GOAL]\n\u22a2 CompactSpace \u2191I\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nx : \u2191I\n\u22a2 0 \u2264 1 - \u2191x\n[PROOFSTEP]\nsimpa using x.2.2\n[GOAL]\nx : \u2191I\n\u22a2 1 - \u2191x \u2264 1\n[PROOFSTEP]\nsimpa using x.2.1\n[GOAL]\na t : \u211d\nha : 0 < a\n\u22a2 a * t \u2208 I \u2194 t \u2208 Icc 0 (1 / a)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\na t : \u211d\nha : 0 < a\n\u22a2 a * t \u2208 I \u2192 t \u2208 Icc 0 (1 / a)\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mpr\na t : \u211d\nha : 0 < a\n\u22a2 t \u2208 Icc 0 (1 / a) \u2192 a * t \u2208 I\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mp.intro\na t : \u211d\nha : 0 < a\nh\u2081 : 0 \u2264 a * t\nh\u2082 : a * t \u2264 1\n\u22a2 t \u2208 Icc 0 (1 / a)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro\na t : \u211d\nha : 0 < a\nh\u2081 : 0 \u2264 t\nh\u2082 : t \u2264 1 / a\n\u22a2 a * t \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.intro.left\na t : \u211d\nha : 0 < a\nh\u2081 : 0 \u2264 a * t\nh\u2082 : a * t \u2264 1\n\u22a2 0 \u2264 t\n[PROOFSTEP]\nexact nonneg_of_mul_nonneg_right h\u2081 ha\n[GOAL]\ncase mp.intro.right\na t : \u211d\nha : 0 < a\nh\u2081 : 0 \u2264 a * t\nh\u2082 : a * t \u2264 1\n\u22a2 t \u2264 1 / a\n[PROOFSTEP]\nrwa [le_div_iff ha, mul_comm]\n[GOAL]\ncase mpr.intro.left\na t : \u211d\nha : 0 < a\nh\u2081 : 0 \u2264 t\nh\u2082 : t \u2264 1 / a\n\u22a2 0 \u2264 a * t\n[PROOFSTEP]\nexact mul_nonneg ha.le h\u2081\n[GOAL]\ncase mpr.intro.right\na t : \u211d\nha : 0 < a\nh\u2081 : 0 \u2264 t\nh\u2082 : t \u2264 1 / a\n\u22a2 a * t \u2264 1\n[PROOFSTEP]\nrwa [le_div_iff ha, mul_comm] at h\u2082 \n[GOAL]\nt : \u211d\n\u22a2 2 * t - 1 \u2208 I \u2194 t \u2208 Icc (1 / 2) 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nt : \u211d\n\u22a2 2 * t - 1 \u2208 I \u2192 t \u2208 Icc (1 / 2) 1\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mpr\nt : \u211d\n\u22a2 t \u2208 Icc (1 / 2) 1 \u2192 2 * t - 1 \u2208 I\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mp.intro\nt : \u211d\nh\u2081 : 0 \u2264 2 * t - 1\nh\u2082 : 2 * t - 1 \u2264 1\n\u22a2 t \u2208 Icc (1 / 2) 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro\nt : \u211d\nh\u2081 : 1 / 2 \u2264 t\nh\u2082 : t \u2264 1\n\u22a2 2 * t - 1 \u2208 I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.intro.left\nt : \u211d\nh\u2081 : 0 \u2264 2 * t - 1\nh\u2082 : 2 * t - 1 \u2264 1\n\u22a2 1 / 2 \u2264 t\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mp.intro.right\nt : \u211d\nh\u2081 : 0 \u2264 2 * t - 1\nh\u2082 : 2 * t - 1 \u2264 1\n\u22a2 t \u2264 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr.intro.left\nt : \u211d\nh\u2081 : 1 / 2 \u2264 t\nh\u2082 : t \u2264 1\n\u22a2 0 \u2264 2 * t - 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr.intro.right\nt : \u211d\nh\u2081 : 1 / 2 \u2264 t\nh\u2082 : t \u2264 1\n\u22a2 2 * t - 1 \u2264 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u2191unitInterval\n\u22a2 0 \u2264 \u2191x\n[PROOFSTEP]\nunit_interval\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : 0 < a\n\u22a2 \u2191(affineHomeomorph a b (_ : a \u2260 0)) '' Icc 0 1 = Icc b (a + b)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a < b\n\u22a2 \u2191(Icc a b) \u2243\u209c \u2191(Icc 0 1)\n[PROOFSTEP]\nlet e := Homeomorph.image (affineHomeomorph (b - a) a (sub_pos.mpr h).ne.symm) (Set.Icc 0 1)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a < b\ne : \u2191(Icc 0 1) \u2243\u209c \u2191(\u2191(affineHomeomorph (b - a) a (_ : b - a \u2260 0)) '' Icc 0 1) :=\n  Homeomorph.image (affineHomeomorph (b - a) a (_ : b - a \u2260 0)) (Icc 0 1)\n\u22a2 \u2191(Icc a b) \u2243\u209c \u2191(Icc 0 1)\n[PROOFSTEP]\nrefine' (e.trans _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a < b\ne : \u2191(Icc 0 1) \u2243\u209c \u2191(\u2191(affineHomeomorph (b - a) a (_ : b - a \u2260 0)) '' Icc 0 1) :=\n  Homeomorph.image (affineHomeomorph (b - a) a (_ : b - a \u2260 0)) (Icc 0 1)\n\u22a2 \u2191(\u2191(affineHomeomorph (b - a) a (_ : b - a \u2260 0)) '' Icc 0 1) \u2243\u209c \u2191(Icc a b)\n[PROOFSTEP]\napply Homeomorph.setCongr\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a < b\ne : \u2191(Icc 0 1) \u2243\u209c \u2191(\u2191(affineHomeomorph (b - a) a (_ : b - a \u2260 0)) '' Icc 0 1) :=\n  Homeomorph.image (affineHomeomorph (b - a) a (_ : b - a \u2260 0)) (Icc 0 1)\n\u22a2 \u2191(affineHomeomorph (b - a) a (_ : b - a \u2260 0)) '' Icc 0 1 = Icc a b\n[PROOFSTEP]\nrw [affineHomeomorph_image_I _ _ (sub_pos.2 h)]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : TopologicalRing \ud835\udd5c\na b : \ud835\udd5c\nh : a < b\ne : \u2191(Icc 0 1) \u2243\u209c \u2191(\u2191(affineHomeomorph (b - a) a (_ : b - a \u2260 0)) '' Icc 0 1) :=\n  Homeomorph.image (affineHomeomorph (b - a) a (_ : b - a \u2260 0)) (Icc 0 1)\n\u22a2 Icc a (b - a + a) = Icc a b\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Topology.UnitInterval", "llama_tokens": 3057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199592797929, "lm_q2_score": 0.6959583250334527, "lm_q1q2_score": 0.5829485838749535}}
{"text": "[GOAL]\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\n\u22a2 M \u2243\u2097[R] M\n[PROOFSTEP]\nrefine'\n  { toLin b b A with\n    toFun := toLin b b A\n    invFun := toLin b b A\u207b\u00b9\n    left_inv := fun x => _\n    right_inv := fun x => _ }\n[GOAL]\ncase refine'_1\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\nsrc\u271d : (fun x => M \u2192\u2097[R] M) A := \u2191(toLin b b) A\nx : M\n\u22a2 \u2191(\u2191(toLin b b) A\u207b\u00b9)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191(toLin b b) A),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : 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 : 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        x) =\n    x\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase refine'_2\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\nsrc\u271d : (fun x => M \u2192\u2097[R] M) A := \u2191(toLin b b) A\nx : M\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191(\u2191(toLin b b) A),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : 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 : 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      (\u2191(\u2191(toLin b b) A\u207b\u00b9) x) =\n    x\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase refine'_1\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\nsrc\u271d : (fun x => M \u2192\u2097[R] M) A := \u2191(toLin b b) A\nx : M\n\u22a2 \u2191(\u2191(toLin b b) A\u207b\u00b9) (\u2191(\u2191(toLin b b) A) x) = x\n[PROOFSTEP]\nrw [\u2190 LinearMap.comp_apply]\n[GOAL]\ncase refine'_2\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\nsrc\u271d : (fun x => M \u2192\u2097[R] M) A := \u2191(toLin b b) A\nx : M\n\u22a2 \u2191(\u2191(toLin b b) A) (\u2191(\u2191(toLin b b) A\u207b\u00b9) x) = x\n[PROOFSTEP]\nrw [\u2190 LinearMap.comp_apply]\n[GOAL]\ncase refine'_1\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\nsrc\u271d : (fun x => M \u2192\u2097[R] M) A := \u2191(toLin b b) A\nx : M\n\u22a2 \u2191(comp (\u2191(toLin b b) A\u207b\u00b9) (\u2191(toLin b b) A)) x = x\n[PROOFSTEP]\nsimp only [\u2190 Matrix.toLin_mul b b b, Matrix.nonsing_inv_mul _ hA, Matrix.mul_nonsing_inv _ hA, toLin_one,\n  LinearMap.id_apply]\n[GOAL]\ncase refine'_2\nn : Type u_1\ninst\u271d\u2074 : Fintype n\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nb : Basis n R M\ninst\u271d : DecidableEq n\nA : Matrix n n R\nhA : IsUnit (det A)\nsrc\u271d : (fun x => M \u2192\u2097[R] M) A := \u2191(toLin b b) A\nx : M\n\u22a2 \u2191(comp (\u2191(toLin b b) A) (\u2191(toLin b b) A\u207b\u00b9)) x = x\n[PROOFSTEP]\nsimp only [\u2190 Matrix.toLin_mul b b b, Matrix.nonsing_inv_mul _ hA, Matrix.mul_nonsing_inv _ hA, toLin_one,\n  LinearMap.id_apply]\n[GOAL]\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2194 det M = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2192 det M = 0\n[PROOFSTEP]\nrintro \u27e8v, hv, mul_eq\u27e9\n[GOAL]\ncase mp.intro.intro\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec M v = 0\n\u22a2 det M = 0\n[PROOFSTEP]\ncontrapose! hv\n[GOAL]\ncase mp.intro.intro\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nv : n \u2192 K\nmul_eq : mulVec M v = 0\nhv : det M \u2260 0\n\u22a2 v = 0\n[PROOFSTEP]\nexact eq_zero_of_mulVec_eq_zero hv mul_eq\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\n\u22a2 det M = 0 \u2192 \u2203 v x, mulVec M v = 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\n\u22a2 (\u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0) \u2192 det M \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\n\u22a2 det M \u2260 0\n[PROOFSTEP]\nhave : Function.Injective (Matrix.toLin' M) := by\n  simpa only [\u2190 LinearMap.ker_eq_bot, ker_toLin'_eq_bot_iff, not_imp_not] using h\n[GOAL]\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\n\u22a2 Function.Injective \u2191(\u2191toLin' M)\n[PROOFSTEP]\nsimpa only [\u2190 LinearMap.ker_eq_bot, ker_toLin'_eq_bot_iff, not_imp_not] using h\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\nthis : Function.Injective \u2191(\u2191toLin' M)\n\u22a2 det M \u2260 0\n[PROOFSTEP]\nhave : M * LinearMap.toMatrix' ((LinearEquiv.ofInjectiveEndo (Matrix.toLin' M) this).symm : (n \u2192 K) \u2192\u2097[K] n \u2192 K) = 1 :=\n  by\n  refine' Matrix.toLin'.injective (LinearMap.ext fun v => _)\n  rw [Matrix.toLin'_mul, Matrix.toLin'_one, Matrix.toLin'_toMatrix', LinearMap.comp_apply]\n  exact (LinearEquiv.ofInjectiveEndo (Matrix.toLin' M) this).apply_symm_apply v\n[GOAL]\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\nthis : Function.Injective \u2191(\u2191toLin' M)\n\u22a2 M * \u2191toMatrix' \u2191(LinearEquiv.symm (LinearEquiv.ofInjectiveEndo (\u2191toLin' M) this)) = 1\n[PROOFSTEP]\nrefine' Matrix.toLin'.injective (LinearMap.ext fun v => _)\n[GOAL]\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\nthis : Function.Injective \u2191(\u2191toLin' M)\nv : n \u2192 K\n\u22a2 \u2191(\u2191toLin' (M * \u2191toMatrix' \u2191(LinearEquiv.symm (LinearEquiv.ofInjectiveEndo (\u2191toLin' M) this)))) v = \u2191(\u2191toLin' 1) v\n[PROOFSTEP]\nrw [Matrix.toLin'_mul, Matrix.toLin'_one, Matrix.toLin'_toMatrix', LinearMap.comp_apply]\n[GOAL]\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\nthis : Function.Injective \u2191(\u2191toLin' M)\nv : n \u2192 K\n\u22a2 \u2191(\u2191toLin' M) (\u2191\u2191(LinearEquiv.symm (LinearEquiv.ofInjectiveEndo (\u2191toLin' M) this)) v) = \u2191LinearMap.id v\n[PROOFSTEP]\nexact (LinearEquiv.ofInjectiveEndo (Matrix.toLin' M) this).apply_symm_apply v\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2075 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\u271d\ninst\u271d\u00b2 : Module R M\u271d\nK : Type u_4\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Field K\nM : Matrix n n K\nh : \u2200 (v : n \u2192 K), v \u2260 0 \u2192 mulVec M v \u2260 0\nthis\u271d : Function.Injective \u2191(\u2191toLin' M)\nthis : M * \u2191toMatrix' \u2191(LinearEquiv.symm (LinearEquiv.ofInjectiveEndo (\u2191toLin' M) this\u271d)) = 1\n\u22a2 det M \u2260 0\n[PROOFSTEP]\nexact Matrix.det_ne_zero_of_right_inverse this\n[GOAL]\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2194 det M = 0\n[PROOFSTEP]\nhave : (\u2203 (v : _) (_ : v \u2260 0), mulVec ((algebraMap A K).mapMatrix M) v = 0) \u2194 _ := exists_mulVec_eq_zero_iff_aux\n[GOAL]\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis :\n  (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det (\u2191(RingHom.mapMatrix (algebraMap A K)) M) = 0\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2194 det M = 0\n[PROOFSTEP]\nrw [\u2190 RingHom.map_det, IsFractionRing.to_map_eq_zero_iff] at this \n[GOAL]\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2194 det M = 0\n[PROOFSTEP]\nrefine' Iff.trans _ this\n[GOAL]\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2194 \u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\n\u22a2 (\u2203 v x, mulVec M v = 0) \u2192 \u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\n[PROOFSTEP]\nrintro \u27e8v, hv, mul_eq\u27e9\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\n\u22a2 (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2192 \u2203 v x, mulVec M v = 0\n[PROOFSTEP]\nrintro \u27e8v, hv, mul_eq\u27e9\n[GOAL]\ncase mp.intro.intro\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 A\nhv : v \u2260 0\nmul_eq : mulVec M v = 0\n\u22a2 \u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\n[PROOFSTEP]\nrefine' \u27e8fun i => algebraMap _ _ (v i), mt (fun h => funext fun i => _) hv, _\u27e9\n[GOAL]\ncase mp.intro.intro.refine'_1\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 A\nhv : v \u2260 0\nmul_eq : mulVec M v = 0\nh : (fun i => \u2191(algebraMap A K) (v i)) = 0\ni : n\n\u22a2 v i = OfNat.ofNat 0 i\n[PROOFSTEP]\nexact IsFractionRing.to_map_eq_zero_iff.mp (congr_fun h i)\n[GOAL]\ncase mp.intro.intro.refine'_2\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 A\nhv : v \u2260 0\nmul_eq : mulVec M v = 0\n\u22a2 (mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) fun i => \u2191(algebraMap A K) (v i)) = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase mp.intro.intro.refine'_2.h\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 A\nhv : v \u2260 0\nmul_eq : mulVec M v = 0\ni : n\n\u22a2 mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) (fun i => \u2191(algebraMap A K) (v i)) i = OfNat.ofNat 0 i\n[PROOFSTEP]\nrefine' (RingHom.map_mulVec _ _ _ i).symm.trans _\n[GOAL]\ncase mp.intro.intro.refine'_2.h\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 A\nhv : v \u2260 0\nmul_eq : mulVec M v = 0\ni : n\n\u22a2 \u2191(algebraMap A K) (mulVec (fun i j => M i j) (fun i => v i) i) = OfNat.ofNat 0 i\n[PROOFSTEP]\nrw [mul_eq, Pi.zero_apply, RingHom.map_zero, Pi.zero_apply]\n[GOAL]\ncase mpr.intro.intro\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\n\u22a2 \u2203 v x, mulVec M v = 0\n[PROOFSTEP]\nletI := Classical.decEq K\n[GOAL]\ncase mpr.intro.intro\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\n\u22a2 \u2203 v x, mulVec M v = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8b, hb\u27e9, ba_eq\u27e9 := IsLocalization.exist_integer_multiples_of_finset (nonZeroDivisors A) (Finset.univ.image v)\n[GOAL]\ncase mpr.intro.intro.intro.mk\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nba_eq : \u2200 (a : K), a \u2208 Finset.image v Finset.univ \u2192 IsLocalization.IsInteger A (\u2191{ val := b, property := hb } \u2022 a)\n\u22a2 \u2203 v x, mulVec M v = 0\n[PROOFSTEP]\nchoose f hf using ba_eq\n[GOAL]\ncase mpr.intro.intro.intro.mk\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\n\u22a2 \u2203 v x, mulVec M v = 0\n[PROOFSTEP]\nrefine' \u27e8fun i => f _ (Finset.mem_image.mpr \u27e8i, Finset.mem_univ i, rfl\u27e9), mt (fun h => funext fun i => _) hv, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_1\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\nh : (fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) = 0\ni : n\n\u22a2 v i = OfNat.ofNat 0 i\n[PROOFSTEP]\nhave := congr_arg (algebraMap A K) (congr_fun h i)\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_1\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d\u00b9 : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis\u271d : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\nh : (fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) = 0\ni : n\nthis : \u2191(algebraMap A K) (f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) = \u2191(algebraMap A K) (OfNat.ofNat 0 i)\n\u22a2 v i = OfNat.ofNat 0 i\n[PROOFSTEP]\nrw [hf, Subtype.coe_mk, Pi.zero_apply, RingHom.map_zero, Algebra.smul_def, mul_eq_zero,\n  IsFractionRing.to_map_eq_zero_iff] at this \n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_1\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d\u00b9 : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis\u271d : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\nh : (fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) = 0\ni : n\nthis : b = 0 \u2228 v i = 0\n\u22a2 v i = OfNat.ofNat 0 i\n[PROOFSTEP]\nexact this.resolve_left (nonZeroDivisors.ne_zero hb)\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_2\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\n\u22a2 (mulVec M fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_2.h\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\ni : n\n\u22a2 mulVec M (fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) i = OfNat.ofNat 0 i\n[PROOFSTEP]\nrefine' IsFractionRing.injective A K _\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_2.h\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\ni : n\n\u22a2 \u2191(algebraMap A K) (mulVec M (fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) i) =\n    \u2191(algebraMap A K) (OfNat.ofNat 0 i)\n[PROOFSTEP]\ncalc\n  algebraMap A K (M.mulVec (fun i : n => f (v i) _) i) =\n      ((algebraMap A K).mapMatrix M).mulVec (algebraMap _ K b \u2022 v) i :=\n    ?_\n  _ = 0 := ?_\n  _ = algebraMap A K 0 := (RingHom.map_zero _).symm\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_2.h.calc_1\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\ni : n\n\u22a2 \u2191(algebraMap A K) (mulVec M (fun i => f (v i) (_ : v i \u2208 Finset.image v Finset.univ)) i) =\n    mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) (\u2191(algebraMap A K) b \u2022 v) i\n[PROOFSTEP]\nsimp_rw [RingHom.map_mulVec, mulVec, dotProduct, Function.comp_apply, hf, RingHom.mapMatrix_apply, Pi.smul_apply,\n  smul_eq_mul, Algebra.smul_def]\n[GOAL]\ncase mpr.intro.intro.intro.mk.refine'_2.h.calc_2\nn : Type u_1\ninst\u271d\u2079 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : Module R M\u271d\nA : Type u_4\nK : Type u_5\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Nontrivial A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nM : Matrix n n A\nthis\u271d : (\u2203 v x, mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0) \u2194 det M = 0\nv : n \u2192 K\nhv : v \u2260 0\nmul_eq : mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) v = 0\nthis : DecidableEq K := Classical.decEq K\nb : A\nhb : b \u2208 nonZeroDivisors A\nf : (a : K) \u2192 a \u2208 Finset.image v Finset.univ \u2192 A\nhf : \u2200 (a : K) (a_1 : a \u2208 Finset.image v Finset.univ), \u2191(algebraMap A K) (f a a_1) = \u2191{ val := b, property := hb } \u2022 a\ni : n\n\u22a2 mulVec (\u2191(RingHom.mapMatrix (algebraMap A K)) M) (\u2191(algebraMap A K) b \u2022 v) i = 0\n[PROOFSTEP]\nrw [mulVec_smul, mul_eq, Pi.smul_apply, Pi.zero_apply, smul_zero]\n[GOAL]\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\n\u22a2 (\u2203 v x, vecMul v M = 0) \u2194 det M = 0\n[PROOFSTEP]\nsimpa only [\u2190 M.det_transpose, \u2190 mulVec_transpose] using exists_mulVec_eq_zero_iff\n[GOAL]\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\n\u22a2 Nondegenerate M \u2194 det M \u2260 0\n[PROOFSTEP]\nrefine' Iff.trans _ (not_iff_not.mpr exists_vecMul_eq_zero_iff)\n[GOAL]\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\n\u22a2 Nondegenerate M \u2194 \u00ac\u2203 v x, vecMul v M = 0\n[PROOFSTEP]\nsimp only [not_exists]\n[GOAL]\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\n\u22a2 Nondegenerate M \u2194 \u2200 (x : n \u2192 A), x \u2260 0 \u2192 \u00acvecMul x M = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\n\u22a2 Nondegenerate M \u2192 \u2200 (x : n \u2192 A), x \u2260 0 \u2192 \u00acvecMul x M = 0\n[PROOFSTEP]\nintro hM v hv hMv\n[GOAL]\ncase mp\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\nhM : Nondegenerate M\nv : n \u2192 A\nhv : v \u2260 0\nhMv : vecMul v M = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8w, hwMv\u27e9 := hM.exists_not_ortho_of_ne_zero hv\n[GOAL]\ncase mp.intro\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\nhM : Nondegenerate M\nv : n \u2192 A\nhv : v \u2260 0\nhMv : vecMul v M = 0\nw : n \u2192 A\nhwMv : v \u2b1d\u1d65 mulVec M w \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp [dotProduct_mulVec, hMv, zero_dotProduct, ne_eq, not_true] at hwMv \n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\n\u22a2 (\u2200 (x : n \u2192 A), x \u2260 0 \u2192 \u00acvecMul x M = 0) \u2192 Nondegenerate M\n[PROOFSTEP]\nintro h v hv\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\nh : \u2200 (x : n \u2192 A), x \u2260 0 \u2192 \u00acvecMul x M = 0\nv : n \u2192 A\nhv : \u2200 (w : n \u2192 A), v \u2b1d\u1d65 mulVec M w = 0\n\u22a2 v = 0\n[PROOFSTEP]\nrefine' not_imp_not.mp (h v) (funext fun i => _)\n[GOAL]\ncase mpr\nn : Type u_1\ninst\u271d\u2076 : Fintype n\nR : Type u_2\nM\u271d : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\u271d\ninst\u271d\u00b3 : Module R M\u271d\nA : Type u_4\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing A\ninst\u271d : IsDomain A\nM : Matrix n n A\nh : \u2200 (x : n \u2192 A), x \u2260 0 \u2192 \u00acvecMul x M = 0\nv : n \u2192 A\nhv : \u2200 (w : n \u2192 A), v \u2b1d\u1d65 mulVec M w = 0\ni : n\n\u22a2 vecMul v M i = OfNat.ofNat 0 i\n[PROOFSTEP]\nsimpa only [dotProduct_mulVec, dotProduct_single, mul_one] using hv (Pi.single i 1)\n[GOAL]\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (j : n), 0 < \u2211 i : n, A i j\n\u22a2 det A \u2260 0\n[PROOFSTEP]\ncases isEmpty_or_nonempty n\n[GOAL]\ncase inl\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (j : n), 0 < \u2211 i : n, A i j\nh\u271d : IsEmpty n\n\u22a2 det A \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (j : n), 0 < \u2211 i : n, A i j\nh\u271d : Nonempty n\n\u22a2 det A \u2260 0\n[PROOFSTEP]\ncontrapose! h2\n[GOAL]\ncase inr\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\n\u22a2 \u2203 j, \u2211 i : n, A i j \u2264 0\n[PROOFSTEP]\nobtain \u27e8v, \u27e8h_vnz, h_vA\u27e9\u27e9 := Matrix.exists_vecMul_eq_zero_iff.mpr h2\n[GOAL]\ncase inr.intro.intro\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\n\u22a2 \u2203 j, \u2211 i : n, A i j \u2264 0\n[PROOFSTEP]\nwlog h_sup : 0 < Finset.sup' Finset.univ Finset.univ_nonempty v\n[GOAL]\ncase inr.intro.intro.inr\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\nh_sup : \u00ac0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\n\u22a2 \u2203 j, \u2211 i : n, A i j \u2264 0\n[PROOFSTEP]\nrefine this h1 inferInstance h2 (-1 \u2022 v) ?_ ?_ ?_\n[GOAL]\ncase inr.intro.intro.inr.refine_1\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\nh_sup : \u00ac0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\n\u22a2 -1 \u2022 v \u2260 0\n[PROOFSTEP]\nexact smul_ne_zero (by norm_num) h_vnz\n[GOAL]\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\nh_sup : \u00ac0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\n\u22a2 -1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.intro.intro.inr.refine_2\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\nh_sup : \u00ac0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\n\u22a2 (vecMul (-1 \u2022 v) fun i j => A i j) = 0\n[PROOFSTEP]\nrw [Matrix.vecMul_smul, h_vA, smul_zero]\n[GOAL]\ncase inr.intro.intro.inr.refine_3\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\nh_sup : \u00ac0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\n\u22a2 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) (-1 \u2022 v)\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Function.ne_iff.mp h_vnz\n[GOAL]\ncase inr.intro.intro.inr.refine_3.intro\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\nh_sup : \u00ac0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\ni : n\nhi : v i \u2260 OfNat.ofNat 0 i\n\u22a2 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) (-1 \u2022 v)\n[PROOFSTEP]\nsimp_rw [Finset.lt_sup'_iff, Finset.mem_univ, true_and] at h_sup \u22a2\n[GOAL]\ncase inr.intro.intro.inr.refine_3.intro\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\ni : n\nhi : v i \u2260 OfNat.ofNat 0 i\nh_sup : \u00ac\u2203 b, 0 < v b\n\u22a2 \u2203 b, 0 < (-1 \u2022 v) b\n[PROOFSTEP]\nsimp_rw [not_exists, not_lt] at h_sup \n[GOAL]\ncase inr.intro.intro.inr.refine_3.intro\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\ni : n\nhi : v i \u2260 OfNat.ofNat 0 i\nh_sup : \u2200 (x : n), v x \u2264 0\n\u22a2 \u2203 b, 0 < (-1 \u2022 v) b\n[PROOFSTEP]\nrefine \u27e8i, ?_\u27e9\n[GOAL]\ncase inr.intro.intro.inr.refine_3.intro\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\ni : n\nhi : v i \u2260 OfNat.ofNat 0 i\nh_sup : \u2200 (x : n), v x \u2264 0\n\u22a2 0 < (-1 \u2022 v) i\n[PROOFSTEP]\nrw [Pi.smul_apply, neg_smul, one_smul, Left.neg_pos_iff]\n[GOAL]\ncase inr.intro.intro.inr.refine_3.intro\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nthis :\n  \u2200 {n : Type u_1} [inst : Fintype n] [inst_1 : DecidableEq n] {S : Type u_2} [inst_2 : LinearOrderedCommRing S]\n    {A : Matrix n n S},\n    (\u2200 (i j : n), i \u2260 j \u2192 A i j < 0) \u2192\n      \u2200 (h : Nonempty n),\n        det A = 0 \u2192\n          \u2200 (v : n \u2192 S),\n            v \u2260 0 \u2192\n              vecMul v A = 0 \u2192 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v \u2192 \u2203 j, \u2211 i : n, A i j \u2264 0\ni : n\nhi : v i \u2260 OfNat.ofNat 0 i\nh_sup : \u2200 (x : n), v x \u2264 0\n\u22a2 v i < 0\n[PROOFSTEP]\nrefine Ne.lt_of_le hi (h_sup i)\n[GOAL]\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\n\u22a2 \u2203 j, \u2211 i : n, A i j \u2264 0\n[PROOFSTEP]\nobtain \u27e8j\u2080, -, h_j\u2080\u27e9 := Finset.exists_mem_eq_sup' Finset.univ_nonempty v\n[GOAL]\ncase intro.intro\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\n\u22a2 \u2203 j, \u2211 i : n, A i j \u2264 0\n[PROOFSTEP]\nrefine \u27e8j\u2080, ?_\u27e9\n[GOAL]\ncase intro.intro\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\n\u22a2 \u2211 i : n, A i j\u2080 \u2264 0\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_left (h_j\u2080 \u25b8 h_sup), Finset.mul_sum, mul_zero]\n[GOAL]\ncase intro.intro\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\n\u22a2 \u2211 x : n, v j\u2080 * A x j\u2080 \u2264 0\n[PROOFSTEP]\nrw [show 0 = \u2211 i, v i * A i j\u2080 from (congrFun h_vA j\u2080).symm]\n[GOAL]\ncase intro.intro\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\n\u22a2 \u2211 x : n, v j\u2080 * A x j\u2080 \u2264 \u2211 i : n, v i * A i j\u2080\n[PROOFSTEP]\nrefine Finset.sum_le_sum (fun i hi => ?_)\n[GOAL]\ncase intro.intro\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\ni : n\nhi : i \u2208 Finset.univ\n\u22a2 v j\u2080 * A i j\u2080 \u2264 v i * A i j\u2080\n[PROOFSTEP]\nby_cases h : i = j\u2080\n[GOAL]\ncase pos\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\ni : n\nhi : i \u2208 Finset.univ\nh : i = j\u2080\n\u22a2 v j\u2080 * A i j\u2080 \u2264 v i * A i j\u2080\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase neg\nn\u271d : Type u_1\ninst\u271d\u00b3 : Fintype n\u271d\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh\u271d : Nonempty n\nh2 : det A = 0\nv : n \u2192 S\nh_vnz : v \u2260 0\nh_vA : vecMul v A = 0\nh_sup : 0 < Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v\nj\u2080 : n\nh_j\u2080 : Finset.sup' Finset.univ (_ : Finset.Nonempty Finset.univ) v = v j\u2080\ni : n\nhi : i \u2208 Finset.univ\nh : \u00aci = j\u2080\n\u22a2 v j\u2080 * A i j\u2080 \u2264 v i * A i j\u2080\n[PROOFSTEP]\nexact (mul_le_mul_right_of_neg (h1 i j\u2080 h)).mpr (h_j\u2080 \u25b8 Finset.le_sup' v hi)\n[GOAL]\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (i : n), 0 < \u2211 j : n, A i j\n\u22a2 det A \u2260 0\n[PROOFSTEP]\nrw [\u2190 Matrix.det_transpose]\n[GOAL]\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (i : n), 0 < \u2211 j : n, A i j\n\u22a2 det A\u1d40 \u2260 0\n[PROOFSTEP]\nrefine det_ne_zero_of_sum_col_pos ?_ ?_\n[GOAL]\ncase refine_1\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (i : n), 0 < \u2211 j : n, A i j\n\u22a2 \u2200 (i j : n), i \u2260 j \u2192 A\u1d40 i j < 0\n[PROOFSTEP]\nsimp_rw [Matrix.transpose_apply]\n[GOAL]\ncase refine_1\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (i : n), 0 < \u2211 j : n, A i j\n\u22a2 \u2200 (i j : n), i \u2260 j \u2192 A j i < 0\n[PROOFSTEP]\nexact fun i j h => h1 j i h.symm\n[GOAL]\ncase refine_2\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (i : n), 0 < \u2211 j : n, A i j\n\u22a2 \u2200 (j : n), 0 < \u2211 i : n, A\u1d40 i j\n[PROOFSTEP]\nsimp_rw [Matrix.transpose_apply]\n[GOAL]\ncase refine_2\nn : Type u_1\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nS : Type u_2\ninst\u271d : LinearOrderedCommRing S\nA : Matrix n n S\nh1 : \u2200 (i j : n), i \u2260 j \u2192 A i j < 0\nh2 : \u2200 (i : n), 0 < \u2211 j : n, A i j\n\u22a2 \u2200 (j : n), 0 < \u2211 x : n, A j x\n[PROOFSTEP]\nexact h2\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "llama_tokens": 22978, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.800691997339971, "lm_q2_score": 0.727975443004307, "lm_q1q2_score": 0.5828841114735688}}
{"text": "[GOAL]\nF : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nf g : Homotopy f\u2080 f\u2081\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\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\ng : Homotopy f\u2080 f\u2081\ntoFun\u271d : \u2191I \u00d7 X \u2192 Y\ncontinuous_toFun\u271d : Continuous toFun\u271d\nmap_zero_left\u271d : \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk toFun\u271d) (0, x) = \u2191f\u2080 x\nmap_one_left\u271d : \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk toFun\u271d) (1, x) = \u2191f\u2081 x\nh :\n  (fun f => f.toFun)\n      { toContinuousMap := ContinuousMap.mk toFun\u271d, map_zero_left := map_zero_left\u271d, map_one_left := map_one_left\u271d } =\n    (fun f => f.toFun) g\n\u22a2 { toContinuousMap := ContinuousMap.mk toFun\u271d, map_zero_left := map_zero_left\u271d, map_one_left := map_one_left\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\ntoFun\u271d\u00b9 : \u2191I \u00d7 X \u2192 Y\ncontinuous_toFun\u271d\u00b9 : Continuous toFun\u271d\u00b9\nmap_zero_left\u271d\u00b9 : \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk toFun\u271d\u00b9) (0, x) = \u2191f\u2080 x\nmap_one_left\u271d\u00b9 : \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk toFun\u271d\u00b9) (1, x) = \u2191f\u2081 x\ntoFun\u271d : \u2191I \u00d7 X \u2192 Y\ncontinuous_toFun\u271d : Continuous toFun\u271d\nmap_zero_left\u271d : \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk toFun\u271d) (0, x) = \u2191f\u2080 x\nmap_one_left\u271d : \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk toFun\u271d) (1, x) = \u2191f\u2081 x\nh :\n  (fun f => f.toFun)\n      { toContinuousMap := ContinuousMap.mk toFun\u271d\u00b9, map_zero_left := map_zero_left\u271d\u00b9,\n        map_one_left := map_one_left\u271d\u00b9 } =\n    (fun f => f.toFun)\n      { toContinuousMap := ContinuousMap.mk toFun\u271d, map_zero_left := map_zero_left\u271d, map_one_left := map_one_left\u271d }\n\u22a2 { toContinuousMap := ContinuousMap.mk toFun\u271d\u00b9, map_zero_left := map_zero_left\u271d\u00b9, map_one_left := map_one_left\u271d\u00b9 } =\n    { toContinuousMap := ContinuousMap.mk toFun\u271d, map_zero_left := map_zero_left\u271d, map_one_left := map_one_left\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nt : \u211d\nht : t \u2264 0\nx : X\n\u22a2 \u2191(\u2191(extend F) t) x = \u2191f\u2080 x\n[PROOFSTEP]\nrw [\u2190 F.apply_zero]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nt : \u211d\nht : t \u2264 0\nx : X\n\u22a2 \u2191(\u2191(extend F) t) x = \u2191F (0, x)\n[PROOFSTEP]\nexact ContinuousMap.congr_fun (Set.IccExtend_of_le_left (zero_le_one' \u211d) F.curry ht) x\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nt : \u211d\nht : 1 \u2264 t\nx : X\n\u22a2 \u2191(\u2191(extend F) t) x = \u2191f\u2081 x\n[PROOFSTEP]\nrw [\u2190 F.apply_one]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nt : \u211d\nht : 1 \u2264 t\nx : X\n\u22a2 \u2191(\u2191(extend F) t) x = \u2191F (1, x)\n[PROOFSTEP]\nexact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' \u211d) F.curry ht) x\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\n\u22a2 \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk fun x => \u2191F (\u03c3 x.fst, x.snd)) (0, x) = \u2191f\u2081 x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\n\u22a2 \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk fun x => \u2191F (\u03c3 x.fst, x.snd)) (1, x) = \u2191f\u2080 x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\n\u22a2 symm (symm F) = F\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nx\u271d : \u2191I \u00d7 X\n\u22a2 \u2191(symm (symm F)) x\u271d = \u2191F x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\n\u22a2 Continuous fun x =>\n    if \u2191x.fst \u2264 1 / 2 then \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd else \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd\n[PROOFSTEP]\nrefine'\n  continuous_if_le (continuous_induced_dom.comp continuous_fst) continuous_const\n    (F.continuous.comp (by continuity)).continuousOn (G.continuous.comp (by continuity)).continuousOn _\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\n\u22a2 Continuous fun x => (Set.projIcc 0 1 extend.proof_1 (2 * \u2191x.fst), x.snd)\n[PROOFSTEP]\ncontinuity\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\n\u22a2 Continuous fun x => (Set.projIcc 0 1 extend.proof_1 (2 * \u2191x.fst - 1), x.snd)\n[PROOFSTEP]\ncontinuity\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\n\u22a2 \u2200 (x : \u2191I \u00d7 X), \u2191x.fst = 1 / 2 \u2192 \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd = \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd\n[PROOFSTEP]\nrintro x hx\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nhx : \u2191x.fst = 1 / 2\n\u22a2 \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd = \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd\n[PROOFSTEP]\nrw [hx]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nhx : \u2191x.fst = 1 / 2\n\u22a2 \u2191(\u2191(extend F) (2 * (1 / 2))) x.snd = \u2191(\u2191(extend G) (2 * (1 / 2) - 1)) x.snd\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : X\n\u22a2 ContinuousMap.toFun\n      (ContinuousMap.mk fun x =>\n        if \u2191x.fst \u2264 1 / 2 then \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd else \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd)\n      (0, x) =\n    \u2191f\u2080 x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : X\n\u22a2 ContinuousMap.toFun\n      (ContinuousMap.mk fun x =>\n        if \u2191x.fst \u2264 1 / 2 then \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd else \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd)\n      (1, x) =\n    \u2191f\u2082 x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\n\u22a2 (if \u2191x.fst \u2264 1 / 2 then \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd else \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd) =\n    if h : \u2191x.fst \u2264 1 / 2 then \u2191F ({ val := 2 * \u2191x.fst, property := (_ : 2 * \u2191x.fst \u2208 I) }, x.snd)\n    else \u2191G ({ val := 2 * \u2191x.fst - 1, property := (_ : 2 * \u2191x.fst - 1 \u2208 I) }, x.snd)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nh\u271d : \u2191x.fst \u2264 1 / 2\n\u22a2 \u2191(\u2191(extend F) (2 * \u2191x.fst)) x.snd = \u2191F ({ val := 2 * \u2191x.fst, property := (_ : 2 * \u2191x.fst \u2208 I) }, x.snd)\n[PROOFSTEP]\nrw [extend, ContinuousMap.coe_IccExtend, Set.IccExtend_of_mem]\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nh\u271d : \u2191x.fst \u2264 1 / 2\n\u22a2 \u2191(\u2191(curry F) { val := 2 * \u2191x.fst, property := ?pos.hx\u271d }) x.snd =\n    \u2191F ({ val := 2 * \u2191x.fst, property := (_ : 2 * \u2191x.fst \u2208 I) }, x.snd)\ncase pos.hx\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nh\u271d : \u2191x.fst \u2264 1 / 2\n\u22a2 2 * \u2191x.fst \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nh\u271d : \u00ac\u2191x.fst \u2264 1 / 2\n\u22a2 \u2191(\u2191(extend G) (2 * \u2191x.fst - 1)) x.snd = \u2191G ({ val := 2 * \u2191x.fst - 1, property := (_ : 2 * \u2191x.fst - 1 \u2208 I) }, x.snd)\n[PROOFSTEP]\nrw [extend, ContinuousMap.coe_IccExtend, Set.IccExtend_of_mem]\n[GOAL]\ncase neg\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nh\u271d : \u00ac\u2191x.fst \u2264 1 / 2\n\u22a2 \u2191(\u2191(curry G) { val := 2 * \u2191x.fst - 1, property := ?neg.hx\u271d }) x.snd =\n    \u2191G ({ val := 2 * \u2191x.fst - 1, property := (_ : 2 * \u2191x.fst - 1 \u2208 I) }, x.snd)\ncase neg.hx\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nx : \u2191I \u00d7 X\nh\u271d : \u00ac\u2191x.fst \u2264 1 / 2\n\u22a2 2 * \u2191x.fst - 1 \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrfl\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\n\u22a2 symm (trans F G) = trans (symm G) (symm F)\n[PROOFSTEP]\next \u27e8t, _\u27e9\n[GOAL]\ncase h.mk\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\n\u22a2 \u2191(symm (trans F G)) (t, snd\u271d) = \u2191(trans (symm G) (symm F)) (t, snd\u271d)\n[PROOFSTEP]\nrw [trans_apply, symm_apply, trans_apply]\n[GOAL]\ncase h.mk\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\n\u22a2 (if h : \u2191(\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).fst \u2264 1 / 2 then\n      \u2191F\n        ({ val := 2 * \u2191(\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).fst,\n            property := (_ : 2 * \u2191(\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).fst \u2208 I) },\n          (\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).snd)\n    else\n      \u2191G\n        ({ val := 2 * \u2191(\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).fst - 1,\n            property := (_ : 2 * \u2191(\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).fst - 1 \u2208 I) },\n          (\u03c3 (t, snd\u271d).fst, (t, snd\u271d).snd).snd)) =\n    if h : \u2191(t, snd\u271d).fst \u2264 1 / 2 then\n      \u2191(symm G) ({ val := 2 * \u2191(t, snd\u271d).fst, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) }, (t, snd\u271d).snd)\n    else \u2191(symm F) ({ val := 2 * \u2191(t, snd\u271d).fst - 1, property := (_ : 2 * \u2191(t, snd\u271d).fst - 1 \u2208 I) }, (t, snd\u271d).snd)\n[PROOFSTEP]\nsimp only [coe_symm_eq, symm_apply]\n[GOAL]\ncase h.mk\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\n\u22a2 (if h : 1 - \u2191t \u2264 1 / 2 then \u2191F ({ val := 2 * (1 - \u2191t), property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t))) }, snd\u271d)\n    else \u2191G ({ val := 2 * (1 - \u2191t) - 1, property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t) - 1)) }, snd\u271d)) =\n    if h : \u2191t \u2264 1 / 2 then \u2191G (\u03c3 { val := 2 * \u2191t, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) }, snd\u271d)\n    else \u2191F (\u03c3 { val := 2 * \u2191t - 1, property := (_ : 2 * \u2191(t, snd\u271d).fst - 1 \u2208 I) }, snd\u271d)\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082 h\u2082\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\n\u22a2 \u2191F ({ val := 2 * (1 - \u2191t), property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t))) }, snd\u271d) =\n    \u2191G (\u03c3 { val := 2 * \u2191t, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) }, snd\u271d)\n[PROOFSTEP]\nhave ht : (t : \u211d) = 1 / 2 := by linarith\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\n\u22a2 \u2191t = 1 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\nht : \u2191t = 1 / 2\n\u22a2 \u2191F ({ val := 2 * (1 - \u2191t), property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t))) }, snd\u271d) =\n    \u2191G (\u03c3 { val := 2 * \u2191t, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) }, snd\u271d)\n[PROOFSTEP]\nsimp only [ht]\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\nht : \u2191t = 1 / 2\n\u22a2 \u2191F ({ val := 2 * (1 - 1 / 2), property := (_ : (fun x => x \u2208 I) (2 * (1 - 1 / 2))) }, snd\u271d) =\n    \u2191G (\u03c3 { val := 2 * (1 / 2), property := (_ : (fun x => x \u2208 I) (2 * (1 / 2))) }, snd\u271d)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191F ({ val := 2 * (1 - \u2191t), property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t))) }, snd\u271d) =\n    \u2191F (\u03c3 { val := 2 * \u2191t - 1, property := (_ : 2 * \u2191(t, snd\u271d).fst - 1 \u2208 I) }, snd\u271d)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase neg.h.e_6.h.e_fst\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u00ac\u2191t \u2264 1 / 2\n\u22a2 { val := 2 * (1 - \u2191t), property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t))) } =\n    \u03c3 { val := 2 * \u2191t - 1, property := (_ : 2 * \u2191(t, snd\u271d).fst - 1 \u2208 I) }\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase neg.h.e_6.h.e_fst.a\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191{ val := 2 * (1 - \u2191t), property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t))) } =\n    \u2191(\u03c3 { val := 2 * \u2191t - 1, property := (_ : 2 * \u2191(t, snd\u271d).fst - 1 \u2208 I) })\n[PROOFSTEP]\nsimp only [coe_symm_eq]\n[GOAL]\ncase neg.h.e_6.h.e_fst.a\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : 1 - \u2191t \u2264 1 / 2\nh\u2082 : \u00ac\u2191t \u2264 1 / 2\n\u22a2 2 * (1 - \u2191t) = 1 - (2 * \u2191t - 1)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : \u00ac1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\n\u22a2 \u2191G ({ val := 2 * (1 - \u2191t) - 1, property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t) - 1)) }, snd\u271d) =\n    \u2191G (\u03c3 { val := 2 * \u2191t, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) }, snd\u271d)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase pos.h.e_6.h.e_fst\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : \u00ac1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\n\u22a2 { val := 2 * (1 - \u2191t) - 1, property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t) - 1)) } =\n    \u03c3 { val := 2 * \u2191t, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) }\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase pos.h.e_6.h.e_fst.a\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : \u00ac1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\n\u22a2 \u2191{ val := 2 * (1 - \u2191t) - 1, property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t) - 1)) } =\n    \u2191(\u03c3 { val := 2 * \u2191t, property := (_ : 2 * \u2191(t, snd\u271d).fst \u2208 I) })\n[PROOFSTEP]\nsimp only [coe_symm_eq]\n[GOAL]\ncase pos.h.e_6.h.e_fst.a\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : \u00ac1 - \u2191t \u2264 1 / 2\nh\u2082 : \u2191t \u2264 1 / 2\n\u22a2 2 * (1 - \u2191t) - 1 = 1 - 2 * \u2191t\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : \u00ac1 - \u2191t \u2264 1 / 2\nh\u2082 : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191G ({ val := 2 * (1 - \u2191t) - 1, property := (_ : (fun x => x \u2208 I) (2 * (1 - \u2191t) - 1)) }, snd\u271d) =\n    \u2191F (\u03c3 { val := 2 * \u2191t - 1, property := (_ : 2 * \u2191(t, snd\u271d).fst - 1 \u2208 I) }, snd\u271d)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy f\u2081 f\u2082\nt : \u2191I\nsnd\u271d : X\nh\u2081 : \u00ac1 - \u2191t \u2264 1 / 2\nh\u2082 : \u00ac\u2191t \u2264 1 / 2\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 g\u2080 g\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nh\u2080 : f\u2080 = g\u2080\nh\u2081 : f\u2081 = g\u2081\n\u22a2 \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk \u2191F) (0, x) = \u2191g\u2080 x\n[PROOFSTEP]\nsimp [\u2190 h\u2080]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 g\u2080 g\u2081 : C(X, Y)\nF : Homotopy f\u2080 f\u2081\nh\u2080 : f\u2080 = g\u2080\nh\u2081 : f\u2081 = g\u2081\n\u22a2 \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk \u2191F) (1, x) = \u2191g\u2081 x\n[PROOFSTEP]\nsimp [\u2190 h\u2081]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\ng\u2080 g\u2081 : C(Y, Z)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy g\u2080 g\u2081\n\u22a2 \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk fun x => \u2191G (x.fst, \u2191F x)) (0, x) = \u2191(comp g\u2080 f\u2080) x\n[PROOFSTEP]\nsimp\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 : C(X, Y)\ng\u2080 g\u2081 : C(Y, Z)\nF : Homotopy f\u2080 f\u2081\nG : Homotopy g\u2080 g\u2081\n\u22a2 \u2200 (x : X), ContinuousMap.toFun (ContinuousMap.mk fun x => \u2191G (x.fst, \u2191F x)) (1, x) = \u2191(comp g\u2081 f\u2081) x\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\n\u22a2 \u2200 {x y : C(X, Y)}, Homotopic x y \u2192 Homotopic y x\n[PROOFSTEP]\napply symm\n[GOAL]\nF : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\n\u22a2 \u2200 {x y z : C(X, Y)}, Homotopic x y \u2192 Homotopic y z \u2192 Homotopic x z\n[PROOFSTEP]\napply trans\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nP : C(X, Y) \u2192 Prop\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : HomotopyWith f\u2080 f\u2081 P\nG : HomotopyWith f\u2081 f\u2082 P\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\n\u22a2 P\n    (ContinuousMap.mk fun x =>\n      ContinuousMap.toFun\n        { toContinuousMap := src\u271d.toContinuousMap,\n            map_zero_left := (_ : \u2200 (x : X), ContinuousMap.toFun src\u271d.toContinuousMap (0, x) = \u2191f\u2080 x),\n            map_one_left := (_ : \u2200 (x : X), ContinuousMap.toFun src\u271d.toContinuousMap (1, x) = \u2191f\u2082 x) }.toContinuousMap\n        (t, x))\n[PROOFSTEP]\nsimp only [Homotopy.trans]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nP : C(X, Y) \u2192 Prop\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : HomotopyWith f\u2080 f\u2081 P\nG : HomotopyWith f\u2081 f\u2082 P\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\n\u22a2 P\n    (ContinuousMap.mk fun x =>\n      if \u2191t \u2264 1 / 2 then \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x\n      else \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n[PROOFSTEP]\nchange P \u27e8fun _ => ite ((t : \u211d) \u2264 _) _ _, _\u27e9\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nP : C(X, Y) \u2192 Prop\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : HomotopyWith f\u2080 f\u2081 P\nG : HomotopyWith f\u2081 f\u2082 P\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\n\u22a2 P\n    (ContinuousMap.mk fun x =>\n      if \u2191t \u2264 1 / 2 then \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x\n      else \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nP : C(X, Y) \u2192 Prop\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : HomotopyWith f\u2080 f\u2081 P\nG : HomotopyWith f\u2081 f\u2082 P\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 P (ContinuousMap.mk fun x => \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x)\n[PROOFSTEP]\nexact F.extendProp _\n[GOAL]\ncase neg\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nP : C(X, Y) \u2192 Prop\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nF : HomotopyWith f\u2080 f\u2081 P\nG : HomotopyWith f\u2081 f\u2082 P\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 P (ContinuousMap.mk fun x => \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n[PROOFSTEP]\nexact G.extendProp _\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nS : Set X\nF : HomotopyRel f\u2080 f\u2081 S\nG : HomotopyRel f\u2081 f\u2082 S\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nx : X\nhx : x \u2208 S\n\u22a2 \u2191(mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := src\u271d.toContinuousMap,\n                  map_zero_left := (_ : \u2200 (x : X), ContinuousMap.toFun src\u271d.toContinuousMap (0, x) = \u2191f\u2080 x),\n                  map_one_left :=\n                    (_ : \u2200 (x : X), ContinuousMap.toFun src\u271d.toContinuousMap (1, x) = \u2191f\u2082 x) }.toContinuousMap\n              (t, x))\n        x =\n      \u2191f\u2080 x \u2227\n    \u2191(mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := src\u271d.toContinuousMap,\n                  map_zero_left := (_ : \u2200 (x : X), ContinuousMap.toFun src\u271d.toContinuousMap (0, x) = \u2191f\u2080 x),\n                  map_one_left :=\n                    (_ : \u2200 (x : X), ContinuousMap.toFun src\u271d.toContinuousMap (1, x) = \u2191f\u2082 x) }.toContinuousMap\n              (t, x))\n        x =\n      \u2191f\u2082 x\n[PROOFSTEP]\nsimp only [Homotopy.trans]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nS : Set X\nF : HomotopyRel f\u2080 f\u2081 S\nG : HomotopyRel f\u2081 f\u2082 S\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nx : X\nhx : x \u2208 S\n\u22a2 \u2191(mk fun x =>\n            if \u2191t \u2264 1 / 2 then \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x\n            else \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n        x =\n      \u2191f\u2080 x \u2227\n    \u2191(mk fun x =>\n            if \u2191t \u2264 1 / 2 then \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x\n            else \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n        x =\n      \u2191f\u2082 x\n[PROOFSTEP]\nchange (\u27e8fun _ => ite ((t : \u211d) \u2264 _) _ _, _\u27e9 : C(X, Y)) _ = _ \u2227 _ = _\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nS : Set X\nF : HomotopyRel f\u2080 f\u2081 S\nG : HomotopyRel f\u2081 f\u2082 S\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nx : X\nhx : x \u2208 S\n\u22a2 \u2191(mk fun x =>\n            if \u2191t \u2264 1 / 2 then \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x\n            else \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n        x =\n      \u2191f\u2080 x \u2227\n    \u2191(mk fun x =>\n            if \u2191t \u2264 1 / 2 then \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x\n            else \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x)\n        x =\n      \u2191f\u2082 x\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nS : Set X\nF : HomotopyRel f\u2080 f\u2081 S\nG : HomotopyRel f\u2081 f\u2082 S\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nx : X\nhx : x \u2208 S\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 \u2191(mk fun x => \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x) x = \u2191f\u2080 x \u2227\n    \u2191(mk fun x => \u2191(\u2191(Homotopy.extend F.toHomotopy) (2 * \u2191t)) x) x = \u2191f\u2082 x\n[PROOFSTEP]\nsimp [(HomotopyWith.extendProp F (2 * t) x hx).1, F.fst_eq_snd hx, G.fst_eq_snd hx]\n[GOAL]\ncase neg\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080 f\u2081 f\u2082 : C(X, Y)\nS : Set X\nF : HomotopyRel f\u2080 f\u2081 S\nG : HomotopyRel f\u2081 f\u2082 S\nsrc\u271d : Homotopy f\u2080 f\u2082 := Homotopy.trans F.toHomotopy G.toHomotopy\nt : \u2191I\nx : X\nhx : x \u2208 S\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191(mk fun x => \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x) x = \u2191f\u2080 x \u2227\n    \u2191(mk fun x => \u2191(\u2191(Homotopy.extend G.toHomotopy) (2 * \u2191t - 1)) x) x = \u2191f\u2082 x\n[PROOFSTEP]\nsimp [(HomotopyWith.extendProp G (2 * t - 1) x hx).1, F.fst_eq_snd hx, G.fst_eq_snd hx]\n[GOAL]\nF\u271d : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nf\u2080\u271d f\u2081\u271d f\u2082 : C(X, Y)\nS : Set X\nf\u2080 f\u2081 g\u2080 g\u2081 : C(X, Y)\nF : HomotopyRel f\u2080 f\u2081 S\nh\u2080 : f\u2080 = g\u2080\nh\u2081 : f\u2081 = g\u2081\nt : \u2191I\nx : X\nhx : x \u2208 S\n\u22a2 \u2191(mk fun x => ContinuousMap.toFun (Homotopy.cast F.toHomotopy h\u2080 h\u2081).toContinuousMap (t, x)) x = \u2191g\u2080 x \u2227\n    \u2191(mk fun x => ContinuousMap.toFun (Homotopy.cast F.toHomotopy h\u2080 h\u2081).toContinuousMap (t, x)) x = \u2191g\u2081 x\n[PROOFSTEP]\nsimpa only [\u2190 h\u2080, \u2190 h\u2081] using F.prop t x hx\n[GOAL]\nF : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nS : Set X\n\u22a2 \u2200 {x y : C(X, Y)}, HomotopicRel x y S \u2192 HomotopicRel y x S\n[PROOFSTEP]\napply symm\n[GOAL]\nF : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace Z'\nS : Set X\n\u22a2 \u2200 {x y z : C(X, Y)}, HomotopicRel x y S \u2192 HomotopicRel y z S \u2192 HomotopicRel x z S\n[PROOFSTEP]\napply trans\n", "meta": {"mathlib_filename": "Mathlib.Topology.Homotopy.Basic", "llama_tokens": 15468, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245870332531, "lm_q2_score": 0.6992544210587585, "lm_q1q2_score": 0.5827059016599664}}
{"text": "[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d c\u271d d\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 (f \u226b g) \u226b h = f \u226b g \u226b h\n[PROOFSTEP]\napply Discrete.ext\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d c\u271d d\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 ((f \u226b g) \u226b h).as = (f \u226b g \u226b h).as\n[PROOFSTEP]\nchange (f.as \u226b g.as) \u226b h.as = f.as \u226b (g.as \u226b h.as)\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d c\u271d d\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\ng : b\u271d \u27f6 c\u271d\nh : c\u271d \u27f6 d\u271d\n\u22a2 (f.as \u226b g.as) \u226b h.as = f.as \u226b g.as \u226b h.as\n[PROOFSTEP]\nrw [Category.assoc]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\n\u22a2 \ud835\udfd9 a\u271d \u226b f = f\n[PROOFSTEP]\napply Discrete.ext\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\n\u22a2 (\ud835\udfd9 a\u271d \u226b f).as = f.as\n[PROOFSTEP]\nchange \ud835\udfd9 _ \u226b _ = _\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\n\u22a2 \ud835\udfd9 a\u271d \u226b f.as = f.as\n[PROOFSTEP]\nrw [Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\n\u22a2 f \u226b \ud835\udfd9 b\u271d = f\n[PROOFSTEP]\napply Discrete.ext\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\n\u22a2 (f \u226b \ud835\udfd9 b\u271d).as = f.as\n[PROOFSTEP]\nchange _ \u226b \ud835\udfd9 _ = _\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf : a\u271d \u27f6 b\u271d\n\u22a2 f.as \u226b \ud835\udfd9 b\u271d = f.as\n[PROOFSTEP]\nrw [Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 \u2200 {a b : LocallyDiscrete C} (f : a \u27f6 b), \ud835\udfd9 a \u226b f = f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 \ud835\udfd9 a\u271d \u226b f\u271d = f\u271d\n[PROOFSTEP]\napply Discrete.ext\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (\ud835\udfd9 a\u271d \u226b f\u271d).as = f\u271d.as\n[PROOFSTEP]\napply Category.id_comp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 \u2200 {a b : LocallyDiscrete C} (f : a \u27f6 b), f \u226b \ud835\udfd9 b = f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 f\u271d \u226b \ud835\udfd9 b\u271d = f\u271d\n[PROOFSTEP]\napply Discrete.ext\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d : LocallyDiscrete C\nf\u271d : a\u271d \u27f6 b\u271d\n\u22a2 (f\u271d \u226b \ud835\udfd9 b\u271d).as = f\u271d.as\n[PROOFSTEP]\napply Category.comp_id\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\n\u22a2 \u2200 {a b c d : LocallyDiscrete C} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d), (f \u226b g) \u226b h = f \u226b g \u226b h\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d c\u271d d\u271d : LocallyDiscrete C\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\n\u22a2 (f\u271d \u226b g\u271d) \u226b h\u271d = f\u271d \u226b g\u271d \u226b h\u271d\n[PROOFSTEP]\napply Discrete.ext\n[GOAL]\ncase as\nC : Type u\ninst\u271d : Category.{v, u} C\na\u271d b\u271d c\u271d d\u271d : LocallyDiscrete C\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d : b\u271d \u27f6 c\u271d\nh\u271d : c\u271d \u27f6 d\u271d\n\u22a2 ((f\u271d \u226b g\u271d) \u226b h\u271d).as = (f\u271d \u226b g\u271d \u226b h\u271d).as\n[PROOFSTEP]\napply Category.assoc\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.LocallyDiscrete", "llama_tokens": 1839, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744939732856, "lm_q2_score": 0.7122321781307375, "lm_q1q2_score": 0.5823028626267287}}
{"text": "[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 { obj := fun U => CommRingCat.of (Localization (obj G U)),\n          map := fun {U V} i =>\n            CommRingCat.ofHom\n              (IsLocalization.map (Localization (obj G V)) (F.map i)\n                (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V))) }.map\n      (\ud835\udfd9 U) =\n    \ud835\udfd9\n      ({ obj := fun U => CommRingCat.of (Localization (obj G U)),\n            map := fun {U V} i =>\n              CommRingCat.ofHom\n                (IsLocalization.map (Localization (obj G V)) (F.map i)\n                  (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V))) }.obj\n        U)\n[PROOFSTEP]\nsimp_rw [F.map_id]\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 CommRingCat.ofHom\n      (IsLocalization.map (Localization (obj G U)) (\ud835\udfd9 (F.obj U))\n        (_ : obj G U \u2264 Submonoid.comap (\ud835\udfd9 (F.obj U)) (obj G U))) =\n    \ud835\udfd9 (CommRingCat.of (Localization (obj G U)))\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nU : (Opens \u2191X)\u1d52\u1d56\nx : (forget CommRingCat).obj (CommRingCat.of (Localization (obj G U)))\n\u22a2 \u2191(CommRingCat.ofHom\n          (IsLocalization.map (Localization (obj G U)) (\ud835\udfd9 (F.obj U))\n            (_ : obj G U \u2264 Submonoid.comap (\ud835\udfd9 (F.obj U)) (obj G U))))\n      x =\n    \u2191(\ud835\udfd9 (CommRingCat.of (Localization (obj G U)))) x\n[PROOFSTEP]\nexact IsLocalization.map_id (M := G.obj U) (S := Localization (G.obj U)) x\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nU V W : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 { obj := fun U => CommRingCat.of (Localization (obj G U)),\n          map := fun {U V} i =>\n            CommRingCat.ofHom\n              (IsLocalization.map (Localization (obj G V)) (F.map i)\n                (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V))) }.map\n      (i \u226b j) =\n    { obj := fun U => CommRingCat.of (Localization (obj G U)),\n            map := fun {U V} i =>\n              CommRingCat.ofHom\n                (IsLocalization.map (Localization (obj G V)) (F.map i)\n                  (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V))) }.map\n        i \u226b\n      { obj := fun U => CommRingCat.of (Localization (obj G U)),\n            map := fun {U V} i =>\n              CommRingCat.ofHom\n                (IsLocalization.map (Localization (obj G V)) (F.map i)\n                  (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V))) }.map\n        j\n[PROOFSTEP]\ndelta CommRingCat.ofHom CommRingCat.of Bundled.of\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nU V W : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 { obj := fun U => Bundled.mk (Localization (obj G U)),\n          map := fun {U V} i =>\n            IsLocalization.map (Localization (obj G V)) (F.map i)\n              (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V)) }.map\n      (i \u226b j) =\n    { obj := fun U => Bundled.mk (Localization (obj G U)),\n            map := fun {U V} i =>\n              IsLocalization.map (Localization (obj G V)) (F.map i)\n                (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V)) }.map\n        i \u226b\n      { obj := fun U => Bundled.mk (Localization (obj G U)),\n            map := fun {U V} i =>\n              IsLocalization.map (Localization (obj G V)) (F.map i)\n                (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V)) }.map\n        j\n[PROOFSTEP]\nsimp_rw [F.map_comp, CommRingCat.comp_eq_ring_hom_comp]\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nU V W : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\nj : V \u27f6 W\n\u22a2 IsLocalization.map (Localization (obj G W)) (RingHom.comp (F.map j) (F.map i))\n      (_ : obj G U \u2264 Submonoid.comap (RingHom.comp (F.map j) (F.map i)) (obj G W)) =\n    RingHom.comp\n      (IsLocalization.map (Localization (obj G W)) (F.map j) (_ : obj G V \u2264 Submonoid.comap (F.map j) (obj G W)))\n      (IsLocalization.map (Localization (obj G V)) (F.map i) (_ : obj G U \u2264 Submonoid.comap (F.map i) (obj G V)))\n[PROOFSTEP]\nrw [IsLocalization.map_comp_map]\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nS : (x : \u2191X) \u2192 Submonoid \u2191(stalk F x)\nU V : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\n\u22a2 (fun U => \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ F x) (S \u2191x)) U \u2264\n    Submonoid.comap (F.map i) ((fun U => \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ F x) (S \u2191x)) V)\n[PROOFSTEP]\nintro s hs\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nS : (x : \u2191X) \u2192 Submonoid \u2191(stalk F x)\nU V : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\ns : (forget CommRingCat).obj (F.obj U)\nhs : s \u2208 (fun U => \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ F x) (S \u2191x)) U\n\u22a2 s \u2208 Submonoid.comap (F.map i) ((fun U => \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ F x) (S \u2191x)) V)\n[PROOFSTEP]\nsimp only [Submonoid.mem_comap, Submonoid.mem_iInf] at hs \u22a2\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nS : (x : \u2191X) \u2192 Submonoid \u2191(stalk F x)\nU V : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\ns : (forget CommRingCat).obj (F.obj U)\nhs : \u2200 (i : { x // x \u2208 U.unop }), \u2191(germ F i) s \u2208 S \u2191i\n\u22a2 \u2200 (i_1 : { x // x \u2208 V.unop }), \u2191(germ F i_1) (\u2191(F.map i) s) \u2208 S \u2191i_1\n[PROOFSTEP]\nintro x\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nS : (x : \u2191X) \u2192 Submonoid \u2191(stalk F x)\nU V : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\ns : (forget CommRingCat).obj (F.obj U)\nhs : \u2200 (i : { x // x \u2208 U.unop }), \u2191(germ F i) s \u2208 S \u2191i\nx : { x // x \u2208 V.unop }\n\u22a2 \u2191(germ F x) (\u2191(F.map i) s) \u2208 S \u2191x\n[PROOFSTEP]\nchange (F.map i.unop.op \u226b F.germ x) s \u2208 _\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nS : (x : \u2191X) \u2192 Submonoid \u2191(stalk F x)\nU V : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\ns : (forget CommRingCat).obj (F.obj U)\nhs : \u2200 (i : { x // x \u2208 U.unop }), \u2191(germ F i) s \u2208 S \u2191i\nx : { x // x \u2208 V.unop }\n\u22a2 \u2191(F.map i.unop.op \u226b germ F x) s \u2208 S \u2191x\n[PROOFSTEP]\nrw [F.germ_res]\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\nS : (x : \u2191X) \u2192 Submonoid \u2191(stalk F x)\nU V : (Opens \u2191X)\u1d52\u1d56\ni : U \u27f6 V\ns : (forget CommRingCat).obj (F.obj U)\nhs : \u2200 (i : { x // x \u2208 U.unop }), \u2191(germ F i) s \u2208 S \u2191i\nx : { x // x \u2208 V.unop }\n\u22a2 \u2191(germ F ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U.unop) }) x)) s \u2208 S \u2191x\n[PROOFSTEP]\nexact hs _\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\n\u22a2 Mono (toTotalQuotientPresheaf (Sheaf.presheaf F))\n[PROOFSTEP]\nsuffices : \u2200 (U : (Opens \u2191X)\u1d52\u1d56), Mono (F.presheaf.toTotalQuotientPresheaf.app U)\n[GOAL]\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nthis : \u2200 (U : (Opens \u2191X)\u1d52\u1d56), Mono (NatTrans.app (toTotalQuotientPresheaf (Sheaf.presheaf F)) U)\n\u22a2 Mono (toTotalQuotientPresheaf (Sheaf.presheaf F))\n[PROOFSTEP]\napply NatTrans.mono_of_mono_app\n[GOAL]\ncase this\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\n\u22a2 \u2200 (U : (Opens \u2191X)\u1d52\u1d56), Mono (NatTrans.app (toTotalQuotientPresheaf (Sheaf.presheaf F)) U)\n[PROOFSTEP]\nintro U\n[GOAL]\ncase this\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 Mono (NatTrans.app (toTotalQuotientPresheaf (Sheaf.presheaf F)) U)\n[PROOFSTEP]\napply ConcreteCategory.mono_of_injective\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 Function.Injective \u2191(NatTrans.app (toTotalQuotientPresheaf (Sheaf.presheaf F)) U)\n[PROOFSTEP]\ndsimp [toTotalQuotientPresheaf, CommRingCat.ofHom]\n  -- Porting note : this is a hack to make the `refine` below works\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 Function.Injective\n    \u2191(algebraMap (\u2191((Sheaf.presheaf F).obj U))\n        (Localization\n          (\u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070)))\n[PROOFSTEP]\nset m := _\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : ?m.65485 := ?m.65486\n\u22a2 Function.Injective\n    \u2191(algebraMap (\u2191((Sheaf.presheaf F).obj U))\n        (Localization\n          (\u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070)))\n[PROOFSTEP]\nchange Function.Injective (algebraMap _ (Localization m))\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\n\u22a2 Function.Injective \u2191(algebraMap ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) (Localization m))\n[PROOFSTEP]\nchange Function.Injective (algebraMap (F.presheaf.obj U) _)\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\n\u22a2 Function.Injective \u2191(algebraMap (\u2191((Sheaf.presheaf F).obj U)) (Localization m))\n[PROOFSTEP]\nhaveI : IsLocalization _ (Localization m) := Localization.isLocalization\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\n\u22a2 Function.Injective \u2191(algebraMap (\u2191((Sheaf.presheaf F).obj U)) (Localization m))\n[PROOFSTEP]\nrefine IsLocalization.injective (M := m) (S := Localization m) ?_\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\n\u22a2 m \u2264 ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U))\u2070\n[PROOFSTEP]\nintro s hs t e\n[GOAL]\ncase this.i\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\ns : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\nhs : s \u2208 m\nt : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\ne : t * s = 0\n\u22a2 t = 0\n[PROOFSTEP]\napply section_ext F (unop U)\n[GOAL]\ncase this.i.h\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\ns : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\nhs : s \u2208 m\nt : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\ne : t * s = 0\n\u22a2 \u2200 (x : { x // x \u2208 U.unop }), \u2191(germ (Sheaf.presheaf F) x) t = \u2191(germ (Sheaf.presheaf F) x) 0\n[PROOFSTEP]\nintro x\n[GOAL]\ncase this.i.h\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\ns : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\nhs : s \u2208 m\nt : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\ne : t * s = 0\nx : { x // x \u2208 U.unop }\n\u22a2 \u2191(germ (Sheaf.presheaf F) x) t = \u2191(germ (Sheaf.presheaf F) x) 0\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\ncase this.i.h\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\ns : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\nhs : s \u2208 m\nt : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\ne : t * s = 0\nx : { x // x \u2208 U.unop }\n\u22a2 \u2191(germ (Sheaf.presheaf F) x) t = 0\n[PROOFSTEP]\napply Submonoid.mem_iInf.mp hs x\n[GOAL]\ncase this.i.h.a\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\ns : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\nhs : s \u2208 m\nt : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\ne : t * s = 0\nx : { x // x \u2208 U.unop }\n\u22a2 \u2191(germ (Sheaf.presheaf F) x) t * \u2191(germ (Sheaf.presheaf F) x) s = 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase this.i.h.a\nX : TopCat\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : ConcreteCategory C\nF\u271d : Presheaf CommRingCat X\nG : SubmonoidPresheaf F\u271d\nF : Sheaf CommRingCat X\nU : (Opens \u2191X)\u1d52\u1d56\nm : Submonoid ((forget CommRingCat).obj ((Sheaf.presheaf F).obj U)) :=\n  \u2a05 (x : { x // x \u2208 U.unop }), Submonoid.comap (germ (Sheaf.presheaf F) x) (\u2191(stalk (Sheaf.presheaf F) \u2191x))\u2070\nthis : IsLocalization m (Localization m)\ns : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\nhs : s \u2208 m\nt : (forget CommRingCat).obj ((Sheaf.presheaf F).obj U)\ne : t * s = 0\nx : { x // x \u2208 U.unop }\n\u22a2 \u2191(germ (Sheaf.presheaf F) x) t * \u2191(germ (Sheaf.presheaf F) x) s = 0\n[PROOFSTEP]\nrw [\u2190 map_mul, e, map_zero]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.Operations", "llama_tokens": 7412, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8615382200964034, "lm_q2_score": 0.6757645944891558, "lm_q1q2_score": 0.582197025940355}}
{"text": "[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 4 * b\u2088 W = b\u2082 W * b\u2086 W - b\u2084 W ^ 2\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, b\u2088]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 4 * (W.a\u2081 ^ 2 * W.a\u2086 + 4 * W.a\u2082 * W.a\u2086 - W.a\u2081 * W.a\u2083 * W.a\u2084 + W.a\u2082 * W.a\u2083 ^ 2 - W.a\u2084 ^ 2) =\n    (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (W.a\u2083 ^ 2 + 4 * W.a\u2086) - (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) ^ 2\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 1728 * \u0394 W = c\u2084 W ^ 3 - c\u2086 W ^ 2\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, b\u2088, c\u2084, c\u2086, \u0394]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 1728 *\n      (-(W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 2 *\n              (W.a\u2081 ^ 2 * W.a\u2086 + 4 * W.a\u2082 * W.a\u2086 - W.a\u2081 * W.a\u2083 * W.a\u2084 + W.a\u2082 * W.a\u2083 ^ 2 - W.a\u2084 ^ 2) -\n            8 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) ^ 3 -\n          27 * (W.a\u2083 ^ 2 + 4 * W.a\u2086) ^ 2 +\n        9 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) * (W.a\u2083 ^ 2 + 4 * W.a\u2086)) =\n    ((W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 2 - 24 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083)) ^ 3 -\n      (-(W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 3 + 36 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) -\n          216 * (W.a\u2083 ^ 2 + 4 * W.a\u2086)) ^\n        2\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 c\u2084 (ofJ0 R) = 0\n[PROOFSTEP]\nrw [ofJ0, c\u2084, b\u2082, b\u2084]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 ^ 2 + 4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2082) ^\n        2 -\n      24 *\n        (2 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2084 +\n          { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083) =\n    0\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 \u0394 (ofJ0 R) = -27\n[PROOFSTEP]\nrw [ofJ0, \u0394, b\u2082, b\u2084, b\u2086, b\u2088]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 -({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 ^ 2 + 4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2082) ^\n                2 *\n            ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 ^ 2 *\n                      { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2086 +\n                    4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2082 *\n                      { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2086 -\n                  { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 *\n                      { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083 *\n                    { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2084 +\n                { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2082 *\n                  { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083 ^ 2 -\n              { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2084 ^ 2) -\n          8 *\n            (2 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2084 +\n                { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 *\n                  { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083) ^\n              3 -\n        27 *\n          ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083 ^ 2 +\n              4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2086) ^\n            2 +\n      9 *\n            ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 ^ 2 +\n              4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2082) *\n          (2 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2084 +\n            { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2081 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083) *\n        ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2083 ^ 2 +\n          4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 1, a\u2084 := 0, a\u2086 := 0 }.a\u2086) =\n    -27\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 c\u2084 (ofJ1728 R) = -48\n[PROOFSTEP]\nrw [ofJ1728, c\u2084, b\u2082, b\u2084]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 ^ 2 + 4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2082) ^\n        2 -\n      24 *\n        (2 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2084 +\n          { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083) =\n    -48\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 \u0394 (ofJ1728 R) = -64\n[PROOFSTEP]\nrw [ofJ1728, \u0394, b\u2082, b\u2084, b\u2086, b\u2088]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\n\u22a2 -({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 ^ 2 + 4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2082) ^\n                2 *\n            ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 ^ 2 *\n                      { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2086 +\n                    4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2082 *\n                      { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2086 -\n                  { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 *\n                      { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083 *\n                    { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2084 +\n                { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2082 *\n                  { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083 ^ 2 -\n              { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2084 ^ 2) -\n          8 *\n            (2 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2084 +\n                { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 *\n                  { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083) ^\n              3 -\n        27 *\n          ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083 ^ 2 +\n              4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2086) ^\n            2 +\n      9 *\n            ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 ^ 2 +\n              4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2082) *\n          (2 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2084 +\n            { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2081 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083) *\n        ({ a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2083 ^ 2 +\n          4 * { a\u2081 := 0, a\u2082 := 0, a\u2083 := 0, a\u2084 := 1, a\u2086 := 0 }.a\u2086) =\n    -64\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nj : R\n\u22a2 c\u2084 (ofJ j) = j * (j - 1728) ^ 3\n[PROOFSTEP]\nsimp only [ofJ, c\u2084, b\u2082, b\u2084]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nj : R\n\u22a2 ((j - 1728) ^ 2 + 4 * 0) ^ 2 - 24 * (2 * (-36 * (j - 1728) ^ 3) + (j - 1728) * 0) = j * (j - 1728) ^ 3\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nj : R\n\u22a2 \u0394 (ofJ j) = j ^ 2 * (j - 1728) ^ 9\n[PROOFSTEP]\nsimp only [ofJ, \u0394, b\u2082, b\u2084, b\u2086, b\u2088]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nj : R\n\u22a2 -((j - 1728) ^ 2 + 4 * 0) ^ 2 *\n            ((j - 1728) ^ 2 * -(j - 1728) ^ 5 + 4 * 0 * -(j - 1728) ^ 5 - (j - 1728) * 0 * (-36 * (j - 1728) ^ 3) +\n                0 * 0 ^ 2 -\n              (-36 * (j - 1728) ^ 3) ^ 2) -\n          8 * (2 * (-36 * (j - 1728) ^ 3) + (j - 1728) * 0) ^ 3 -\n        27 * (0 ^ 2 + 4 * -(j - 1728) ^ 5) ^ 2 +\n      9 * ((j - 1728) ^ 2 + 4 * 0) * (2 * (-36 * (j - 1728) ^ 3) + (j - 1728) * 0) * (0 ^ 2 + 4 * -(j - 1728) ^ 5) =\n    j ^ 2 * (j - 1728) ^ 9\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 comp id C = C\n[PROOFSTEP]\nsimp only [comp, id, zero_add, zero_mul, mul_zero, one_mul]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 comp C id = C\n[PROOFSTEP]\nsimp only [comp, id, add_zero, mul_zero, one_mul, mul_one, one_pow, Units.val_one]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 comp (inv C) C = id\n[PROOFSTEP]\nrw [comp, id, inv]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 { u := { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.u * C.u,\n      r :=\n        { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * \u2191C.u ^ 2 +\n          C.r,\n      s :=\n        \u2191C.u * { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.s + C.s,\n      t :=\n        { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.t * \u2191C.u ^ 3 +\n            { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * C.s *\n              \u2191C.u ^ 2 +\n          C.t } =\n    { u := 1, r := 0, s := 0, t := 0 }\n[PROOFSTEP]\next\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 \u2191{ u := { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.u * C.u,\n          r :=\n            { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r *\n                \u2191C.u ^ 2 +\n              C.r,\n          s :=\n            \u2191C.u * { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.s +\n              C.s,\n          t :=\n            { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.t *\n                  \u2191C.u ^ 3 +\n                { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r *\n                    C.s *\n                  \u2191C.u ^ 2 +\n              C.t }.u =\n    \u2191{ u := 1, r := 0, s := 0, t := 0 }.u\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase r\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 { u := { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.u * C.u,\n        r :=\n          { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * \u2191C.u ^ 2 +\n            C.r,\n        s :=\n          \u2191C.u * { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.s +\n            C.s,\n        t :=\n          { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.t * \u2191C.u ^ 3 +\n              { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * C.s *\n                \u2191C.u ^ 2 +\n            C.t }.r =\n    { u := 1, r := 0, s := 0, t := 0 }.r\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase s\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 { u := { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.u * C.u,\n        r :=\n          { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * \u2191C.u ^ 2 +\n            C.r,\n        s :=\n          \u2191C.u * { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.s +\n            C.s,\n        t :=\n          { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.t * \u2191C.u ^ 3 +\n              { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * C.s *\n                \u2191C.u ^ 2 +\n            C.t }.s =\n    { u := 1, r := 0, s := 0, t := 0 }.s\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase t\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 { u := { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.u * C.u,\n        r :=\n          { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * \u2191C.u ^ 2 +\n            C.r,\n        s :=\n          \u2191C.u * { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.s +\n            C.s,\n        t :=\n          { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.t * \u2191C.u ^ 3 +\n              { u := C.u\u207b\u00b9, r := -C.r * \u2191C.u\u207b\u00b9 ^ 2, s := -C.s * \u2191C.u\u207b\u00b9, t := (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 }.r * C.s *\n                \u2191C.u ^ 2 +\n            C.t }.t =\n    { u := 1, r := 0, s := 0, t := 0 }.t\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 \u2191(C.u\u207b\u00b9 * C.u) = \u21911\n[PROOFSTEP]\nexact C.u.inv_mul\n[GOAL]\ncase r\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 -C.r * \u2191C.u\u207b\u00b9 ^ 2 * \u2191C.u ^ 2 + C.r = 0\n[PROOFSTEP]\nlinear_combination (norm := ring1) -C.r * pow_mul_pow_eq_one 2 C.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 -C.r * \u2191C.u\u207b\u00b9 ^ 2 * \u2191C.u ^ 2 + C.r - 0 - (-C.r * (\u2191C.u\u207b\u00b9 ^ 2 * \u2191C.u ^ 2) - -C.r * 1) = 0\n[PROOFSTEP]\nring1\n[GOAL]\ncase s\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 \u2191C.u * (-C.s * \u2191C.u\u207b\u00b9) + C.s = 0\n[PROOFSTEP]\nlinear_combination (norm := ring1) -C.s * C.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 \u2191C.u * (-C.s * \u2191C.u\u207b\u00b9) + C.s - 0 - (-C.s * (\u2191C.u\u207b\u00b9 * \u2191C.u) - -C.s * 1) = 0\n[PROOFSTEP]\nring1\n[GOAL]\ncase t\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 * \u2191C.u ^ 3 + -C.r * \u2191C.u\u207b\u00b9 ^ 2 * C.s * \u2191C.u ^ 2 + C.t = 0\n[PROOFSTEP]\nlinear_combination (norm := ring1)\n  (C.r * C.s - C.t) * pow_mul_pow_eq_one 3 C.u.inv_mul + -C.r * C.s * pow_mul_pow_eq_one 2 C.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C' C'' C : VariableChange R\n\u22a2 (C.r * C.s - C.t) * \u2191C.u\u207b\u00b9 ^ 3 * \u2191C.u ^ 3 + -C.r * \u2191C.u\u207b\u00b9 ^ 2 * C.s * \u2191C.u ^ 2 + C.t - 0 -\n      ((C.r * C.s - C.t) * (\u2191C.u\u207b\u00b9 ^ 3 * \u2191C.u ^ 3) + -C.r * C.s * (\u2191C.u\u207b\u00b9 ^ 2 * \u2191C.u ^ 2) -\n        ((C.r * C.s - C.t) * 1 + -C.r * C.s * 1)) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 comp (comp C C') C'' = comp C (comp C' C'')\n[PROOFSTEP]\next\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 \u2191(comp (comp C C') C'').u = \u2191(comp C (comp C' C'')).u\n[PROOFSTEP]\nsimp only [comp, Units.val_mul]\n[GOAL]\ncase r\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 (comp (comp C C') C'').r = (comp C (comp C' C'')).r\n[PROOFSTEP]\nsimp only [comp, Units.val_mul]\n[GOAL]\ncase s\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 (comp (comp C C') C'').s = (comp C (comp C' C'')).s\n[PROOFSTEP]\nsimp only [comp, Units.val_mul]\n[GOAL]\ncase t\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 (comp (comp C C') C'').t = (comp C (comp C' C'')).t\n[PROOFSTEP]\nsimp only [comp, Units.val_mul]\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 \u2191C.u * \u2191C'.u * \u2191C''.u = \u2191C.u * (\u2191C'.u * \u2191C''.u)\n[PROOFSTEP]\nring1\n[GOAL]\ncase r\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 (C.r * \u2191C'.u ^ 2 + C'.r) * \u2191C''.u ^ 2 + C''.r = C.r * (\u2191C'.u * \u2191C''.u) ^ 2 + (C'.r * \u2191C''.u ^ 2 + C''.r)\n[PROOFSTEP]\nring1\n[GOAL]\ncase s\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 \u2191C''.u * (\u2191C'.u * C.s + C'.s) + C''.s = \u2191C'.u * \u2191C''.u * C.s + (\u2191C''.u * C'.s + C''.s)\n[PROOFSTEP]\nring1\n[GOAL]\ncase t\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC\u271d C'\u271d C''\u271d C C' C'' : VariableChange R\n\u22a2 (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * \u2191C''.u ^ 3 + (C.r * \u2191C'.u ^ 2 + C'.r) * C''.s * \u2191C''.u ^ 2 +\n      C''.t =\n    C.t * (\u2191C'.u * \u2191C''.u) ^ 3 + C.r * (\u2191C''.u * C'.s + C''.s) * (\u2191C'.u * \u2191C''.u) ^ 2 +\n      (C'.t * \u2191C''.u ^ 3 + C'.r * C''.s * \u2191C''.u ^ 2 + C''.t)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 variableChange W VariableChange.id = W\n[PROOFSTEP]\nrw [VariableChange.id, variableChange, inv_one, Units.val_one]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 { a\u2081 := 1 * (W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),\n      a\u2082 :=\n        1 ^ 2 *\n          (W.a\u2082 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2081 + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -\n            { u := 1, r := 0, s := 0, t := 0 }.s ^ 2),\n      a\u2083 := 1 ^ 3 * (W.a\u2083 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),\n      a\u2084 :=\n        1 ^ 4 *\n          (W.a\u2084 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2083 + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2082 -\n                ({ u := 1, r := 0, s := 0, t := 0 }.t +\n                    { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *\n                  W.a\u2081 +\n              3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -\n            2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),\n      a\u2086 :=\n        1 ^ 6 *\n          (W.a\u2086 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2084 + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a\u2082 +\n                  { u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -\n                { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2083 -\n              { u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -\n            { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2081) } =\n    W\n[PROOFSTEP]\next\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 { a\u2081 := 1 * (W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),\n        a\u2082 :=\n          1 ^ 2 *\n            (W.a\u2082 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2081 + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -\n              { u := 1, r := 0, s := 0, t := 0 }.s ^ 2),\n        a\u2083 := 1 ^ 3 * (W.a\u2083 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2084 :=\n          1 ^ 4 *\n            (W.a\u2084 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2083 + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2082 -\n                  ({ u := 1, r := 0, s := 0, t := 0 }.t +\n                      { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *\n                    W.a\u2081 +\n                3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -\n              2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2086 :=\n          1 ^ 6 *\n            (W.a\u2086 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2084 + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a\u2082 +\n                    { u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -\n                  { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2083 -\n                { u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -\n              { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2081) }.a\u2081 =\n    W.a\u2081\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 1 * (W.a\u2081 + 2 * 0) = W.a\u2081\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 { a\u2081 := 1 * (W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),\n        a\u2082 :=\n          1 ^ 2 *\n            (W.a\u2082 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2081 + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -\n              { u := 1, r := 0, s := 0, t := 0 }.s ^ 2),\n        a\u2083 := 1 ^ 3 * (W.a\u2083 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2084 :=\n          1 ^ 4 *\n            (W.a\u2084 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2083 + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2082 -\n                  ({ u := 1, r := 0, s := 0, t := 0 }.t +\n                      { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *\n                    W.a\u2081 +\n                3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -\n              2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2086 :=\n          1 ^ 6 *\n            (W.a\u2086 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2084 + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a\u2082 +\n                    { u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -\n                  { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2083 -\n                { u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -\n              { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2081) }.a\u2082 =\n    W.a\u2082\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 1 ^ 2 * (W.a\u2082 - 0 * W.a\u2081 + 3 * 0 - 0 ^ 2) = W.a\u2082\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 { a\u2081 := 1 * (W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),\n        a\u2082 :=\n          1 ^ 2 *\n            (W.a\u2082 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2081 + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -\n              { u := 1, r := 0, s := 0, t := 0 }.s ^ 2),\n        a\u2083 := 1 ^ 3 * (W.a\u2083 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2084 :=\n          1 ^ 4 *\n            (W.a\u2084 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2083 + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2082 -\n                  ({ u := 1, r := 0, s := 0, t := 0 }.t +\n                      { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *\n                    W.a\u2081 +\n                3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -\n              2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2086 :=\n          1 ^ 6 *\n            (W.a\u2086 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2084 + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a\u2082 +\n                    { u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -\n                  { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2083 -\n                { u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -\n              { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2081) }.a\u2083 =\n    W.a\u2083\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 1 ^ 3 * (W.a\u2083 + 0 * W.a\u2081 + 2 * 0) = W.a\u2083\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 { a\u2081 := 1 * (W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),\n        a\u2082 :=\n          1 ^ 2 *\n            (W.a\u2082 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2081 + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -\n              { u := 1, r := 0, s := 0, t := 0 }.s ^ 2),\n        a\u2083 := 1 ^ 3 * (W.a\u2083 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2084 :=\n          1 ^ 4 *\n            (W.a\u2084 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2083 + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2082 -\n                  ({ u := 1, r := 0, s := 0, t := 0 }.t +\n                      { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *\n                    W.a\u2081 +\n                3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -\n              2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2086 :=\n          1 ^ 6 *\n            (W.a\u2086 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2084 + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a\u2082 +\n                    { u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -\n                  { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2083 -\n                { u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -\n              { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2081) }.a\u2084 =\n    W.a\u2084\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 1 ^ 4 * (W.a\u2084 - 0 * W.a\u2083 + 2 * 0 * W.a\u2082 - (0 + 0 * 0) * W.a\u2081 + 3 * 0 ^ 2 - 2 * 0 * 0) = W.a\u2084\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 { a\u2081 := 1 * (W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.s),\n        a\u2082 :=\n          1 ^ 2 *\n            (W.a\u2082 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2081 + 3 * { u := 1, r := 0, s := 0, t := 0 }.r -\n              { u := 1, r := 0, s := 0, t := 0 }.s ^ 2),\n        a\u2083 := 1 ^ 3 * (W.a\u2083 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2081 + 2 * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2084 :=\n          1 ^ 4 *\n            (W.a\u2084 - { u := 1, r := 0, s := 0, t := 0 }.s * W.a\u2083 + 2 * { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2082 -\n                  ({ u := 1, r := 0, s := 0, t := 0 }.t +\n                      { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.s) *\n                    W.a\u2081 +\n                3 * { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 -\n              2 * { u := 1, r := 0, s := 0, t := 0 }.s * { u := 1, r := 0, s := 0, t := 0 }.t),\n        a\u2086 :=\n          1 ^ 6 *\n            (W.a\u2086 + { u := 1, r := 0, s := 0, t := 0 }.r * W.a\u2084 + { u := 1, r := 0, s := 0, t := 0 }.r ^ 2 * W.a\u2082 +\n                    { u := 1, r := 0, s := 0, t := 0 }.r ^ 3 -\n                  { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2083 -\n                { u := 1, r := 0, s := 0, t := 0 }.t ^ 2 -\n              { u := 1, r := 0, s := 0, t := 0 }.r * { u := 1, r := 0, s := 0, t := 0 }.t * W.a\u2081) }.a\u2086 =\n    W.a\u2086\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 1 ^ 6 * (W.a\u2086 + 0 * W.a\u2084 + 0 ^ 2 * W.a\u2082 + 0 ^ 3 - 0 * W.a\u2083 - 0 ^ 2 - 0 * 0 * W.a\u2081) = W.a\u2086\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 variableChange W (VariableChange.comp C C') = variableChange (variableChange W C') C\n[PROOFSTEP]\nsimp only [VariableChange.comp, variableChange]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 { a\u2081 := \u2191(C.u * C'.u)\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)),\n      a\u2082 :=\n        \u2191(C.u * C'.u)\u207b\u00b9 ^ 2 *\n          (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2),\n      a\u2083 :=\n        \u2191(C.u * C'.u)\u207b\u00b9 ^ 3 *\n          (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n      a\u2084 :=\n        \u2191(C.u * C'.u)\u207b\u00b9 ^ 4 *\n          (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                  W.a\u2081 +\n              3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n            2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n      a\u2086 :=\n        \u2191(C.u * C'.u)\u207b\u00b9 ^ 6 *\n          (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n              (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n            (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) } =\n    { a\u2081 := \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s),\n      a\u2082 :=\n        \u2191C.u\u207b\u00b9 ^ 2 *\n          (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n            C.s ^ 2),\n      a\u2083 :=\n        \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t),\n      a\u2084 :=\n        \u2191C.u\u207b\u00b9 ^ 4 *\n          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                      (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                        2 * C'.s * C'.t) -\n                    C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                  2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n              3 * C.r ^ 2 -\n            2 * C.s * C.t),\n      a\u2086 :=\n        \u2191C.u\u207b\u00b9 ^ 6 *\n          (\u2191C'.u\u207b\u00b9 ^ 6 *\n                        (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                          C'.r * C'.t * W.a\u2081) +\n                      C.r *\n                        (\u2191C'.u\u207b\u00b9 ^ 4 *\n                          (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                            2 * C'.s * C'.t)) +\n                    C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                  C.r ^ 3 -\n                C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n              C.t ^ 2 -\n            C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) }\n[PROOFSTEP]\next\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 { a\u2081 := \u2191(C.u * C'.u)\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)),\n        a\u2082 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 2 *\n            (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2),\n        a\u2083 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 3 *\n            (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2084 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 4 *\n            (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                    W.a\u2081 +\n                3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n              2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2086 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 6 *\n            (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 +\n                    (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n              (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) }.a\u2081 =\n    { a\u2081 := \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s),\n        a\u2082 :=\n          \u2191C.u\u207b\u00b9 ^ 2 *\n            (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n              C.s ^ 2),\n        a\u2083 :=\n          \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t),\n        a\u2084 :=\n          \u2191C.u\u207b\u00b9 ^ 4 *\n            (\u2191C'.u\u207b\u00b9 ^ 4 *\n                        (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                          2 * C'.s * C'.t) -\n                      C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                    2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                  (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n                3 * C.r ^ 2 -\n              2 * C.s * C.t),\n        a\u2086 :=\n          \u2191C.u\u207b\u00b9 ^ 6 *\n            (\u2191C'.u\u207b\u00b9 ^ 6 *\n                          (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                            C'.r * C'.t * W.a\u2081) +\n                        C.r *\n                          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                            (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                              2 * C'.s * C'.t)) +\n                      C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                    C.r ^ 3 -\n                  C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n                C.t ^ 2 -\n              C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) }.a\u2081\n[PROOFSTEP]\nsimp only [mul_inv, Units.val_mul]\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 { a\u2081 := \u2191(C.u * C'.u)\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)),\n        a\u2082 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 2 *\n            (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2),\n        a\u2083 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 3 *\n            (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2084 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 4 *\n            (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                    W.a\u2081 +\n                3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n              2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2086 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 6 *\n            (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 +\n                    (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n              (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) }.a\u2082 =\n    { a\u2081 := \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s),\n        a\u2082 :=\n          \u2191C.u\u207b\u00b9 ^ 2 *\n            (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n              C.s ^ 2),\n        a\u2083 :=\n          \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t),\n        a\u2084 :=\n          \u2191C.u\u207b\u00b9 ^ 4 *\n            (\u2191C'.u\u207b\u00b9 ^ 4 *\n                        (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                          2 * C'.s * C'.t) -\n                      C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                    2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                  (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n                3 * C.r ^ 2 -\n              2 * C.s * C.t),\n        a\u2086 :=\n          \u2191C.u\u207b\u00b9 ^ 6 *\n            (\u2191C'.u\u207b\u00b9 ^ 6 *\n                          (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                            C'.r * C'.t * W.a\u2081) +\n                        C.r *\n                          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                            (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                              2 * C'.s * C'.t)) +\n                      C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                    C.r ^ 3 -\n                  C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n                C.t ^ 2 -\n              C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) }.a\u2082\n[PROOFSTEP]\nsimp only [mul_inv, Units.val_mul]\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 { a\u2081 := \u2191(C.u * C'.u)\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)),\n        a\u2082 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 2 *\n            (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2),\n        a\u2083 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 3 *\n            (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2084 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 4 *\n            (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                    W.a\u2081 +\n                3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n              2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2086 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 6 *\n            (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 +\n                    (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n              (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) }.a\u2083 =\n    { a\u2081 := \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s),\n        a\u2082 :=\n          \u2191C.u\u207b\u00b9 ^ 2 *\n            (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n              C.s ^ 2),\n        a\u2083 :=\n          \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t),\n        a\u2084 :=\n          \u2191C.u\u207b\u00b9 ^ 4 *\n            (\u2191C'.u\u207b\u00b9 ^ 4 *\n                        (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                          2 * C'.s * C'.t) -\n                      C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                    2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                  (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n                3 * C.r ^ 2 -\n              2 * C.s * C.t),\n        a\u2086 :=\n          \u2191C.u\u207b\u00b9 ^ 6 *\n            (\u2191C'.u\u207b\u00b9 ^ 6 *\n                          (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                            C'.r * C'.t * W.a\u2081) +\n                        C.r *\n                          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                            (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                              2 * C'.s * C'.t)) +\n                      C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                    C.r ^ 3 -\n                  C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n                C.t ^ 2 -\n              C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) }.a\u2083\n[PROOFSTEP]\nsimp only [mul_inv, Units.val_mul]\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 { a\u2081 := \u2191(C.u * C'.u)\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)),\n        a\u2082 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 2 *\n            (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2),\n        a\u2083 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 3 *\n            (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2084 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 4 *\n            (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                    W.a\u2081 +\n                3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n              2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2086 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 6 *\n            (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 +\n                    (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n              (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) }.a\u2084 =\n    { a\u2081 := \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s),\n        a\u2082 :=\n          \u2191C.u\u207b\u00b9 ^ 2 *\n            (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n              C.s ^ 2),\n        a\u2083 :=\n          \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t),\n        a\u2084 :=\n          \u2191C.u\u207b\u00b9 ^ 4 *\n            (\u2191C'.u\u207b\u00b9 ^ 4 *\n                        (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                          2 * C'.s * C'.t) -\n                      C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                    2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                  (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n                3 * C.r ^ 2 -\n              2 * C.s * C.t),\n        a\u2086 :=\n          \u2191C.u\u207b\u00b9 ^ 6 *\n            (\u2191C'.u\u207b\u00b9 ^ 6 *\n                          (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                            C'.r * C'.t * W.a\u2081) +\n                        C.r *\n                          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                            (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                              2 * C'.s * C'.t)) +\n                      C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                    C.r ^ 3 -\n                  C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n                C.t ^ 2 -\n              C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) }.a\u2084\n[PROOFSTEP]\nsimp only [mul_inv, Units.val_mul]\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 { a\u2081 := \u2191(C.u * C'.u)\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)),\n        a\u2082 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 2 *\n            (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2),\n        a\u2083 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 3 *\n            (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2084 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 4 *\n            (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                    W.a\u2081 +\n                3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n              2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)),\n        a\u2086 :=\n          \u2191(C.u * C'.u)\u207b\u00b9 ^ 6 *\n            (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 +\n                    (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                  (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n              (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) }.a\u2086 =\n    { a\u2081 := \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s),\n        a\u2082 :=\n          \u2191C.u\u207b\u00b9 ^ 2 *\n            (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n              C.s ^ 2),\n        a\u2083 :=\n          \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t),\n        a\u2084 :=\n          \u2191C.u\u207b\u00b9 ^ 4 *\n            (\u2191C'.u\u207b\u00b9 ^ 4 *\n                        (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                          2 * C'.s * C'.t) -\n                      C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                    2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                  (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n                3 * C.r ^ 2 -\n              2 * C.s * C.t),\n        a\u2086 :=\n          \u2191C.u\u207b\u00b9 ^ 6 *\n            (\u2191C'.u\u207b\u00b9 ^ 6 *\n                          (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                            C'.r * C'.t * W.a\u2081) +\n                        C.r *\n                          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                            (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                              2 * C'.s * C'.t)) +\n                      C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                    C.r ^ 3 -\n                  C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n                C.t ^ 2 -\n              C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) }.a\u2086\n[PROOFSTEP]\nsimp only [mul_inv, Units.val_mul]\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 \u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)) = \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s)\n[PROOFSTEP]\nlinear_combination (norm := ring1) \u2191C.u\u207b\u00b9 * C.s * 2 * C'.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 \u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * (\u2191C'.u * C.s + C'.s)) - \u2191C.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s) + 2 * C.s) -\n      (\u2191C.u\u207b\u00b9 * C.s * 2 * (\u2191C'.u\u207b\u00b9 * \u2191C'.u) - \u2191C.u\u207b\u00b9 * C.s * 2 * 1) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 2 *\n      (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2) =\n    \u2191C.u\u207b\u00b9 ^ 2 *\n      (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n        C.s ^ 2)\n[PROOFSTEP]\nlinear_combination (norm := ring1)\n  C.s * (-C'.s * 2 - W.a\u2081) * (\u2191C.u\u207b\u00b9 : R) ^ 2 * \u2191C'.u\u207b\u00b9 * C'.u.inv_mul +\n    (C.r * 3 - C.s ^ 2) * (\u2191C.u\u207b\u00b9 : R) ^ 2 * pow_mul_pow_eq_one 2 C'.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 2 *\n          (W.a\u2082 - (\u2191C'.u * C.s + C'.s) * W.a\u2081 + 3 * (C.r * \u2191C'.u ^ 2 + C'.r) - (\u2191C'.u * C.s + C'.s) ^ 2) -\n        \u2191C.u\u207b\u00b9 ^ 2 *\n          (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2) - C.s * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 3 * C.r -\n            C.s ^ 2) -\n      (C.s * (-C'.s * 2 - W.a\u2081) * \u2191C.u\u207b\u00b9 ^ 2 * \u2191C'.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 * \u2191C'.u) +\n          (C.r * 3 - C.s ^ 2) * \u2191C.u\u207b\u00b9 ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * \u2191C'.u ^ 2) -\n        (C.s * (-C'.s * 2 - W.a\u2081) * \u2191C.u\u207b\u00b9 ^ 2 * \u2191C'.u\u207b\u00b9 * 1 + (C.r * 3 - C.s ^ 2) * \u2191C.u\u207b\u00b9 ^ 2 * 1)) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 3 *\n      (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)) =\n    \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t)\n[PROOFSTEP]\nlinear_combination (norm := ring1)\n  C.r * (C'.s * 2 + W.a\u2081) * (\u2191C.u\u207b\u00b9 : R) ^ 3 * \u2191C'.u\u207b\u00b9 * pow_mul_pow_eq_one 2 C'.u.inv_mul +\n    C.t * 2 * (\u2191C.u\u207b\u00b9 : R) ^ 3 * pow_mul_pow_eq_one 3 C'.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 3 *\n          (W.a\u2083 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2081 + 2 * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)) -\n        \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t) + C.r * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) + 2 * C.t) -\n      (C.r * (C'.s * 2 + W.a\u2081) * \u2191C.u\u207b\u00b9 ^ 3 * \u2191C'.u\u207b\u00b9 * (\u2191C'.u\u207b\u00b9 ^ 2 * \u2191C'.u ^ 2) +\n          C.t * 2 * \u2191C.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 ^ 3 * \u2191C'.u ^ 3) -\n        (C.r * (C'.s * 2 + W.a\u2081) * \u2191C.u\u207b\u00b9 ^ 3 * \u2191C'.u\u207b\u00b9 * 1 + C.t * 2 * \u2191C.u\u207b\u00b9 ^ 3 * 1)) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 4 *\n      (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n            (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) * W.a\u2081 +\n          3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n        2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)) =\n    \u2191C.u\u207b\u00b9 ^ 4 *\n      (\u2191C'.u\u207b\u00b9 ^ 4 *\n                  (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                    2 * C'.s * C'.t) -\n                C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n              2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n            (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n          3 * C.r ^ 2 -\n        2 * C.s * C.t)\n[PROOFSTEP]\nlinear_combination (norm := ring1)\n  C.s * (-W.a\u2083 - C'.r * W.a\u2081 - C'.t * 2) * (\u2191C.u\u207b\u00b9 : R) ^ 4 * (\u2191C'.u\u207b\u00b9 : R) ^ 3 * C'.u.inv_mul +\n        (\u2191C.u\u207b\u00b9 : R) ^ 4 * (\u2191C'.u\u207b\u00b9 : R) ^ 2 *\n            (C.r * C'.r * 6 + C.r * W.a\u2082 * 2 - C'.s * C.r * W.a\u2081 * 2 - C'.s ^ 2 * C.r * 2) *\n          pow_mul_pow_eq_one 2 C'.u.inv_mul +\n      -(\u2191C.u\u207b\u00b9 : R) ^ 4 * \u2191C'.u\u207b\u00b9 * (C.s * C'.s * C.r * 2 + C.s * C.r * W.a\u2081 + C'.s * C.t * 2 + C.t * W.a\u2081) *\n        pow_mul_pow_eq_one 3 C'.u.inv_mul +\n    (\u2191C.u\u207b\u00b9 : R) ^ 4 * (C.r ^ 2 * 3 - C.s * C.t * 2) * pow_mul_pow_eq_one 4 C'.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 4 *\n          (W.a\u2084 - (\u2191C'.u * C.s + C'.s) * W.a\u2083 + 2 * (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2082 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t + (C.r * \u2191C'.u ^ 2 + C'.r) * (\u2191C'.u * C.s + C'.s)) *\n                  W.a\u2081 +\n              3 * (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 -\n            2 * (\u2191C'.u * C.s + C'.s) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t)) -\n        \u2191C.u\u207b\u00b9 ^ 4 *\n          (\u2191C'.u\u207b\u00b9 ^ 4 *\n                      (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                        2 * C'.s * C'.t) -\n                    C.s * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) +\n                  2 * C.r * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) -\n                (C.t + C.r * C.s) * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)) +\n              3 * C.r ^ 2 -\n            2 * C.s * C.t) -\n      (C.s * (-W.a\u2083 - C'.r * W.a\u2081 - C'.t * 2) * \u2191C.u\u207b\u00b9 ^ 4 * \u2191C'.u\u207b\u00b9 ^ 3 * (\u2191C'.u\u207b\u00b9 * \u2191C'.u) +\n              \u2191C.u\u207b\u00b9 ^ 4 * \u2191C'.u\u207b\u00b9 ^ 2 *\n                  (C.r * C'.r * 6 + C.r * W.a\u2082 * 2 - C'.s * C.r * W.a\u2081 * 2 - C'.s ^ 2 * C.r * 2) *\n                (\u2191C'.u\u207b\u00b9 ^ 2 * \u2191C'.u ^ 2) +\n            -\u2191C.u\u207b\u00b9 ^ 4 * \u2191C'.u\u207b\u00b9 * (C.s * C'.s * C.r * 2 + C.s * C.r * W.a\u2081 + C'.s * C.t * 2 + C.t * W.a\u2081) *\n              (\u2191C'.u\u207b\u00b9 ^ 3 * \u2191C'.u ^ 3) +\n          \u2191C.u\u207b\u00b9 ^ 4 * (C.r ^ 2 * 3 - C.s * C.t * 2) * (\u2191C'.u\u207b\u00b9 ^ 4 * \u2191C'.u ^ 4) -\n        (C.s * (-W.a\u2083 - C'.r * W.a\u2081 - C'.t * 2) * \u2191C.u\u207b\u00b9 ^ 4 * \u2191C'.u\u207b\u00b9 ^ 3 * 1 +\n              \u2191C.u\u207b\u00b9 ^ 4 * \u2191C'.u\u207b\u00b9 ^ 2 *\n                  (C.r * C'.r * 6 + C.r * W.a\u2082 * 2 - C'.s * C.r * W.a\u2081 * 2 - C'.s ^ 2 * C.r * 2) *\n                1 +\n            -\u2191C.u\u207b\u00b9 ^ 4 * \u2191C'.u\u207b\u00b9 * (C.s * C'.s * C.r * 2 + C.s * C.r * W.a\u2081 + C'.s * C.t * 2 + C.t * W.a\u2081) * 1 +\n          \u2191C.u\u207b\u00b9 ^ 4 * (C.r ^ 2 * 3 - C.s * C.t * 2) * 1)) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 6 *\n      (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n            (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n          (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n        (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) =\n    \u2191C.u\u207b\u00b9 ^ 6 *\n      (\u2191C'.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 - C'.r * C'.t * W.a\u2081) +\n                  C.r *\n                    (\u2191C'.u\u207b\u00b9 ^ 4 *\n                      (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                        2 * C'.s * C'.t)) +\n                C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n              C.r ^ 3 -\n            C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n          C.t ^ 2 -\n        C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s)))\n[PROOFSTEP]\nlinear_combination (norm := ring1)\n  C.r * (\u2191C.u\u207b\u00b9 : R) ^ 6 * (\u2191C'.u\u207b\u00b9 : R) ^ 4 *\n              (C'.r * W.a\u2082 * 2 - C'.r * C'.s * W.a\u2081 + C'.r ^ 2 * 3 + W.a\u2084 - C'.s * C'.t * 2 - C'.s * W.a\u2083 -\n                C'.t * W.a\u2081) *\n            pow_mul_pow_eq_one 2 C'.u.inv_mul +\n          -(\u2191C.u\u207b\u00b9 : R) ^ 6 * (\u2191C'.u\u207b\u00b9 : R) ^ 3 * C.t * (C'.r * W.a\u2081 + C'.t * 2 + W.a\u2083) *\n            pow_mul_pow_eq_one 3 C'.u.inv_mul +\n        C.r ^ 2 * (\u2191C.u\u207b\u00b9 : R) ^ 6 * (\u2191C'.u\u207b\u00b9 : R) ^ 2 * (C'.r * 3 + W.a\u2082 - C'.s * W.a\u2081 - C'.s ^ 2) *\n          pow_mul_pow_eq_one 4 C'.u.inv_mul +\n      -C.r * C.t * (\u2191C.u\u207b\u00b9 : R) ^ 6 * \u2191C'.u\u207b\u00b9 * (C'.s * 2 + W.a\u2081) * pow_mul_pow_eq_one 5 C'.u.inv_mul +\n    (\u2191C.u\u207b\u00b9 : R) ^ 6 * (C.r ^ 3 - C.t ^ 2) * pow_mul_pow_eq_one 6 C'.u.inv_mul\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nW\u271d : WeierstrassCurve R\nC\u271d C C' : VariableChange R\nW : WeierstrassCurve R\n\u22a2 (\u2191C.u\u207b\u00b9 * \u2191C'.u\u207b\u00b9) ^ 6 *\n          (W.a\u2086 + (C.r * \u2191C'.u ^ 2 + C'.r) * W.a\u2084 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 2 * W.a\u2082 + (C.r * \u2191C'.u ^ 2 + C'.r) ^ 3 -\n                (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2083 -\n              (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) ^ 2 -\n            (C.r * \u2191C'.u ^ 2 + C'.r) * (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) * W.a\u2081) -\n        \u2191C.u\u207b\u00b9 ^ 6 *\n          (\u2191C'.u\u207b\u00b9 ^ 6 *\n                        (W.a\u2086 + C'.r * W.a\u2084 + C'.r ^ 2 * W.a\u2082 + C'.r ^ 3 - C'.t * W.a\u2083 - C'.t ^ 2 -\n                          C'.r * C'.t * W.a\u2081) +\n                      C.r *\n                        (\u2191C'.u\u207b\u00b9 ^ 4 *\n                          (W.a\u2084 - C'.s * W.a\u2083 + 2 * C'.r * W.a\u2082 - (C'.t + C'.r * C'.s) * W.a\u2081 + 3 * C'.r ^ 2 -\n                            2 * C'.s * C'.t)) +\n                    C.r ^ 2 * (\u2191C'.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C'.s * W.a\u2081 + 3 * C'.r - C'.s ^ 2)) +\n                  C.r ^ 3 -\n                C.t * (\u2191C'.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C'.r * W.a\u2081 + 2 * C'.t)) -\n              C.t ^ 2 -\n            C.r * C.t * (\u2191C'.u\u207b\u00b9 * (W.a\u2081 + 2 * C'.s))) -\n      (C.r * \u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 ^ 4 *\n                    (C'.r * W.a\u2082 * 2 - C'.r * C'.s * W.a\u2081 + C'.r ^ 2 * 3 + W.a\u2084 - C'.s * C'.t * 2 - C'.s * W.a\u2083 -\n                      C'.t * W.a\u2081) *\n                  (\u2191C'.u\u207b\u00b9 ^ 2 * \u2191C'.u ^ 2) +\n                -\u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 ^ 3 * C.t * (C'.r * W.a\u2081 + C'.t * 2 + W.a\u2083) * (\u2191C'.u\u207b\u00b9 ^ 3 * \u2191C'.u ^ 3) +\n              C.r ^ 2 * \u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 ^ 2 * (C'.r * 3 + W.a\u2082 - C'.s * W.a\u2081 - C'.s ^ 2) *\n                (\u2191C'.u\u207b\u00b9 ^ 4 * \u2191C'.u ^ 4) +\n            -C.r * C.t * \u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 * (C'.s * 2 + W.a\u2081) * (\u2191C'.u\u207b\u00b9 ^ 5 * \u2191C'.u ^ 5) +\n          \u2191C.u\u207b\u00b9 ^ 6 * (C.r ^ 3 - C.t ^ 2) * (\u2191C'.u\u207b\u00b9 ^ 6 * \u2191C'.u ^ 6) -\n        (C.r * \u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 ^ 4 *\n                    (C'.r * W.a\u2082 * 2 - C'.r * C'.s * W.a\u2081 + C'.r ^ 2 * 3 + W.a\u2084 - C'.s * C'.t * 2 - C'.s * W.a\u2083 -\n                      C'.t * W.a\u2081) *\n                  1 +\n                -\u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 ^ 3 * C.t * (C'.r * W.a\u2081 + C'.t * 2 + W.a\u2083) * 1 +\n              C.r ^ 2 * \u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 ^ 2 * (C'.r * 3 + W.a\u2082 - C'.s * W.a\u2081 - C'.s ^ 2) * 1 +\n            -C.r * C.t * \u2191C.u\u207b\u00b9 ^ 6 * \u2191C'.u\u207b\u00b9 * (C'.s * 2 + W.a\u2081) * 1 +\n          \u2191C.u\u207b\u00b9 ^ 6 * (C.r ^ 3 - C.t ^ 2) * 1)) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 b\u2082 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 2 * (b\u2082 W + 12 * C.r)\n[PROOFSTEP]\nsimp only [b\u2082, variableChange_a\u2081, variableChange_a\u2082]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)) ^ 2 + 4 * (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)) =\n    \u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2081 ^ 2 + 4 * W.a\u2082 + 12 * C.r)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 b\u2084 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 4 * (b\u2084 W + C.r * b\u2082 W + 6 * C.r ^ 2)\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, variableChange_a\u2081, variableChange_a\u2083, variableChange_a\u2084]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 2 * (\u2191C.u\u207b\u00b9 ^ 4 * (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) +\n      \u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s) * (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) =\n    \u2191C.u\u207b\u00b9 ^ 4 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083 + C.r * (W.a\u2081 ^ 2 + 4 * W.a\u2082) + 6 * C.r ^ 2)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 b\u2086 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 6 * (b\u2086 W + 2 * C.r * b\u2084 W + C.r ^ 2 * b\u2082 W + 4 * C.r ^ 3)\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, variableChange_a\u2083, variableChange_a\u2086]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) ^ 2 +\n      4 * (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) =\n    \u2191C.u\u207b\u00b9 ^ 6 *\n      (W.a\u2083 ^ 2 + 4 * W.a\u2086 + 2 * C.r * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) + C.r ^ 2 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) + 4 * C.r ^ 3)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 b\u2088 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 8 * (b\u2088 W + 3 * C.r * b\u2086 W + 3 * C.r ^ 2 * b\u2084 W + C.r ^ 3 * b\u2082 W + 3 * C.r ^ 4)\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, b\u2088, variableChange_a\u2081, variableChange_a\u2082, variableChange_a\u2083, variableChange_a\u2084,\n  variableChange_a\u2086]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)) ^ 2 *\n              (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) +\n            4 * (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)) *\n              (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) -\n          \u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s) * (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) *\n            (\u2191C.u\u207b\u00b9 ^ 4 *\n              (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) +\n        \u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2) * (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) ^ 2 -\n      (\u2191C.u\u207b\u00b9 ^ 4 * (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) ^ 2 =\n    \u2191C.u\u207b\u00b9 ^ 8 *\n      (W.a\u2081 ^ 2 * W.a\u2086 + 4 * W.a\u2082 * W.a\u2086 - W.a\u2081 * W.a\u2083 * W.a\u2084 + W.a\u2082 * W.a\u2083 ^ 2 - W.a\u2084 ^ 2 +\n              3 * C.r * (W.a\u2083 ^ 2 + 4 * W.a\u2086) +\n            3 * C.r ^ 2 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) +\n          C.r ^ 3 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) +\n        3 * C.r ^ 4)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 c\u2084 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 4 * c\u2084 W\n[PROOFSTEP]\nsimp only [c\u2084, variableChange_b\u2082, variableChange_b\u2084]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 (\u2191C.u\u207b\u00b9 ^ 2 * (b\u2082 W + 12 * C.r)) ^ 2 - 24 * (\u2191C.u\u207b\u00b9 ^ 4 * (b\u2084 W + C.r * b\u2082 W + 6 * C.r ^ 2)) =\n    \u2191C.u\u207b\u00b9 ^ 4 * (b\u2082 W ^ 2 - 24 * b\u2084 W)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 c\u2086 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 6 * c\u2086 W\n[PROOFSTEP]\nsimp only [c\u2086, variableChange_b\u2082, variableChange_b\u2084, variableChange_b\u2086]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 -(\u2191C.u\u207b\u00b9 ^ 2 * (b\u2082 W + 12 * C.r)) ^ 3 +\n        36 * (\u2191C.u\u207b\u00b9 ^ 2 * (b\u2082 W + 12 * C.r)) * (\u2191C.u\u207b\u00b9 ^ 4 * (b\u2084 W + C.r * b\u2082 W + 6 * C.r ^ 2)) -\n      216 * (\u2191C.u\u207b\u00b9 ^ 6 * (b\u2086 W + 2 * C.r * b\u2084 W + C.r ^ 2 * b\u2082 W + 4 * C.r ^ 3)) =\n    \u2191C.u\u207b\u00b9 ^ 6 * (-b\u2082 W ^ 3 + 36 * b\u2082 W * b\u2084 W - 216 * b\u2086 W)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 \u0394 (variableChange W C) = \u2191C.u\u207b\u00b9 ^ 12 * \u0394 W\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, b\u2088, \u0394, variableChange_a\u2081, variableChange_a\u2082, variableChange_a\u2083, variableChange_a\u2084,\n  variableChange_a\u2086]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n\u22a2 -((\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)) ^ 2 + 4 * (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2))) ^ 2 *\n            ((\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)) ^ 2 *\n                      (\u2191C.u\u207b\u00b9 ^ 6 *\n                        (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) +\n                    4 * (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)) *\n                      (\u2191C.u\u207b\u00b9 ^ 6 *\n                        (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) -\n                  \u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s) * (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) *\n                    (\u2191C.u\u207b\u00b9 ^ 4 *\n                      (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) +\n                \u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2) *\n                  (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) ^ 2 -\n              (\u2191C.u\u207b\u00b9 ^ 4 *\n                  (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) ^\n                2) -\n          8 *\n            (2 *\n                  (\u2191C.u\u207b\u00b9 ^ 4 *\n                    (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) +\n                \u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s) * (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t))) ^\n              3 -\n        27 *\n          ((\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) ^ 2 +\n              4 *\n                (\u2191C.u\u207b\u00b9 ^ 6 *\n                  (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081))) ^\n            2 +\n      9 * ((\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)) ^ 2 + 4 * (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2))) *\n          (2 *\n              (\u2191C.u\u207b\u00b9 ^ 4 *\n                (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)) +\n            \u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s) * (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t))) *\n        ((\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)) ^ 2 +\n          4 * (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081))) =\n    \u2191C.u\u207b\u00b9 ^ 12 *\n      (-(W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 2 *\n              (W.a\u2081 ^ 2 * W.a\u2086 + 4 * W.a\u2082 * W.a\u2086 - W.a\u2081 * W.a\u2083 * W.a\u2084 + W.a\u2082 * W.a\u2083 ^ 2 - W.a\u2084 ^ 2) -\n            8 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) ^ 3 -\n          27 * (W.a\u2083 ^ 2 + 4 * W.a\u2086) ^ 2 +\n        9 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) * (W.a\u2083 ^ 2 + 4 * W.a\u2086))\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 b\u2082 (baseChange W A) = \u2191(algebraMap R A) (b\u2082 W)\n[PROOFSTEP]\nsimp only [b\u2082, baseChange_a\u2081, baseChange_a\u2082]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 \u2191(algebraMap R A) W.a\u2081 ^ 2 + 4 * \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) (W.a\u2081 ^ 2 + 4 * W.a\u2082)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 b\u2084 (baseChange W A) = \u2191(algebraMap R A) (b\u2084 W)\n[PROOFSTEP]\nsimp only [b\u2084, baseChange_a\u2081, baseChange_a\u2083, baseChange_a\u2084]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 2 * \u2191(algebraMap R A) W.a\u2084 + \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) W.a\u2083 =\n    \u2191(algebraMap R A) (2 * W.a\u2084 + W.a\u2081 * W.a\u2083)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 b\u2086 (baseChange W A) = \u2191(algebraMap R A) (b\u2086 W)\n[PROOFSTEP]\nsimp only [b\u2086, baseChange_a\u2083, baseChange_a\u2086]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 \u2191(algebraMap R A) W.a\u2083 ^ 2 + 4 * \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) (W.a\u2083 ^ 2 + 4 * W.a\u2086)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 b\u2088 (baseChange W A) = \u2191(algebraMap R A) (b\u2088 W)\n[PROOFSTEP]\nsimp only [b\u2088, baseChange_a\u2081, baseChange_a\u2082, baseChange_a\u2083, baseChange_a\u2084, baseChange_a\u2086]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 \u2191(algebraMap R A) W.a\u2081 ^ 2 * \u2191(algebraMap R A) W.a\u2086 + 4 * \u2191(algebraMap R A) W.a\u2082 * \u2191(algebraMap R A) W.a\u2086 -\n          \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) W.a\u2083 * \u2191(algebraMap R A) W.a\u2084 +\n        \u2191(algebraMap R A) W.a\u2082 * \u2191(algebraMap R A) W.a\u2083 ^ 2 -\n      \u2191(algebraMap R A) W.a\u2084 ^ 2 =\n    \u2191(algebraMap R A) (W.a\u2081 ^ 2 * W.a\u2086 + 4 * W.a\u2082 * W.a\u2086 - W.a\u2081 * W.a\u2083 * W.a\u2084 + W.a\u2082 * W.a\u2083 ^ 2 - W.a\u2084 ^ 2)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 c\u2084 (baseChange W A) = \u2191(algebraMap R A) (c\u2084 W)\n[PROOFSTEP]\nsimp only [c\u2084, baseChange_b\u2082, baseChange_b\u2084]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 \u2191(algebraMap R A) (b\u2082 W) ^ 2 - 24 * \u2191(algebraMap R A) (b\u2084 W) = \u2191(algebraMap R A) (b\u2082 W ^ 2 - 24 * b\u2084 W)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 c\u2086 (baseChange W A) = \u2191(algebraMap R A) (c\u2086 W)\n[PROOFSTEP]\nsimp only [c\u2086, baseChange_b\u2082, baseChange_b\u2084, baseChange_b\u2086]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 -\u2191(algebraMap R A) (b\u2082 W) ^ 3 + 36 * \u2191(algebraMap R A) (b\u2082 W) * \u2191(algebraMap R A) (b\u2084 W) -\n      216 * \u2191(algebraMap R A) (b\u2086 W) =\n    \u2191(algebraMap R A) (-b\u2082 W ^ 3 + 36 * b\u2082 W * b\u2084 W - 216 * b\u2086 W)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 \u0394 (baseChange W A) = \u2191(algebraMap R A) (\u0394 W)\n[PROOFSTEP]\nsimp only [\u0394, baseChange_b\u2082, baseChange_b\u2084, baseChange_b\u2086, baseChange_b\u2088]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 -\u2191(algebraMap R A) (b\u2082 W) ^ 2 * \u2191(algebraMap R A) (b\u2088 W) - 8 * \u2191(algebraMap R A) (b\u2084 W) ^ 3 -\n        27 * \u2191(algebraMap R A) (b\u2086 W) ^ 2 +\n      9 * \u2191(algebraMap R A) (b\u2082 W) * \u2191(algebraMap R A) (b\u2084 W) * \u2191(algebraMap R A) (b\u2086 W) =\n    \u2191(algebraMap R A) (-b\u2082 W ^ 2 * b\u2088 W - 8 * b\u2084 W ^ 3 - 27 * b\u2086 W ^ 2 + 9 * b\u2082 W * b\u2084 W * b\u2086 W)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 baseChange W R = W\n[PROOFSTEP]\next\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange W R).a\u2081 = W.a\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange W R).a\u2082 = W.a\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange W R).a\u2083 = W.a\u2083\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange W R).a\u2084 = W.a\u2084\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange W R).a\u2086 = W.a\u2086\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 baseChange (baseChange W A) B = baseChange W B\n[PROOFSTEP]\next\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange (baseChange W A) B).a\u2081 = (baseChange W B).a\u2081\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange (baseChange W A) B).a\u2082 = (baseChange W B).a\u2082\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange (baseChange W A) B).a\u2083 = (baseChange W B).a\u2083\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange (baseChange W A) B).a\u2084 = (baseChange W B).a\u2084\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (baseChange (baseChange W A) B).a\u2086 = (baseChange W B).a\u2086\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\n\u22a2 W = W'\n[PROOFSTEP]\nrcases mk.inj h1 with \u27e8_, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 W = W'\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.intro.intro.a\u2081\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 W.a\u2081 = W'.a\u2081\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.a\u2082\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 W.a\u2082 = W'.a\u2082\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.a\u2083\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 W.a\u2083 = W'.a\u2083\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.a\u2084\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 W.a\u2084 = W'.a\u2084\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.a\u2086\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 W.a\u2086 = W'.a\u2086\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.a\u2081\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.a\u2082\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.a\u2083\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.a\u2084\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.a\u2086\nR : Type u\ninst\u271d\u2076 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : Function.Injective \u2191(algebraMap R A)\nW W' : WeierstrassCurve R\nh1 : (fun W => baseChange W A) W = (fun W => baseChange W A) W'\nleft\u271d\u00b3 : \u2191(algebraMap R A) W.a\u2081 = \u2191(algebraMap R A) W'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) W.a\u2082 = \u2191(algebraMap R A) W'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) W.a\u2083 = \u2191(algebraMap R A) W'.a\u2083\nleft\u271d : \u2191(algebraMap R A) W.a\u2084 = \u2191(algebraMap R A) W'.a\u2084\nright\u271d : \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n\u22a2 \u2191(algebraMap R A) W.a\u2086 = \u2191(algebraMap R A) W'.a\u2086\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 baseChange A id = id\n[PROOFSTEP]\nsimp only [id, baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) 1, r := \u2191(algebraMap R A) 0, s := \u2191(algebraMap R A) 0,\n      t := \u2191(algebraMap R A) 0 } =\n    { u := 1, r := 0, s := 0, t := 0 }\n[PROOFSTEP]\next\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 \u2191{ u := \u2191(Units.map \u2191(algebraMap R A)) 1, r := \u2191(algebraMap R A) 0, s := \u2191(algebraMap R A) 0,\n          t := \u2191(algebraMap R A) 0 }.u =\n    \u2191{ u := 1, r := 0, s := 0, t := 0 }.u\n[PROOFSTEP]\nsimp only [map_one, Units.val_one, map_zero]\n[GOAL]\ncase r\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) 1, r := \u2191(algebraMap R A) 0, s := \u2191(algebraMap R A) 0,\n        t := \u2191(algebraMap R A) 0 }.r =\n    { u := 1, r := 0, s := 0, t := 0 }.r\n[PROOFSTEP]\nsimp only [map_one, Units.val_one, map_zero]\n[GOAL]\ncase s\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) 1, r := \u2191(algebraMap R A) 0, s := \u2191(algebraMap R A) 0,\n        t := \u2191(algebraMap R A) 0 }.s =\n    { u := 1, r := 0, s := 0, t := 0 }.s\n[PROOFSTEP]\nsimp only [map_one, Units.val_one, map_zero]\n[GOAL]\ncase t\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) 1, r := \u2191(algebraMap R A) 0, s := \u2191(algebraMap R A) 0,\n        t := \u2191(algebraMap R A) 0 }.t =\n    { u := 1, r := 0, s := 0, t := 0 }.t\n[PROOFSTEP]\nsimp only [map_one, Units.val_one, map_zero]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC C' : VariableChange R\n\u22a2 baseChange A (comp C C') = comp (baseChange A C) (baseChange A C')\n[PROOFSTEP]\nsimp only [comp, baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC C' : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) (C.u * C'.u), r := \u2191(algebraMap R A) (C.r * \u2191C'.u ^ 2 + C'.r),\n      s := \u2191(algebraMap R A) (\u2191C'.u * C.s + C'.s),\n      t := \u2191(algebraMap R A) (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) } =\n    { u := \u2191(Units.map \u2191(algebraMap R A)) C.u * \u2191(Units.map \u2191(algebraMap R A)) C'.u,\n      r := \u2191(algebraMap R A) C.r * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 + \u2191(algebraMap R A) C'.r,\n      s := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) * \u2191(algebraMap R A) C.s + \u2191(algebraMap R A) C'.s,\n      t :=\n        \u2191(algebraMap R A) C.t * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 3 +\n            \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C'.s * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 +\n          \u2191(algebraMap R A) C'.t }\n[PROOFSTEP]\next\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC C' : VariableChange R\n\u22a2 \u2191{ u := \u2191(Units.map \u2191(algebraMap R A)) (C.u * C'.u), r := \u2191(algebraMap R A) (C.r * \u2191C'.u ^ 2 + C'.r),\n          s := \u2191(algebraMap R A) (\u2191C'.u * C.s + C'.s),\n          t := \u2191(algebraMap R A) (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) }.u =\n    \u2191{ u := \u2191(Units.map \u2191(algebraMap R A)) C.u * \u2191(Units.map \u2191(algebraMap R A)) C'.u,\n          r := \u2191(algebraMap R A) C.r * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 + \u2191(algebraMap R A) C'.r,\n          s := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) * \u2191(algebraMap R A) C.s + \u2191(algebraMap R A) C'.s,\n          t :=\n            \u2191(algebraMap R A) C.t * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 3 +\n                \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C'.s * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 +\n              \u2191(algebraMap R A) C'.t }.u\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase r\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC C' : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) (C.u * C'.u), r := \u2191(algebraMap R A) (C.r * \u2191C'.u ^ 2 + C'.r),\n        s := \u2191(algebraMap R A) (\u2191C'.u * C.s + C'.s),\n        t := \u2191(algebraMap R A) (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) }.r =\n    { u := \u2191(Units.map \u2191(algebraMap R A)) C.u * \u2191(Units.map \u2191(algebraMap R A)) C'.u,\n        r := \u2191(algebraMap R A) C.r * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 + \u2191(algebraMap R A) C'.r,\n        s := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) * \u2191(algebraMap R A) C.s + \u2191(algebraMap R A) C'.s,\n        t :=\n          \u2191(algebraMap R A) C.t * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 3 +\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C'.s * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 +\n            \u2191(algebraMap R A) C'.t }.r\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase s\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC C' : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) (C.u * C'.u), r := \u2191(algebraMap R A) (C.r * \u2191C'.u ^ 2 + C'.r),\n        s := \u2191(algebraMap R A) (\u2191C'.u * C.s + C'.s),\n        t := \u2191(algebraMap R A) (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) }.s =\n    { u := \u2191(Units.map \u2191(algebraMap R A)) C.u * \u2191(Units.map \u2191(algebraMap R A)) C'.u,\n        r := \u2191(algebraMap R A) C.r * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 + \u2191(algebraMap R A) C'.r,\n        s := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) * \u2191(algebraMap R A) C.s + \u2191(algebraMap R A) C'.s,\n        t :=\n          \u2191(algebraMap R A) C.t * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 3 +\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C'.s * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 +\n            \u2191(algebraMap R A) C'.t }.s\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase t\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC C' : VariableChange R\n\u22a2 { u := \u2191(Units.map \u2191(algebraMap R A)) (C.u * C'.u), r := \u2191(algebraMap R A) (C.r * \u2191C'.u ^ 2 + C'.r),\n        s := \u2191(algebraMap R A) (\u2191C'.u * C.s + C'.s),\n        t := \u2191(algebraMap R A) (C.t * \u2191C'.u ^ 3 + C.r * C'.s * \u2191C'.u ^ 2 + C'.t) }.t =\n    { u := \u2191(Units.map \u2191(algebraMap R A)) C.u * \u2191(Units.map \u2191(algebraMap R A)) C'.u,\n        r := \u2191(algebraMap R A) C.r * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 + \u2191(algebraMap R A) C'.r,\n        s := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) * \u2191(algebraMap R A) C.s + \u2191(algebraMap R A) C'.s,\n        t :=\n          \u2191(algebraMap R A) C.t * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 3 +\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C'.s * \u2191(\u2191(Units.map \u2191(algebraMap R A)) C'.u) ^ 2 +\n            \u2191(algebraMap R A) C'.t }.t\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 baseChange B (baseChange A C) = baseChange B C\n[PROOFSTEP]\next\n[GOAL]\ncase u.a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 \u2191(baseChange B (baseChange A C)).u = \u2191(baseChange B C).u\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase r\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 (baseChange B (baseChange A C)).r = (baseChange B C).r\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase s\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 (baseChange B (baseChange A C)).s = (baseChange B C).s\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\ncase t\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 (baseChange B (baseChange A C)).t = (baseChange B C).t\n[PROOFSTEP]\nexact (IsScalarTower.algebraMap_apply R A B _).symm\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1 : baseChange A C = baseChange A C'\n\u22a2 C = C'\n[PROOFSTEP]\nrcases mk.inj h1 with \u27e8h1, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nh1 : \u2191(Units.map \u2191(algebraMap R A)) C.u = \u2191(Units.map \u2191(algebraMap R A)) C'.u\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\n\u22a2 C = C'\n[PROOFSTEP]\nreplace h1 := (Units.mk.inj h1).left\n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 C = C'\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.intro.u.a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 \u2191C.u = \u2191C'.u\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.r\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 C.r = C'.r\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.s\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 C.s = C'.s\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.t\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 C.t = C'.t\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.u.a\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 \u2191(algebraMap R A) \u2191C.u = \u2191(algebraMap R A) \u2191C'.u\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.r\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.s\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.t\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC\u271d : VariableChange R\nh : Function.Injective \u2191(algebraMap R A)\nC C' : VariableChange R\nh1\u271d : baseChange A C = baseChange A C'\nleft\u271d\u00b9 : \u2191(algebraMap R A) C.r = \u2191(algebraMap R A) C'.r\nleft\u271d : \u2191(algebraMap R A) C.s = \u2191(algebraMap R A) C'.s\nright\u271d : \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\nh1 : \u2191\u2191(algebraMap R A) \u2191C.u = \u2191\u2191(algebraMap R A) \u2191C'.u\n\u22a2 \u2191(algebraMap R A) C.t = \u2191(algebraMap R A) C'.t\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 variableChange (baseChange W A) (VariableChange.baseChange A C) = baseChange (variableChange W C) A\n[PROOFSTEP]\nsimp only [baseChange, variableChange, VariableChange.baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { a\u2081 := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 * (\u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.s),\n      a\u2082 :=\n        \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 2 *\n          (\u2191(algebraMap R A) W.a\u2082 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2081 + 3 * \u2191(algebraMap R A) C.r -\n            \u2191(algebraMap R A) C.s ^ 2),\n      a\u2083 :=\n        \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 3 *\n          (\u2191(algebraMap R A) W.a\u2083 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.t),\n      a\u2084 :=\n        \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 4 *\n          (\u2191(algebraMap R A) W.a\u2084 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2083 +\n                  2 * \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2082 -\n                (\u2191(algebraMap R A) C.t + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.s) * \u2191(algebraMap R A) W.a\u2081 +\n              3 * \u2191(algebraMap R A) C.r ^ 2 -\n            2 * \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) C.t),\n      a\u2086 :=\n        \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 6 *\n          (\u2191(algebraMap R A) W.a\u2086 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2084 +\n                    \u2191(algebraMap R A) C.r ^ 2 * \u2191(algebraMap R A) W.a\u2082 +\n                  \u2191(algebraMap R A) C.r ^ 3 -\n                \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2083 -\n              \u2191(algebraMap R A) C.t ^ 2 -\n            \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2081) } =\n    { a\u2081 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)),\n      a\u2082 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)),\n      a\u2083 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)),\n      a\u2084 :=\n        \u2191(algebraMap R A)\n          (\u2191C.u\u207b\u00b9 ^ 4 * (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)),\n      a\u2086 :=\n        \u2191(algebraMap R A)\n          (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) }\n[PROOFSTEP]\next\n[GOAL]\ncase a\u2081\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { a\u2081 := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 * (\u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.s),\n        a\u2082 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 2 *\n            (\u2191(algebraMap R A) W.a\u2082 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2081 + 3 * \u2191(algebraMap R A) C.r -\n              \u2191(algebraMap R A) C.s ^ 2),\n        a\u2083 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 3 *\n            (\u2191(algebraMap R A) W.a\u2083 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.t),\n        a\u2084 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 4 *\n            (\u2191(algebraMap R A) W.a\u2084 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2083 +\n                    2 * \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2082 -\n                  (\u2191(algebraMap R A) C.t + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.s) * \u2191(algebraMap R A) W.a\u2081 +\n                3 * \u2191(algebraMap R A) C.r ^ 2 -\n              2 * \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) C.t),\n        a\u2086 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 6 *\n            (\u2191(algebraMap R A) W.a\u2086 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2084 +\n                      \u2191(algebraMap R A) C.r ^ 2 * \u2191(algebraMap R A) W.a\u2082 +\n                    \u2191(algebraMap R A) C.r ^ 3 -\n                  \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2083 -\n                \u2191(algebraMap R A) C.t ^ 2 -\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2081) }.a\u2081 =\n    { a\u2081 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)),\n        a\u2082 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)),\n        a\u2083 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)),\n        a\u2084 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 4 *\n              (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)),\n        a\u2086 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) }.a\u2081\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase a\u2082\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { a\u2081 := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 * (\u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.s),\n        a\u2082 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 2 *\n            (\u2191(algebraMap R A) W.a\u2082 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2081 + 3 * \u2191(algebraMap R A) C.r -\n              \u2191(algebraMap R A) C.s ^ 2),\n        a\u2083 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 3 *\n            (\u2191(algebraMap R A) W.a\u2083 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.t),\n        a\u2084 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 4 *\n            (\u2191(algebraMap R A) W.a\u2084 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2083 +\n                    2 * \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2082 -\n                  (\u2191(algebraMap R A) C.t + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.s) * \u2191(algebraMap R A) W.a\u2081 +\n                3 * \u2191(algebraMap R A) C.r ^ 2 -\n              2 * \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) C.t),\n        a\u2086 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 6 *\n            (\u2191(algebraMap R A) W.a\u2086 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2084 +\n                      \u2191(algebraMap R A) C.r ^ 2 * \u2191(algebraMap R A) W.a\u2082 +\n                    \u2191(algebraMap R A) C.r ^ 3 -\n                  \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2083 -\n                \u2191(algebraMap R A) C.t ^ 2 -\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2081) }.a\u2082 =\n    { a\u2081 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)),\n        a\u2082 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)),\n        a\u2083 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)),\n        a\u2084 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 4 *\n              (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)),\n        a\u2086 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) }.a\u2082\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase a\u2083\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { a\u2081 := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 * (\u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.s),\n        a\u2082 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 2 *\n            (\u2191(algebraMap R A) W.a\u2082 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2081 + 3 * \u2191(algebraMap R A) C.r -\n              \u2191(algebraMap R A) C.s ^ 2),\n        a\u2083 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 3 *\n            (\u2191(algebraMap R A) W.a\u2083 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.t),\n        a\u2084 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 4 *\n            (\u2191(algebraMap R A) W.a\u2084 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2083 +\n                    2 * \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2082 -\n                  (\u2191(algebraMap R A) C.t + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.s) * \u2191(algebraMap R A) W.a\u2081 +\n                3 * \u2191(algebraMap R A) C.r ^ 2 -\n              2 * \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) C.t),\n        a\u2086 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 6 *\n            (\u2191(algebraMap R A) W.a\u2086 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2084 +\n                      \u2191(algebraMap R A) C.r ^ 2 * \u2191(algebraMap R A) W.a\u2082 +\n                    \u2191(algebraMap R A) C.r ^ 3 -\n                  \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2083 -\n                \u2191(algebraMap R A) C.t ^ 2 -\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2081) }.a\u2083 =\n    { a\u2081 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)),\n        a\u2082 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)),\n        a\u2083 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)),\n        a\u2084 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 4 *\n              (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)),\n        a\u2086 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) }.a\u2083\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase a\u2084\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { a\u2081 := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 * (\u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.s),\n        a\u2082 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 2 *\n            (\u2191(algebraMap R A) W.a\u2082 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2081 + 3 * \u2191(algebraMap R A) C.r -\n              \u2191(algebraMap R A) C.s ^ 2),\n        a\u2083 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 3 *\n            (\u2191(algebraMap R A) W.a\u2083 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.t),\n        a\u2084 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 4 *\n            (\u2191(algebraMap R A) W.a\u2084 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2083 +\n                    2 * \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2082 -\n                  (\u2191(algebraMap R A) C.t + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.s) * \u2191(algebraMap R A) W.a\u2081 +\n                3 * \u2191(algebraMap R A) C.r ^ 2 -\n              2 * \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) C.t),\n        a\u2086 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 6 *\n            (\u2191(algebraMap R A) W.a\u2086 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2084 +\n                      \u2191(algebraMap R A) C.r ^ 2 * \u2191(algebraMap R A) W.a\u2082 +\n                    \u2191(algebraMap R A) C.r ^ 3 -\n                  \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2083 -\n                \u2191(algebraMap R A) C.t ^ 2 -\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2081) }.a\u2084 =\n    { a\u2081 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)),\n        a\u2082 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)),\n        a\u2083 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)),\n        a\u2084 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 4 *\n              (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)),\n        a\u2086 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) }.a\u2084\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\ncase a\u2086\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nC : VariableChange R\n\u22a2 { a\u2081 := \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 * (\u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.s),\n        a\u2082 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 2 *\n            (\u2191(algebraMap R A) W.a\u2082 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2081 + 3 * \u2191(algebraMap R A) C.r -\n              \u2191(algebraMap R A) C.s ^ 2),\n        a\u2083 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 3 *\n            (\u2191(algebraMap R A) W.a\u2083 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2081 + 2 * \u2191(algebraMap R A) C.t),\n        a\u2084 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 4 *\n            (\u2191(algebraMap R A) W.a\u2084 - \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) W.a\u2083 +\n                    2 * \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2082 -\n                  (\u2191(algebraMap R A) C.t + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.s) * \u2191(algebraMap R A) W.a\u2081 +\n                3 * \u2191(algebraMap R A) C.r ^ 2 -\n              2 * \u2191(algebraMap R A) C.s * \u2191(algebraMap R A) C.t),\n        a\u2086 :=\n          \u2191(\u2191(Units.map \u2191(algebraMap R A)) C.u)\u207b\u00b9 ^ 6 *\n            (\u2191(algebraMap R A) W.a\u2086 + \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) W.a\u2084 +\n                      \u2191(algebraMap R A) C.r ^ 2 * \u2191(algebraMap R A) W.a\u2082 +\n                    \u2191(algebraMap R A) C.r ^ 3 -\n                  \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2083 -\n                \u2191(algebraMap R A) C.t ^ 2 -\n              \u2191(algebraMap R A) C.r * \u2191(algebraMap R A) C.t * \u2191(algebraMap R A) W.a\u2081) }.a\u2086 =\n    { a\u2081 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 * (W.a\u2081 + 2 * C.s)),\n        a\u2082 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 2 * (W.a\u2082 - C.s * W.a\u2081 + 3 * C.r - C.s ^ 2)),\n        a\u2083 := \u2191(algebraMap R A) (\u2191C.u\u207b\u00b9 ^ 3 * (W.a\u2083 + C.r * W.a\u2081 + 2 * C.t)),\n        a\u2084 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 4 *\n              (W.a\u2084 - C.s * W.a\u2083 + 2 * C.r * W.a\u2082 - (C.t + C.r * C.s) * W.a\u2081 + 3 * C.r ^ 2 - 2 * C.s * C.t)),\n        a\u2086 :=\n          \u2191(algebraMap R A)\n            (\u2191C.u\u207b\u00b9 ^ 6 * (W.a\u2086 + C.r * W.a\u2084 + C.r ^ 2 * W.a\u2082 + C.r ^ 3 - C.t * W.a\u2083 - C.t ^ 2 - C.r * C.t * W.a\u2081)) }.a\u2086\n[PROOFSTEP]\nsimp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe, map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 Cubic.disc (twoTorsionPolynomial W) = 16 * \u0394 W\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, b\u2088, \u0394, twoTorsionPolynomial, Cubic.disc]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 (W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 2 * (2 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083)) ^ 2 - 4 * 4 * (2 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083)) ^ 3 -\n          4 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 3 * (W.a\u2083 ^ 2 + 4 * W.a\u2086) -\n        27 * 4 ^ 2 * (W.a\u2083 ^ 2 + 4 * W.a\u2086) ^ 2 +\n      18 * 4 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (2 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083)) * (W.a\u2083 ^ 2 + 4 * W.a\u2086) =\n    16 *\n      (-(W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 2 *\n              (W.a\u2081 ^ 2 * W.a\u2086 + 4 * W.a\u2082 * W.a\u2086 - W.a\u2081 * W.a\u2083 * W.a\u2084 + W.a\u2082 * W.a\u2083 ^ 2 - W.a\u2084 ^ 2) -\n            8 * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) ^ 3 -\n          27 * (W.a\u2083 ^ 2 + 4 * W.a\u2086) ^ 2 +\n        9 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (2 * W.a\u2084 + W.a\u2081 * W.a\u2083) * (W.a\u2083 ^ 2 + 4 * W.a\u2086))\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Invertible 2\n\u22a2 IsUnit (Cubic.disc (twoTorsionPolynomial W)) \u2194 IsUnit (\u0394 W)\n[PROOFSTEP]\nrw [twoTorsionPolynomial_disc, IsUnit.mul_iff, show (16 : R) = 2 ^ 4 by norm_num1]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Invertible 2\n\u22a2 16 = 2 ^ 4\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Invertible 2\n\u22a2 IsUnit (2 ^ 4) \u2227 IsUnit (\u0394 W) \u2194 IsUnit (\u0394 W)\n[PROOFSTEP]\nexact and_iff_right <| isUnit_of_invertible <| 2 ^ 4\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 WeierstrassCurve.polynomial W =\n    Cubic.toPoly\n      { a := 0, b := 1, c := Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 },\n        d := Cubic.toPoly { a := -1, b := -W.a\u2082, c := -W.a\u2084, d := -W.a\u2086 } }\n[PROOFSTEP]\nsimp only [WeierstrassCurve.polynomial, Cubic.toPoly]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) =\n    \u2191C 0 * Y ^ 3 + \u2191C 1 * Y ^ 2 + \u2191C (\u2191C 0 * Y ^ 3 + \u2191C 0 * Y ^ 2 + \u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y +\n      \u2191C (\u2191C (-1) * Y ^ 3 + \u2191C (-W.a\u2082) * Y ^ 2 + \u2191C (-W.a\u2084) * Y + \u2191C (-W.a\u2086))\n[PROOFSTEP]\nC_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 Y ^ 2 + (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * Y -\n      (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086)) =\n    0 * Y ^ 3 + 1 * Y ^ 2 + (0 * \u2191C Y ^ 3 + 0 * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * Y +\n      (-1 * \u2191C Y ^ 3 + -\u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + -\u2191C (\u2191C W.a\u2084) * \u2191C Y + -\u2191C (\u2191C W.a\u2086))\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Nontrivial R\n\u22a2 WeierstrassCurve.polynomial W \u2260 0\n[PROOFSTEP]\nrw [polynomial_eq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Nontrivial R\n\u22a2 Cubic.toPoly\n      { a := 0, b := 1, c := Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 },\n        d := Cubic.toPoly { a := -1, b := -W.a\u2082, c := -W.a\u2084, d := -W.a\u2086 } } \u2260\n    0\n[PROOFSTEP]\nexact Cubic.ne_zero_of_b_ne_zero one_ne_zero\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Nontrivial R\n\u22a2 degree (WeierstrassCurve.polynomial W) = 2\n[PROOFSTEP]\nrw [polynomial_eq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Nontrivial R\n\u22a2 degree\n      (Cubic.toPoly\n        { a := 0, b := 1, c := Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 },\n          d := Cubic.toPoly { a := -1, b := -W.a\u2082, c := -W.a\u2084, d := -W.a\u2086 } }) =\n    2\n[PROOFSTEP]\nexact Cubic.degree_of_b_ne_zero' one_ne_zero\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Nontrivial R\n\u22a2 natDegree (WeierstrassCurve.polynomial W) = 2\n[PROOFSTEP]\nrw [polynomial_eq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : Nontrivial R\n\u22a2 natDegree\n      (Cubic.toPoly\n        { a := 0, b := 1, c := Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 },\n          d := Cubic.toPoly { a := -1, b := -W.a\u2082, c := -W.a\u2084, d := -W.a\u2086 } }) =\n    2\n[PROOFSTEP]\nexact Cubic.natDegree_of_b_ne_zero' one_ne_zero\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 Monic (WeierstrassCurve.polynomial W)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u271d : Nontrivial R\n\u22a2 Monic (WeierstrassCurve.polynomial W)\n[PROOFSTEP]\nsimpa only [polynomial_eq] using Cubic.monic_of_b_eq_one'\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\n\u22a2 Irreducible (WeierstrassCurve.polynomial W)\n[PROOFSTEP]\nby_contra h\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh : \u00acIrreducible (WeierstrassCurve.polynomial W)\n\u22a2 False\n[PROOFSTEP]\nrcases(W.monic_polynomial.not_irreducible_iff_exists_add_mul_eq_coeff W.natDegree_polynomial).mp h with \u27e8f, g, h0, h1\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : coeff (WeierstrassCurve.polynomial W) 0 = f * g\nh1 : coeff (WeierstrassCurve.polynomial W) 1 = f + g\n\u22a2 False\n[PROOFSTEP]\nsimp only [polynomial_eq, Cubic.coeff_eq_c, Cubic.coeff_eq_d] at h0 h1 \n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : Cubic.toPoly { a := -1, b := -W.a\u2082, c := -W.a\u2084, d := -W.a\u2086 } = f * g\nh1 : Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 } = f + g\n\u22a2 False\n[PROOFSTEP]\napply_fun degree at h0 h1 \n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : degree (Cubic.toPoly { a := -1, b := -W.a\u2082, c := -W.a\u2084, d := -W.a\u2086 }) = degree (f * g)\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\n\u22a2 False\n[PROOFSTEP]\nrw [Cubic.degree_of_a_ne_zero' <| neg_ne_zero.mpr <| one_ne_zero' R, degree_mul] at h0 \n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\n\u22a2 False\n[PROOFSTEP]\napply (h1.symm.le.trans Cubic.degree_of_b_eq_zero').not_lt\n[GOAL]\ncase intro.intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\nrcases Nat.WithBot.add_eq_three_iff.mp h0.symm with h | h | h | h\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 0 \u2227 degree g = 3\n\u22a2 1 < degree (f + g)\ncase intro.intro.intro.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 1 \u2227 degree g = 2\n\u22a2 1 < degree (f + g)\ncase intro.intro.intro.inr.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 2 \u2227 degree g = 1\n\u22a2 1 < degree (f + g)\ncase intro.intro.intro.inr.inr.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 3 \u2227 degree g = 0\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\niterate 2 rw [degree_add_eq_right_of_degree_lt] <;> simp only [h]\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 0 \u2227 degree g = 3\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\nrw [degree_add_eq_right_of_degree_lt]\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 0 \u2227 degree g = 3\n\u22a2 1 < degree g\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 0 \u2227 degree g = 3\n\u22a2 degree f < degree g\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 1 \u2227 degree g = 2\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\nrw [degree_add_eq_right_of_degree_lt]\n[GOAL]\ncase intro.intro.intro.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 1 \u2227 degree g = 2\n\u22a2 1 < degree g\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 1 \u2227 degree g = 2\n\u22a2 degree f < degree g\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 2 \u2227 degree g = 1\n\u22a2 1 < degree (f + g)\ncase intro.intro.intro.inr.inr.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 3 \u2227 degree g = 0\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\niterate 2 rw [degree_add_eq_left_of_degree_lt] <;> simp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 2 \u2227 degree g = 1\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\nrw [degree_add_eq_left_of_degree_lt]\n[GOAL]\ncase intro.intro.intro.inr.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 2 \u2227 degree g = 1\n\u22a2 1 < degree f\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inr.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 2 \u2227 degree g = 1\n\u22a2 degree g < degree f\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inr.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 3 \u2227 degree g = 0\n\u22a2 1 < degree (f + g)\n[PROOFSTEP]\nrw [degree_add_eq_left_of_degree_lt]\n[GOAL]\ncase intro.intro.intro.inr.inr.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 3 \u2227 degree g = 0\n\u22a2 1 < degree f\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase intro.intro.intro.inr.inr.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsDomain R\nh\u271d : \u00acIrreducible (WeierstrassCurve.polynomial W)\nf g : R[X]\nh0 : 3 = degree f + degree g\nh1 : degree (Cubic.toPoly { a := 0, b := 0, c := W.a\u2081, d := W.a\u2083 }) = degree (f + g)\nh : degree f = 3 \u2227 degree g = 0\n\u22a2 degree g < degree f\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 eval x (eval (\u2191C y) (WeierstrassCurve.polynomial W)) =\n    y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)\n[PROOFSTEP]\nsimp only [WeierstrassCurve.polynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 eval x (eval (\u2191C y) (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))) =\n    y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 y ^ 2 + (W.a\u2081 * x + W.a\u2083) * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086) =\n    y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)\n[PROOFSTEP]\nrw [add_mul, \u2190 add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 eval 0 (eval 0 (WeierstrassCurve.polynomial W)) = -W.a\u2086\n[PROOFSTEP]\nsimp only [\u2190 C_0, eval_polynomial, zero_add, zero_sub, mul_zero, zero_pow <| Nat.zero_lt_succ _]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.equation W x y \u2194 y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086) = 0\n[PROOFSTEP]\nrw [WeierstrassCurve.equation, eval_polynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.equation W x y \u2194 y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n[PROOFSTEP]\nrw [equation_iff', sub_eq_zero]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 WeierstrassCurve.equation W 0 0 \u2194 W.a\u2086 = 0\n[PROOFSTEP]\nrw [WeierstrassCurve.equation, C_0, eval_polynomial_zero, neg_eq_zero]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.equation W x y \u2194 WeierstrassCurve.equation (variableChange W { u := 1, r := x, s := 0, t := y }) 0 0\n[PROOFSTEP]\nrw [equation_iff', \u2190 neg_eq_zero, equation_zero, variableChange_a\u2086, inv_one, Units.val_one]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 -(y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)) = 0 \u2194\n    1 ^ 6 *\n        (W.a\u2086 + { u := 1, r := x, s := 0, t := y }.r * W.a\u2084 + { u := 1, r := x, s := 0, t := y }.r ^ 2 * W.a\u2082 +\n                { u := 1, r := x, s := 0, t := y }.r ^ 3 -\n              { u := 1, r := x, s := 0, t := y }.t * W.a\u2083 -\n            { u := 1, r := x, s := 0, t := y }.t ^ 2 -\n          { u := 1, r := x, s := 0, t := y }.r * { u := 1, r := x, s := 0, t := y }.t * W.a\u2081) =\n      0\n[PROOFSTEP]\ncongr! 1\n[GOAL]\ncase a.h.e'_2\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 -(y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y - (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)) =\n    1 ^ 6 *\n      (W.a\u2086 + { u := 1, r := x, s := 0, t := y }.r * W.a\u2084 + { u := 1, r := x, s := 0, t := y }.r ^ 2 * W.a\u2082 +\n              { u := 1, r := x, s := 0, t := y }.r ^ 3 -\n            { u := 1, r := x, s := 0, t := y }.t * W.a\u2083 -\n          { u := 1, r := x, s := 0, t := y }.t ^ 2 -\n        { u := 1, r := x, s := 0, t := y }.r * { u := 1, r := x, s := 0, t := y }.t * W.a\u2081)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\n\u22a2 WeierstrassCurve.equation W x y \u2194\n    WeierstrassCurve.equation (baseChange W A) (\u2191(algebraMap R A) x) (\u2191(algebraMap R A) y)\n[PROOFSTEP]\nsimp only [equation_iff]\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\n\u22a2 y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086 \u2194\n    \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n        (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n      \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n          (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n        (baseChange W A).a\u2086\n[PROOFSTEP]\nexact\n  \u27e8fun h => by convert congr_arg (algebraMap R A) h <;> map_simp <;> rfl, fun h => by\n    apply NoZeroSMulDivisors.algebraMap_injective R A; map_simp; exact h\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n\u22a2 \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n        (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n      (baseChange W A).a\u2086\n[PROOFSTEP]\nconvert congr_arg (algebraMap R A) h\n[GOAL]\ncase h.e'_2\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n\u22a2 \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) (y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n\u22a2 \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 + (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n      (baseChange W A).a\u2086 =\n    \u2191(algebraMap R A) (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase h.e'_2\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n\u22a2 \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) y ^ 2 + \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      \u2191(algebraMap R A) W.a\u2083 * \u2191(algebraMap R A) y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n\u22a2 \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 + (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n      (baseChange W A).a\u2086 =\n    \u2191(algebraMap R A) x ^ 3 + \u2191(algebraMap R A) W.a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n        \u2191(algebraMap R A) W.a\u2084 * \u2191(algebraMap R A) x +\n      \u2191(algebraMap R A) W.a\u2086\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n        (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n      (baseChange W A).a\u2086\n\u22a2 y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y = x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective R A\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n        (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n      (baseChange W A).a\u2086\n\u22a2 \u2191(algebraMap R A) (y ^ 2 + W.a\u2081 * x * y + W.a\u2083 * y) = \u2191(algebraMap R A) (x ^ 3 + W.a\u2082 * x ^ 2 + W.a\u2084 * x + W.a\u2086)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase a\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  \u2191(algebraMap R A) y ^ 2 + (baseChange W A).a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      (baseChange W A).a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) x ^ 3 + (baseChange W A).a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n        (baseChange W A).a\u2084 * \u2191(algebraMap R A) x +\n      (baseChange W A).a\u2086\n\u22a2 \u2191(algebraMap R A) y ^ 2 + \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) x * \u2191(algebraMap R A) y +\n      \u2191(algebraMap R A) W.a\u2083 * \u2191(algebraMap R A) y =\n    \u2191(algebraMap R A) x ^ 3 + \u2191(algebraMap R A) W.a\u2082 * \u2191(algebraMap R A) x ^ 2 +\n        \u2191(algebraMap R A) W.a\u2084 * \u2191(algebraMap R A) x +\n      \u2191(algebraMap R A) W.a\u2086\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial B\ninst\u271d : NoZeroSMulDivisors A B\nx y : A\n\u22a2 WeierstrassCurve.equation (baseChange W A) x y \u2194\n    WeierstrassCurve.equation (baseChange W B) (\u2191(algebraMap A B) x) (\u2191(algebraMap A B) y)\n[PROOFSTEP]\nrw [equation_iff_baseChange (W.baseChange A) B, baseChange_baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 eval x (eval (\u2191C y) (WeierstrassCurve.polynomialX W)) = W.a\u2081 * y - (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)\n[PROOFSTEP]\nsimp only [WeierstrassCurve.polynomialX]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 eval x (eval (\u2191C y) (\u2191C (\u2191C W.a\u2081) * Y - \u2191C (\u2191C 3 * Y ^ 2 + \u2191C (2 * W.a\u2082) * Y + \u2191C W.a\u2084))) =\n    W.a\u2081 * y - (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 eval 0 (eval 0 (WeierstrassCurve.polynomialX W)) = -W.a\u2084\n[PROOFSTEP]\nsimp only [\u2190 C_0, eval_polynomialX, zero_add, zero_sub, mul_zero, zero_pow zero_lt_two]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 eval x (eval (\u2191C y) (WeierstrassCurve.polynomialY W)) = 2 * y + W.a\u2081 * x + W.a\u2083\n[PROOFSTEP]\nsimp only [WeierstrassCurve.polynomialY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 eval x (eval (\u2191C y) (\u2191C (\u2191C 2) * Y + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) = 2 * y + W.a\u2081 * x + W.a\u2083\n[PROOFSTEP]\neval_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 2 * y + (W.a\u2081 * x + W.a\u2083) = 2 * y + W.a\u2081 * x + W.a\u2083\n[PROOFSTEP]\nrw [\u2190 add_assoc]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 eval 0 (eval 0 (WeierstrassCurve.polynomialY W)) = W.a\u2083\n[PROOFSTEP]\nsimp only [\u2190 C_0, eval_polynomialY, zero_add, mul_zero]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.nonsingular W x y \u2194\n    WeierstrassCurve.equation W x y \u2227 (W.a\u2081 * y - (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084) \u2260 0 \u2228 2 * y + W.a\u2081 * x + W.a\u2083 \u2260 0)\n[PROOFSTEP]\nrw [WeierstrassCurve.nonsingular, equation_iff', eval_polynomialX, eval_polynomialY]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.nonsingular W x y \u2194\n    WeierstrassCurve.equation W x y \u2227 (W.a\u2081 * y \u2260 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084 \u2228 y \u2260 -y - W.a\u2081 * x - W.a\u2083)\n[PROOFSTEP]\nrw [nonsingular_iff', sub_ne_zero, \u2190 @sub_ne_zero _ _ y]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.equation W x y \u2227 (W.a\u2081 * y \u2260 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084 \u2228 2 * y + W.a\u2081 * x + W.a\u2083 \u2260 0) \u2194\n    WeierstrassCurve.equation W x y \u2227 (W.a\u2081 * y \u2260 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084 \u2228 y - (-y - W.a\u2081 * x - W.a\u2083) \u2260 0)\n[PROOFSTEP]\ncongr! 3\n[GOAL]\ncase a.h.e'_2.h.e'_2.h.e'_2\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 2 * y + W.a\u2081 * x + W.a\u2083 = y - (-y - W.a\u2081 * x - W.a\u2083)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\n\u22a2 WeierstrassCurve.nonsingular W 0 0 \u2194 W.a\u2086 = 0 \u2227 (W.a\u2083 \u2260 0 \u2228 W.a\u2084 \u2260 0)\n[PROOFSTEP]\nrw [WeierstrassCurve.nonsingular, equation_zero, C_0, eval_polynomialX_zero, neg_ne_zero, eval_polynomialY_zero,\n  or_comm]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 WeierstrassCurve.nonsingular W x y \u2194\n    WeierstrassCurve.nonsingular (variableChange W { u := 1, r := x, s := 0, t := y }) 0 0\n[PROOFSTEP]\nrw [nonsingular_iff', equation_iff_variableChange, equation_zero, \u2190 neg_ne_zero, or_comm, nonsingular_zero,\n  variableChange_a\u2083, variableChange_a\u2084, inv_one, Units.val_one]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 (variableChange W { u := 1, r := x, s := 0, t := y }).a\u2086 = 0 \u2227\n      (2 * y + W.a\u2081 * x + W.a\u2083 \u2260 0 \u2228 -(W.a\u2081 * y - (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)) \u2260 0) \u2194\n    (variableChange W { u := 1, r := x, s := 0, t := y }).a\u2086 = 0 \u2227\n      (1 ^ 3 * (W.a\u2083 + { u := 1, r := x, s := 0, t := y }.r * W.a\u2081 + 2 * { u := 1, r := x, s := 0, t := y }.t) \u2260 0 \u2228\n        1 ^ 4 *\n            (W.a\u2084 - { u := 1, r := x, s := 0, t := y }.s * W.a\u2083 + 2 * { u := 1, r := x, s := 0, t := y }.r * W.a\u2082 -\n                  ({ u := 1, r := x, s := 0, t := y }.t +\n                      { u := 1, r := x, s := 0, t := y }.r * { u := 1, r := x, s := 0, t := y }.s) *\n                    W.a\u2081 +\n                3 * { u := 1, r := x, s := 0, t := y }.r ^ 2 -\n              2 * { u := 1, r := x, s := 0, t := y }.s * { u := 1, r := x, s := 0, t := y }.t) \u2260\n          0)\n[PROOFSTEP]\nsimp only [variableChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 \u21911\u207b\u00b9 ^ 6 * (W.a\u2086 + x * W.a\u2084 + x ^ 2 * W.a\u2082 + x ^ 3 - y * W.a\u2083 - y ^ 2 - x * y * W.a\u2081) = 0 \u2227\n      (2 * y + W.a\u2081 * x + W.a\u2083 \u2260 0 \u2228 -(W.a\u2081 * y - (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)) \u2260 0) \u2194\n    \u21911\u207b\u00b9 ^ 6 * (W.a\u2086 + x * W.a\u2084 + x ^ 2 * W.a\u2082 + x ^ 3 - y * W.a\u2083 - y ^ 2 - x * y * W.a\u2081) = 0 \u2227\n      (1 ^ 3 * (W.a\u2083 + x * W.a\u2081 + 2 * y) \u2260 0 \u2228\n        1 ^ 4 * (W.a\u2084 - 0 * W.a\u2083 + 2 * x * W.a\u2082 - (y + x * 0) * W.a\u2081 + 3 * x ^ 2 - 2 * 0 * y) \u2260 0)\n[PROOFSTEP]\ncongr! 3\n[GOAL]\ncase a.h.e'_2.h.e'_1.h.e'_2\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 2 * y + W.a\u2081 * x + W.a\u2083 = 1 ^ 3 * (W.a\u2083 + x * W.a\u2081 + 2 * y)\n[PROOFSTEP]\nring1\n[GOAL]\ncase a.h.e'_2.h.e'_2.h.e'_2\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\n\u22a2 -(W.a\u2081 * y - (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)) =\n    1 ^ 4 * (W.a\u2084 - 0 * W.a\u2083 + 2 * x * W.a\u2082 - (y + x * 0) * W.a\u2081 + 3 * x ^ 2 - 2 * 0 * y)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\n\u22a2 WeierstrassCurve.nonsingular W x y \u2194\n    WeierstrassCurve.nonsingular (baseChange W A) (\u2191(algebraMap R A) x) (\u2191(algebraMap R A) y)\n[PROOFSTEP]\nrw [nonsingular_iff, nonsingular_iff, and_congr <| W.equation_iff_baseChange A x y]\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\n\u22a2 W.a\u2081 * y \u2260 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084 \u2228 y \u2260 -y - W.a\u2081 * x - W.a\u2083 \u2194\n    (baseChange W A).a\u2081 * \u2191(algebraMap R A) y \u2260\n        3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084 \u2228\n      \u2191(algebraMap R A) y \u2260 -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n[PROOFSTEP]\nrefine\n  \u27e8Or.imp (not_imp_not.mpr fun h => ?_) (not_imp_not.mpr fun h => ?_),\n    Or.imp (not_imp_not.mpr fun h => ?_) (not_imp_not.mpr fun h => ?_)\u27e9\n[GOAL]\ncase refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\n\u22a2 W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\ncase refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n\u22a2 y = -y - W.a\u2081 * x - W.a\u2083\ncase refine_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\ncase refine_4\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y = -y - W.a\u2081 * x - W.a\u2083\n\u22a2 \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n[PROOFSTEP]\nany_goals apply NoZeroSMulDivisors.algebraMap_injective R A; map_simp; exact h\n[GOAL]\ncase refine_1\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\n\u22a2 W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective R A\n[GOAL]\ncase refine_1.a\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\n\u22a2 \u2191(algebraMap R A) (W.a\u2081 * y) = \u2191(algebraMap R A) (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase refine_1.a\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh :\n  (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\n\u22a2 \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * \u2191(algebraMap R A) W.a\u2082 * \u2191(algebraMap R A) x + \u2191(algebraMap R A) W.a\u2084\n[PROOFSTEP]\nexact h\n[GOAL]\ncase refine_2\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n\u22a2 y = -y - W.a\u2081 * x - W.a\u2083\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective R A\n[GOAL]\ncase refine_2.a\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n\u22a2 \u2191(algebraMap R A) y = \u2191(algebraMap R A) (-y - W.a\u2081 * x - W.a\u2083)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase refine_2.a\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n\u22a2 \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) x - \u2191(algebraMap R A) W.a\u2083\n[PROOFSTEP]\nexact h\n[GOAL]\ncase refine_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective R A\n[GOAL]\ncase refine_4\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y = -y - W.a\u2081 * x - W.a\u2083\n\u22a2 \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective R A\n[GOAL]\ncase refine_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\ncase refine_4\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y = -y - W.a\u2081 * x - W.a\u2083\n\u22a2 \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n[PROOFSTEP]\nany_goals convert congr_arg (algebraMap R A) h <;> map_simp <;> rfl\n[GOAL]\ncase refine_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 (baseChange W A).a\u2081 * \u2191(algebraMap R A) y =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084\n[PROOFSTEP]\nconvert congr_arg (algebraMap R A) h\n[GOAL]\ncase h.e'_2\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 (baseChange W A).a\u2081 * \u2191(algebraMap R A) y = \u2191(algebraMap R A) (W.a\u2081 * y)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084 =\n    \u2191(algebraMap R A) (3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase h.e'_2\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 (baseChange W A).a\u2081 * \u2191(algebraMap R A) y = \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : W.a\u2081 * y = 3 * x ^ 2 + 2 * W.a\u2082 * x + W.a\u2084\n\u22a2 3 * \u2191(algebraMap R A) x ^ 2 + 2 * (baseChange W A).a\u2082 * \u2191(algebraMap R A) x + (baseChange W A).a\u2084 =\n    3 * \u2191(algebraMap R A) x ^ 2 + 2 * \u2191(algebraMap R A) W.a\u2082 * \u2191(algebraMap R A) x + \u2191(algebraMap R A) W.a\u2084\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine_4\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y = -y - W.a\u2081 * x - W.a\u2083\n\u22a2 \u2191(algebraMap R A) y = -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083\n[PROOFSTEP]\nconvert congr_arg (algebraMap R A) h\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y = -y - W.a\u2081 * x - W.a\u2083\n\u22a2 -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083 =\n    \u2191(algebraMap R A) (-y - W.a\u2081 * x - W.a\u2083)\n[PROOFSTEP]\nmap_simp\n[GOAL]\ncase h.e'_3\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial A\ninst\u271d : NoZeroSMulDivisors R A\nx y : R\nh : y = -y - W.a\u2081 * x - W.a\u2083\n\u22a2 -\u2191(algebraMap R A) y - (baseChange W A).a\u2081 * \u2191(algebraMap R A) x - (baseChange W A).a\u2083 =\n    -\u2191(algebraMap R A) y - \u2191(algebraMap R A) W.a\u2081 * \u2191(algebraMap R A) x - \u2191(algebraMap R A) W.a\u2083\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : Nontrivial B\ninst\u271d : NoZeroSMulDivisors A B\nx y : A\n\u22a2 WeierstrassCurve.nonsingular (baseChange W A) x y \u2194\n    WeierstrassCurve.nonsingular (baseChange W B) (\u2191(algebraMap A B) x) (\u2191(algebraMap A B) y)\n[PROOFSTEP]\nrw [nonsingular_iff_baseChange (W.baseChange A) B, baseChange_baseChange]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : WeierstrassCurve.equation W 0 0\nh\u0394 : \u0394 W \u2260 0\n\u22a2 WeierstrassCurve.nonsingular W 0 0\n[PROOFSTEP]\nsimp only [equation_zero, nonsingular_zero] at *\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh\u0394 : \u0394 W \u2260 0\nh : W.a\u2086 = 0\n\u22a2 W.a\u2086 = 0 \u2227 (W.a\u2083 \u2260 0 \u2228 W.a\u2084 \u2260 0)\n[PROOFSTEP]\ncontrapose! h\u0394\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : W.a\u2086 = 0\nh\u0394 : W.a\u2086 = 0 \u2192 W.a\u2083 = 0 \u2227 W.a\u2084 = 0\n\u22a2 \u0394 W = 0\n[PROOFSTEP]\nsimp only [b\u2082, b\u2084, b\u2086, b\u2088, \u0394, h, h\u0394]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nh : W.a\u2086 = 0\nh\u0394 : W.a\u2086 = 0 \u2192 W.a\u2083 = 0 \u2227 W.a\u2084 = 0\n\u22a2 -(W.a\u2081 ^ 2 + 4 * W.a\u2082) ^ 2 * (W.a\u2081 ^ 2 * 0 + 4 * W.a\u2082 * 0 - W.a\u2081 * 0 * 0 + W.a\u2082 * 0 ^ 2 - 0 ^ 2) -\n          8 * (2 * 0 + W.a\u2081 * 0) ^ 3 -\n        27 * (0 ^ 2 + 4 * 0) ^ 2 +\n      9 * (W.a\u2081 ^ 2 + 4 * W.a\u2082) * (2 * 0 + W.a\u2081 * 0) * (0 ^ 2 + 4 * 0) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx y : R\nh : WeierstrassCurve.equation W x y\nh\u0394 : \u0394 W \u2260 0\n\u22a2 \u0394 (variableChange W { u := 1, r := x, s := 0, t := y }) \u2260 0\n[PROOFSTEP]\nrwa [variableChange_\u0394, inv_one, Units.val_one, one_pow, one_mul]\n[GOAL]\nR : Type u\ninst\u271d\u2078 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nB : Type w\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsScalarTower R A B\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\n\u22a2 IsPrime (span {WeierstrassCurve.polynomial W})\n[PROOFSTEP]\nsimpa only [span_singleton_prime W.polynomial_ne_zero, \u2190 GCDMonoid.irreducible_iff_prime] using W.irreducible_polynomial\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\n\u22a2 IsDomain (CoordinateRing W)\n[PROOFSTEP]\nclassical exact instIsDomainCoordinateRing W\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW\u271d : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nF : Type u\ninst\u271d : Field F\nW : WeierstrassCurve F\n\u22a2 IsDomain (CoordinateRing W)\n[PROOFSTEP]\nexact instIsDomainCoordinateRing W\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : Nontrivial R\n\u22a2 natDegree (\u2191C (Y - \u2191C x)) < natDegree (WeierstrassCurve.polynomial W)\n[PROOFSTEP]\nrw [natDegree_polynomial, natDegree_C]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : Nontrivial R\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : Nontrivial R\n\u22a2 natDegree (Y - \u2191C y) < natDegree (WeierstrassCurve.polynomial W)\n[PROOFSTEP]\nrw [natDegree_polynomial, natDegree_X_sub_C]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : Nontrivial R\n\u22a2 1 < 2\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u271d : R\ny\u271d : R[X]\nx : R\ny : R[X]\nh : eval x (eval y (WeierstrassCurve.polynomial W)) = 0\n\u22a2 XYIdeal W x y = Ideal.map (Quotient.mk\u2090 R (span {WeierstrassCurve.polynomial W})) (span {\u2191C (Y - \u2191C x), Y - \u2191C y})\n[PROOFSTEP]\nsimp only [XYIdeal, XClass, YClass, \u2190 Set.image_pair, \u2190 map_span]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u271d : R\ny\u271d : R[X]\nx : R\ny : R[X]\nh : eval x (eval y (WeierstrassCurve.polynomial W)) = 0\n\u22a2 Ideal.map (mk W) (span {\u2191C (Y - \u2191C x), Y - \u2191C y}) =\n    Ideal.map (Quotient.mk\u2090 R (span {WeierstrassCurve.polynomial W})) (span {\u2191C (Y - \u2191C x), Y - \u2191C y})\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 Basis (Fin 2) R[X] (CoordinateRing W)\n[PROOFSTEP]\nclassical exact\n  (subsingleton_or_nontrivial R).by_cases (fun _ => default) fun _ =>\n    (AdjoinRoot.powerBasis' W.monic_polynomial).basis.reindex <| finCongr W.natDegree_polynomial\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 Basis (Fin 2) R[X] (CoordinateRing W)\n[PROOFSTEP]\nexact\n  (subsingleton_or_nontrivial R).by_cases (fun _ => default) fun _ =>\n    (AdjoinRoot.powerBasis' W.monic_polynomial).basis.reindex <| finCongr W.natDegree_polynomial\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\nn : Fin 2\n\u22a2 \u2191(CoordinateRing.basis W) n = (AdjoinRoot.powerBasis' (_ : Monic (WeierstrassCurve.polynomial W))).gen ^ \u2191n\n[PROOFSTEP]\nclassical\nnontriviality R\nrw [CoordinateRing.basis, Or.by_cases, dif_neg <| not_subsingleton R, Basis.reindex_apply, PowerBasis.basis_eq_pow]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\nn : Fin 2\n\u22a2 \u2191(CoordinateRing.basis W) n = (AdjoinRoot.powerBasis' (_ : Monic (WeierstrassCurve.polynomial W))).gen ^ \u2191n\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\nn : Fin 2\n\u271d : Nontrivial R\n\u22a2 \u2191(CoordinateRing.basis W) n = (AdjoinRoot.powerBasis' (_ : Monic (WeierstrassCurve.polynomial W))).gen ^ \u2191n\n[PROOFSTEP]\nrw [CoordinateRing.basis, Or.by_cases, dif_neg <| not_subsingleton R, Basis.reindex_apply, PowerBasis.basis_eq_pow]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\nn : Fin 2\n\u271d : Nontrivial R\n\u22a2 (AdjoinRoot.powerBasis' (_ : Monic (WeierstrassCurve.polynomial W))).gen ^\n      \u2191(\u2191(finCongr (_ : natDegree (WeierstrassCurve.polynomial W) = 2)).symm n) =\n    (AdjoinRoot.powerBasis' (_ : Monic (WeierstrassCurve.polynomial W))).gen ^ \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 \u2191(CoordinateRing.basis W) 0 = 1\n[PROOFSTEP]\nsimpa only [basis_apply] using pow_zero _\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 \u2191(CoordinateRing.basis W) 1 = \u2191(mk W) Y\n[PROOFSTEP]\nsimpa only [basis_apply] using pow_one _\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 \u2191(CoordinateRing.basis W) = ![1, \u2191(mk W) Y]\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\nn : Fin 2\n\u22a2 \u2191(CoordinateRing.basis W) n = Matrix.vecCons 1 ![\u2191(mk W) Y] n\n[PROOFSTEP]\nfin_cases n\n[GOAL]\ncase h.head\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 \u2191(CoordinateRing.basis W) { val := 0, isLt := (_ : 0 < 2) } =\n    Matrix.vecCons 1 ![\u2191(mk W) Y] { val := 0, isLt := (_ : 0 < 2) }\ncase h.tail.head\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny : R[X]\n\u22a2 \u2191(CoordinateRing.basis W) { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    Matrix.vecCons 1 ![\u2191(mk W) Y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nexacts [basis_zero W, basis_one W]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\nhpq : p \u2022 1 + q \u2022 \u2191(mk W) Y = 0\n\u22a2 p = 0 \u2227 q = 0\n[PROOFSTEP]\nhave h := Fintype.linearIndependent_iff.mp (CoordinateRing.basis W).linearIndependent ![p, q]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\nhpq : p \u2022 1 + q \u2022 \u2191(mk W) Y = 0\nh :\n  (Finset.sum Finset.univ fun i => Matrix.vecCons p ![q] i \u2022 \u2191(CoordinateRing.basis W) i) = 0 \u2192\n    \u2200 (i : Fin 2), Matrix.vecCons p ![q] i = 0\n\u22a2 p = 0 \u2227 q = 0\n[PROOFSTEP]\nerw [Fin.sum_univ_succ, basis_zero, Fin.sum_univ_one, basis_one] at h \n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\nhpq : p \u2022 1 + q \u2022 \u2191(mk W) Y = 0\nh :\n  Matrix.vecCons p ![q] 0 \u2022 1 + Matrix.vecCons p ![q] (Fin.succ 0) \u2022 \u2191(mk W) Y = 0 \u2192\n    \u2200 (i : Fin 2), Matrix.vecCons p ![q] i = 0\n\u22a2 p = 0 \u2227 q = 0\n[PROOFSTEP]\nexact \u27e8h hpq 0, h hpq 1\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u271d : R\ny : R[X]\nx : CoordinateRing W\n\u22a2 \u2203 p q, p \u2022 1 + q \u2022 \u2191(mk W) Y = x\n[PROOFSTEP]\nhave h := (CoordinateRing.basis W).sum_equivFun x\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u271d : R\ny : R[X]\nx : CoordinateRing W\nh : (Finset.sum Finset.univ fun i => \u2191(Basis.equivFun (CoordinateRing.basis W)) x i \u2022 \u2191(CoordinateRing.basis W) i) = x\n\u22a2 \u2203 p q, p \u2022 1 + q \u2022 \u2191(mk W) Y = x\n[PROOFSTEP]\nerw [Fin.sum_univ_succ, Fin.sum_univ_one, basis_zero, basis_one] at h \n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx\u271d : R\ny : R[X]\nx : CoordinateRing W\nh :\n  \u2191(Basis.equivFun (CoordinateRing.basis W)) x 0 \u2022 1 +\n      \u2191(Basis.equivFun (CoordinateRing.basis W)) x (Fin.succ 0) \u2022 \u2191(mk W) Y =\n    x\n\u22a2 \u2203 p q, p \u2022 1 + q \u2022 \u2191(mk W) Y = x\n[PROOFSTEP]\nexact \u27e8_, _, h\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 (p \u2022 1 + q \u2022 \u2191(mk W) Y) * \u2191(mk W) (\u2191C y) = (p * y) \u2022 1 + (q * y) \u2022 \u2191(mk W) Y\n[PROOFSTEP]\nsimp only [smul, _root_.map_mul]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 (\u2191(mk W) (\u2191C p) * 1 + \u2191(mk W) (\u2191C q) * \u2191(mk W) Y) * \u2191(mk W) (\u2191C y) =\n    \u2191(mk W) (\u2191C p) * \u2191(mk W) (\u2191C y) * 1 + \u2191(mk W) (\u2191C q) * \u2191(mk W) (\u2191C y) * \u2191(mk W) Y\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 (p \u2022 1 + q \u2022 \u2191(mk W) Y) * \u2191(mk W) Y =\n    (q * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) \u2022 1 + (p - q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2022 \u2191(mk W) Y\n[PROOFSTEP]\nhave Y_sq : mk W Y ^ 2 = mk W (C (X ^ 3 + C W.a\u2082 * X ^ 2 + C W.a\u2084 * X + C W.a\u2086) - C (C W.a\u2081 * X + C W.a\u2083) * Y) := by\n  exact AdjoinRoot.mk_eq_mk.mpr \u27e81, by rw [WeierstrassCurve.polynomial]; ring1\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 \u2191(mk W) Y ^ 2 = \u2191(mk W) (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y)\n[PROOFSTEP]\nexact AdjoinRoot.mk_eq_mk.mpr \u27e81, by rw [WeierstrassCurve.polynomial]; ring1\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 (fun x => x ^ 2) Y - (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y) =\n    WeierstrassCurve.polynomial W * 1\n[PROOFSTEP]\nrw [WeierstrassCurve.polynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 (fun x => x ^ 2) Y - (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y) =\n    (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) * 1\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\nY_sq : \u2191(mk W) Y ^ 2 = \u2191(mk W) (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y)\n\u22a2 (p \u2022 1 + q \u2022 \u2191(mk W) Y) * \u2191(mk W) Y =\n    (q * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) \u2022 1 + (p - q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2022 \u2191(mk W) Y\n[PROOFSTEP]\nsimp_rw [smul, add_mul, mul_assoc, \u2190 sq, Y_sq, C_sub, map_sub, C_mul, _root_.map_mul]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\nY_sq : \u2191(mk W) Y ^ 2 = \u2191(mk W) (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y)\n\u22a2 \u2191(mk W) (\u2191C p) * (1 * \u2191(mk W) Y) +\n      \u2191(mk W) (\u2191C q) *\n        (\u2191(mk W) (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) -\n          \u2191(mk W) (\u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) * \u2191(mk W) Y) =\n    \u2191(mk W) (\u2191C q) * \u2191(mk W) (\u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) * 1 +\n      (\u2191(mk W) (\u2191C p) - \u2191(mk W) (\u2191C q) * \u2191(mk W) (\u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) * \u2191(mk W) Y\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 \u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y) =\n    p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)\n[PROOFSTEP]\nsimp_rw [Algebra.norm_eq_matrix_det <| CoordinateRing.basis W, Matrix.det_fin_two, Algebra.leftMulMatrix_eq_repr_mul,\n  basis_zero, mul_one, basis_one, smul_basis_mul_Y, map_add, Finsupp.add_apply, map_smul, Finsupp.smul_apply, \u2190\n  basis_zero, \u2190 basis_one, Basis.repr_self_apply, if_pos, if_neg, smul_eq_mul]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 (p * 1 + q * 0) *\n        (q * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) * 0 + (p - q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) * 1) -\n      (q * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086) * 1 + (p - q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) * 0) *\n        (p * 0 + q * 1) =\n    p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 \u2191C (\u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y)) -\n      (\u2191C p + \u2191C q * Y) * (\u2191C p + \u2191C q * (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) =\n    WeierstrassCurve.polynomial W * \u2191C q ^ 2\n[PROOFSTEP]\nsimp only [norm_smul_basis, WeierstrassCurve.polynomial]\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 \u2191C (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) -\n      (\u2191C p + \u2191C q * Y) * (\u2191C p + \u2191C q * (-Y - \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083))) =\n    (Y ^ 2 + \u2191C (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) * Y - \u2191C (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) * \u2191C q ^ 2\n[PROOFSTEP]\nC_simp\n[GOAL]\nR : Type u\ninst\u271d\u2076 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\nB : Type w\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : R\ny p q : R[X]\n\u22a2 \u2191C p ^ 2 - \u2191C p * \u2191C q * (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) -\n        \u2191C q ^ 2 * (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086)) -\n      (\u2191C p + \u2191C q * Y) * (\u2191C p + \u2191C q * (-Y - (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)))) =\n    (Y ^ 2 + (\u2191C (\u2191C W.a\u2081) * \u2191C Y + \u2191C (\u2191C W.a\u2083)) * Y -\n        (\u2191C Y ^ 3 + \u2191C (\u2191C W.a\u2082) * \u2191C Y ^ 2 + \u2191C (\u2191C W.a\u2084) * \u2191C Y + \u2191C (\u2191C W.a\u2086))) *\n      \u2191C q ^ 2\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 degree (\u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y)) = max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nhave hdp : (p ^ 2).degree = 2 \u2022 p.degree := degree_pow p 2\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\n\u22a2 degree (\u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y)) = max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nhave hdpq : (p * q * (C W.a\u2081 * X + C W.a\u2083)).degree \u2264 p.degree + q.degree + 1 := by\n  simpa only [degree_mul] using add_le_add_left degree_linear_le (p.degree + q.degree)\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\n\u22a2 degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\n[PROOFSTEP]\nsimpa only [degree_mul] using add_le_add_left degree_linear_le (p.degree + q.degree)\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\n\u22a2 degree (\u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y)) = max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nhave hdq : (q ^ 2 * (X ^ 3 + C W.a\u2082 * X ^ 2 + C W.a\u2084 * X + C W.a\u2086)).degree = 2 \u2022 q.degree + 3 := by\n  rw [degree_mul, degree_pow, \u2190 one_mul <| X ^ 3, \u2190 C_1, degree_cubic <| one_ne_zero' R]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\n\u22a2 degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\n[PROOFSTEP]\nrw [degree_mul, degree_pow, \u2190 one_mul <| X ^ 3, \u2190 C_1, degree_cubic <| one_ne_zero' R]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\n\u22a2 degree (\u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y)) = max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nrw [norm_smul_basis]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : p = 0\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nsimpa only [hp, hdq, neg_zero, zero_sub, zero_mul, zero_pow zero_lt_two, degree_neg] using (max_bot_left _).symm\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acp = 0\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acp = 0\nhq : q = 0\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nsimpa only [hq, hdp, sub_zero, zero_mul, mul_zero, zero_pow zero_lt_two] using (max_bot_right _).symm\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acp = 0\nhq : \u00acq = 0\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nrw [\u2190 not_congr degree_eq_bot] at hp hq \n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 degree p) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nrcases hp' : p.degree with _ | dp\n[GOAL]\ncase neg.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\nhp' : degree p = none\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 none) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nexact (hp hp').elim\n[GOAL]\ncase neg.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdp : degree (p ^ 2) = 2 \u2022 degree p\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 degree p + degree q + 1\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhp' : degree p = some dp\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nrw [hp'] at hdp hdpq \n[GOAL]\ncase neg.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + degree q + 1\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 degree q + 3)\n[PROOFSTEP]\nrcases hq' : q.degree with _ | dq\n[GOAL]\ncase neg.some.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + degree q + 1\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\nhq' : degree q = none\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 none + 3)\n[PROOFSTEP]\nexact (hq hq').elim\n[GOAL]\ncase neg.some.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 degree q + 3\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + degree q + 1\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhq' : degree q = some dq\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 some dq + 3)\n[PROOFSTEP]\nrw [hq'] at hdpq hdq \n[GOAL]\ncase neg.some.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 some dq + 3)\n[PROOFSTEP]\nrcases le_or_lt dp (dq + 1) with hpq | hpq\n[GOAL]\ncase neg.some.some.inl\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 some dq + 3)\n[PROOFSTEP]\nconvert\n  (degree_sub_eq_right_of_degree_lt <|\n        (degree_sub_le _ _).trans_lt <| max_lt_iff.mpr \u27e8hdp.trans_lt _, hdpq.trans_lt _\u27e9).trans\n    (max_eq_right_of_lt _).symm\n[GOAL]\ncase h.e'_3.h.e'_4\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 2 \u2022 some dq + 3 = degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))\n[PROOFSTEP]\nrw [hdq]\n[GOAL]\ncase neg.some.some.inl.convert_2\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 2 \u2022 some dp < degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))\n[PROOFSTEP]\nrw [hdq]\n[GOAL]\ncase neg.some.some.inl.convert_3\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 some dp + some dq + 1 < degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))\n[PROOFSTEP]\nrw [hdq]\n[GOAL]\ncase neg.some.some.inl.convert_5\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 2 \u2022 some dp < degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))\n[PROOFSTEP]\nrw [hdq]\n[GOAL]\ncase neg.some.some.inl.convert_2\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 2 \u2022 some dp < 2 \u2022 some dq + 3\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.mpr <| by linarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 (fun x x_1 => x + x_1) dp ((fun x x_1 => x + x_1) dp 0) <\n    (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dq ((fun x x_1 => x + x_1) dq 0)) \u21913\n[PROOFSTEP]\nlinarith only [hpq]\n[GOAL]\ncase neg.some.some.inl.convert_3\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 some dp + some dq + 1 < 2 \u2022 some dq + 3\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.mpr <| by linarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dp dq) 1 <\n    (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dq ((fun x x_1 => x + x_1) dq 0)) \u21913\n[PROOFSTEP]\nlinarith only [hpq]\n[GOAL]\ncase neg.some.some.inl.convert_5\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 2 \u2022 some dp < 2 \u2022 some dq + 3\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.mpr <| by linarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dp \u2264 dq + 1\n\u22a2 (fun x x_1 => x + x_1) dp ((fun x x_1 => x + x_1) dp 0) <\n    (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dq ((fun x x_1 => x + x_1) dq 0)) \u21913\n[PROOFSTEP]\nlinarith only [hpq]\n[GOAL]\ncase neg.some.some.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 degree (p ^ 2 - p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) - q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) =\n    max (2 \u2022 some dp) (2 \u2022 some dq + 3)\n[PROOFSTEP]\nrw [sub_sub]\n[GOAL]\ncase neg.some.some.inr\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 degree (p ^ 2 - (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083) + q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086))) =\n    max (2 \u2022 some dp) (2 \u2022 some dq + 3)\n[PROOFSTEP]\nconvert\n  (degree_sub_eq_left_of_degree_lt <|\n        (degree_add_le _ _).trans_lt <| max_lt_iff.mpr \u27e8hdpq.trans_lt _, hdq.trans_lt _\u27e9).trans\n    (max_eq_left_of_lt _).symm\n[GOAL]\ncase h.e'_3.h.e'_3\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 2 \u2022 some dp = degree (p ^ 2)\n[PROOFSTEP]\nrw [hdp]\n[GOAL]\ncase neg.some.some.inr.convert_2\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 some dp + some dq + 1 < degree (p ^ 2)\n[PROOFSTEP]\nrw [hdp]\n[GOAL]\ncase neg.some.some.inr.convert_3\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 2 \u2022 some dq + 3 < degree (p ^ 2)\n[PROOFSTEP]\nrw [hdp]\n[GOAL]\ncase neg.some.some.inr.convert_5\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 2 \u2022 some dq + 3 < degree (p ^ 2)\n[PROOFSTEP]\nrw [hdp]\n[GOAL]\ncase neg.some.some.inr.convert_2\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 some dp + some dq + 1 < 2 \u2022 some dp\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.mpr <| by linarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dp dq) 1 < (fun x x_1 => x + x_1) dp ((fun x x_1 => x + x_1) dp 0)\n[PROOFSTEP]\nlinarith only [hpq]\n[GOAL]\ncase neg.some.some.inr.convert_3\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 2 \u2022 some dq + 3 < 2 \u2022 some dp\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.mpr <| by linarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dq ((fun x x_1 => x + x_1) dq 0)) \u21913 <\n    (fun x x_1 => x + x_1) dp ((fun x x_1 => x + x_1) dp 0)\n[PROOFSTEP]\nlinarith only [hpq]\n[GOAL]\ncase neg.some.some.inr.convert_5\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 2 \u2022 some dq + 3 < 2 \u2022 some dp\n[PROOFSTEP]\nexact WithBot.coe_lt_coe.mpr <| by linarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nhp : \u00acdegree p = \u22a5\nhq : \u00acdegree q = \u22a5\ndp : \u2115\nhdp : degree (p ^ 2) = 2 \u2022 some dp\nhp' : degree p = some dp\ndq : \u2115\nhdq : degree (q ^ 2 * (Y ^ 3 + \u2191C W.a\u2082 * Y ^ 2 + \u2191C W.a\u2084 * Y + \u2191C W.a\u2086)) = 2 \u2022 some dq + 3\nhdpq : degree (p * q * (\u2191C W.a\u2081 * Y + \u2191C W.a\u2083)) \u2264 some dp + some dq + 1\nhq' : degree q = some dq\nhpq : dq + 1 < dp\n\u22a2 (fun x x_1 => x + x_1) ((fun x x_1 => x + x_1) dq ((fun x x_1 => x + x_1) dq 0)) \u21913 <\n    (fun x x_1 => x + x_1) dp ((fun x x_1 => x + x_1) dp 0)\n[PROOFSTEP]\nlinarith only [hpq]\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx\u271d : R\ny : R[X]\ninst\u271d : IsDomain R\nx : CoordinateRing W\n\u22a2 degree (\u2191(Algebra.norm R[X]) x) \u2260 1\n[PROOFSTEP]\nrcases exists_smul_basis_eq x with \u27e8p, q, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 degree (\u2191(Algebra.norm R[X]) (p \u2022 1 + q \u2022 \u2191(mk W) Y)) \u2260 1\n[PROOFSTEP]\nrw [degree_norm_smul_basis]\n[GOAL]\ncase intro.intro\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 degree p) (2 \u2022 degree q + 3) \u2260 1\n[PROOFSTEP]\nrcases p.degree with (_ | _ | _ | _)\n[GOAL]\ncase intro.intro.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 none) (2 \u2022 degree q + 3) \u2260 1\n[PROOFSTEP]\ncases q.degree\n[GOAL]\ncase intro.intro.some.zero\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 some Nat.zero) (2 \u2022 degree q + 3) \u2260 1\n[PROOFSTEP]\ncases q.degree\n[GOAL]\ncase intro.intro.some.succ.zero\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 some (Nat.succ Nat.zero)) (2 \u2022 degree q + 3) \u2260 1\n[PROOFSTEP]\ncases q.degree\n[GOAL]\ncase intro.intro.some.succ.succ\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ (Nat.succ n\u271d))) (2 \u2022 degree q + 3) \u2260 1\n[PROOFSTEP]\ncases q.degree\n[GOAL]\ncase intro.intro.none.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 none) (2 \u2022 none + 3) \u2260 1\ncase intro.intro.none.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nval\u271d : \u2115\n\u22a2 max (2 \u2022 none) (2 \u2022 some val\u271d + 3) \u2260 1\ncase intro.intro.some.zero.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 some Nat.zero) (2 \u2022 none + 3) \u2260 1\ncase intro.intro.some.zero.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nval\u271d : \u2115\n\u22a2 max (2 \u2022 some Nat.zero) (2 \u2022 some val\u271d + 3) \u2260 1\ncase intro.intro.some.succ.zero.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 some (Nat.succ Nat.zero)) (2 \u2022 none + 3) \u2260 1\ncase intro.intro.some.succ.zero.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nval\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ Nat.zero)) (2 \u2022 some val\u271d + 3) \u2260 1\ncase intro.intro.some.succ.succ.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ (Nat.succ n\u271d))) (2 \u2022 none + 3) \u2260 1\ncase intro.intro.some.succ.succ.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d val\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ (Nat.succ n\u271d))) (2 \u2022 some val\u271d + 3) \u2260 1\n[PROOFSTEP]\nany_goals\n  rintro\n    (_ | _)\n        -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.none.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 none) (2 \u2022 none + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.none.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nval\u271d : \u2115\n\u22a2 max (2 \u2022 none) (2 \u2022 some val\u271d + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.zero.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 some Nat.zero) (2 \u2022 none + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.zero.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nval\u271d : \u2115\n\u22a2 max (2 \u2022 some Nat.zero) (2 \u2022 some val\u271d + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.succ.zero.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 max (2 \u2022 some (Nat.succ Nat.zero)) (2 \u2022 none + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.succ.zero.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nval\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ Nat.zero)) (2 \u2022 some val\u271d + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.succ.succ.none\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ (Nat.succ n\u271d))) (2 \u2022 none + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.succ.succ.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d val\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ (Nat.succ n\u271d))) (2 \u2022 some val\u271d + 3) \u2260 1\n[PROOFSTEP]\nrintro\n  (_ | _)\n      -- porting note: replaced `dec_trivial` with `(cmp_eq_lt_iff _ _).mp rfl` but cannot be inlined\n[GOAL]\ncase intro.intro.some.succ.succ.some\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d val\u271d : \u2115\n\u22a2 max (2 \u2022 some (Nat.succ (Nat.succ n\u271d))) (2 \u2022 some val\u271d + 3) \u2260 1\n[PROOFSTEP]\napply (lt_max_of_lt_right _).ne'\n[GOAL]\nR : Type u\ninst\u271d\u2077 : CommRing R\nW : WeierstrassCurve R\nA : Type v\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Algebra R A\nB : Type w\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : IsScalarTower R A B\nx : R\ny : R[X]\ninst\u271d : IsDomain R\np q : R[X]\nn\u271d val\u271d : \u2115\n\u22a2 1 < 2 \u2022 some val\u271d + 3\n[PROOFSTEP]\nexact (cmp_eq_lt_iff _ _).mp rfl\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 3\nthis : Invertible (-3 ^ 3)\n\u22a2 \u2191(unitOfInvertible (-3 ^ 3)) = WeierstrassCurve.\u0394 (WeierstrassCurve.ofJ0 R)\n[PROOFSTEP]\nrw [unitOfInvertible_val, WeierstrassCurve.ofJ0_\u0394 R]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 3\nthis : Invertible (-3 ^ 3)\n\u22a2 -3 ^ 3 = -27\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 3\n\u22a2 j (ofJ0 R) = 0\n[PROOFSTEP]\nsimp only [j, ofJ0, WeierstrassCurve.ofJ0_c\u2084]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 3\n\u22a2 \u2191(unitOfInvertible (-3 ^ 3))\u207b\u00b9 * 0 ^ 3 = 0\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\nthis : Invertible (-2 ^ 6)\n\u22a2 \u2191(unitOfInvertible (-2 ^ 6)) = WeierstrassCurve.\u0394 (WeierstrassCurve.ofJ1728 R)\n[PROOFSTEP]\nrw [unitOfInvertible_val, WeierstrassCurve.ofJ1728_\u0394 R]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\nthis : Invertible (-2 ^ 6)\n\u22a2 -2 ^ 6 = -64\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\n\u22a2 j (ofJ1728 R) = 1728\n[PROOFSTEP]\nfield_simp [j, ofJ1728, @unitOfInvertible_val _ _ _ <| invertibleNeg _, WeierstrassCurve.ofJ1728_c\u2084]\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\ninst\u271d : Invertible 2\n\u22a2 -(1728 * 2 ^ 6) = (-48) ^ 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nj : R\ninst\u271d\u00b9 : Invertible j\ninst\u271d : Invertible (j - 1728)\n\u22a2 EllipticCurve.j (ofJ' j) = j\n[PROOFSTEP]\nfield_simp [EllipticCurve.j, ofJ', @unitOfInvertible_val _ _ _ <| invertibleMul _ _, WeierstrassCurve.ofJ_c\u2084]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nj : R\ninst\u271d\u00b9 : Invertible j\ninst\u271d : Invertible (j - 1728)\n\u22a2 j * (j ^ 2 * (j - 1728) ^ 9) = (j * (j - 1728) ^ 3) ^ 3\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u00b9 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d : Field F\nj : F\n\u22a2 Nat.coprime 2 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : j = 1728\nh : 2 = 0\n\u22a2 j = 0\n[PROOFSTEP]\nrw [h1728, show (1728 : F) = 2 * 864 by norm_num1, h, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : j = 1728\nh : 2 = 0\n\u22a2 1728 = 2 * 864\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh3 : NeZero 3\n\u22a2 ofJ 0 = ofJ0 F\n[PROOFSTEP]\nrw [ofJ, dif_pos rfl, dif_neg h3.out]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh3 : 3 = 0\n\u22a2 ofJ 0 = ofJ1728 F\n[PROOFSTEP]\nrw [ofJ, dif_pos rfl, dif_pos h3]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh3 : 3 = 0\n\u22a2 ofJ 1728 = ofJ1728 F\n[PROOFSTEP]\nrw [ofJ, dif_pos <| by rw [show (1728 : F) = 3 * 576 by norm_num1, h3, zero_mul], dif_pos h3]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh3 : 3 = 0\n\u22a2 1728 = 0\n[PROOFSTEP]\nrw [show (1728 : F) = 3 * 576 by norm_num1, h3, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh3 : 3 = 0\n\u22a2 1728 = 3 * 576\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : NeZero 2\n\u22a2 ofJ 1728 = ofJ1728 F\n[PROOFSTEP]\nby_cases h3 : (3 : F) = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : NeZero 2\nh3 : 3 = 0\n\u22a2 ofJ 1728 = ofJ1728 F\n[PROOFSTEP]\nexact ofJ_1728_of_three_eq_zero h3\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : NeZero 2\nh3 : \u00ac3 = 0\n\u22a2 ofJ 1728 = ofJ1728 F\n[PROOFSTEP]\nhave h : (1728 : F) \u2260 0 := fun h =>\n  or_iff_not_and_not.mp (mul_eq_zero.mp <| by rwa [show 2 ^ 6 * 3 ^ 3 = (1728 : F) by norm_num1])\n    \u27e8pow_ne_zero 6 h2.out, pow_ne_zero 3 h3\u27e9\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : NeZero 2\nh3 : \u00ac3 = 0\nh : 1728 = 0\n\u22a2 2 ^ 6 * 3 ^ 3 = 0\n[PROOFSTEP]\nrwa [show 2 ^ 6 * 3 ^ 3 = (1728 : F) by norm_num1]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : NeZero 2\nh3 : \u00ac3 = 0\nh : 1728 = 0\n\u22a2 2 ^ 6 * 3 ^ 3 = 1728\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : NeZero 2\nh3 : \u00ac3 = 0\nh : 1728 \u2260 0\n\u22a2 ofJ 1728 = ofJ1728 F\n[PROOFSTEP]\nrw [ofJ, dif_neg h, dif_pos rfl]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : 2 = 0\n\u22a2 ofJ 1728 = ofJ0 F\n[PROOFSTEP]\nrw [ofJ, dif_pos <| by rw [show (1728 : F) = 2 * 864 by norm_num1, h2, zero_mul], dif_neg]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : 2 = 0\n\u22a2 1728 = 0\n[PROOFSTEP]\nrw [show (1728 : F) = 2 * 864 by norm_num1, h2, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh2 : 2 = 0\n\u22a2 1728 = 2 * 864\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : j \u2260 0\nh1728 : j \u2260 1728\n\u22a2 ofJ j = ofJ' j\n[PROOFSTEP]\nrw [ofJ, dif_neg h0, dif_neg h1728]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nby_cases h0 : j = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : j = 0\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nby_cases h3 : (3 : F) = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : j = 0\nh3 : 3 = 0\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nrw [h0, ofJ_0_of_three_eq_zero h3, @ofJ1728_j _ _ <| invertibleOfNonzero <| two_or_three_ne_zero.neg_resolve_right h3,\n  show (1728 : F) = 3 * 576 by norm_num1, h3, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : j = 0\nh3 : 3 = 0\n\u22a2 1728 = 3 * 576\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : j = 0\nh3 : \u00ac3 = 0\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nrw [h0, ofJ_0_of_three_ne_zero (h3 := neZero_iff.2 h3), @ofJ0_j _ _ <| invertibleOfNonzero h3]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nby_cases h1728 : j = 1728\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : j = 1728\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nhave h2 : (2 : F) \u2260 0 := fun h => h0 <| by rw [h1728, show (1728 : F) = 2 * 864 by norm_num1, h, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : j = 1728\nh : 2 = 0\n\u22a2 j = 0\n[PROOFSTEP]\nrw [h1728, show (1728 : F) = 2 * 864 by norm_num1, h, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : j = 1728\nh : 2 = 0\n\u22a2 1728 = 2 * 864\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : j = 1728\nh2 : 2 \u2260 0\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nrw [h1728, ofJ_1728_of_two_ne_zero (h2 := neZero_iff.2 h2), @ofJ1728_j _ _ <| invertibleOfNonzero h2]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nF : Type u\ninst\u271d\u00b9 : Field F\nj : F\ninst\u271d : DecidableEq F\nh0 : \u00acj = 0\nh1728 : \u00acj = 1728\n\u22a2 EllipticCurve.j (ofJ j) = j\n[PROOFSTEP]\nrw [ofJ_ne_0_ne_1728 j h0 h1728,\n  @ofJ'_j _ _ _ (invertibleOfNonzero h0) (invertibleOfNonzero <| sub_ne_zero_of_ne h1728)]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 \u2191(C.u\u207b\u00b9 ^ 12 * E.\u0394') = WeierstrassCurve.\u0394 (WeierstrassCurve.variableChange E.toWeierstrassCurve C)\n[PROOFSTEP]\nrw [Units.val_mul, Units.val_pow_eq_pow_val, coe_\u0394', E.variableChange_\u0394]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 variableChange E WeierstrassCurve.VariableChange.id = E\n[PROOFSTEP]\nsimp only [variableChange, WeierstrassCurve.variableChange_id]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 { toWeierstrassCurve := E.toWeierstrassCurve, \u0394' := WeierstrassCurve.VariableChange.id.u\u207b\u00b9 ^ 12 * E.\u0394',\n      coe_\u0394' :=\n        (_ : \u2191(WeierstrassCurve.VariableChange.id.u\u207b\u00b9 ^ 12 * E.\u0394') = WeierstrassCurve.\u0394 E.toWeierstrassCurve) } =\n    E\n[PROOFSTEP]\nsimp only [WeierstrassCurve.VariableChange.id, inv_one, one_pow, one_mul]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE\u271d : EllipticCurve R\nC\u271d C C' : WeierstrassCurve.VariableChange R\nE : EllipticCurve R\n\u22a2 variableChange E (WeierstrassCurve.VariableChange.comp C C') = variableChange (variableChange E C') C\n[PROOFSTEP]\nsimp only [variableChange, WeierstrassCurve.variableChange_comp]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE\u271d : EllipticCurve R\nC\u271d C C' : WeierstrassCurve.VariableChange R\nE : EllipticCurve R\n\u22a2 { toWeierstrassCurve := WeierstrassCurve.variableChange (WeierstrassCurve.variableChange E.toWeierstrassCurve C') C,\n      \u0394' := (WeierstrassCurve.VariableChange.comp C C').u\u207b\u00b9 ^ 12 * E.\u0394',\n      coe_\u0394' :=\n        (_ :\n          \u2191((WeierstrassCurve.VariableChange.comp C C').u\u207b\u00b9 ^ 12 * E.\u0394') =\n            WeierstrassCurve.\u0394\n              (WeierstrassCurve.variableChange (WeierstrassCurve.variableChange E.toWeierstrassCurve C') C)) } =\n    { toWeierstrassCurve := WeierstrassCurve.variableChange (WeierstrassCurve.variableChange E.toWeierstrassCurve C') C,\n      \u0394' := C.u\u207b\u00b9 ^ 12 * (C'.u\u207b\u00b9 ^ 12 * E.\u0394'),\n      coe_\u0394' :=\n        (_ :\n          \u2191(C.u\u207b\u00b9 ^ 12 * (C'.u\u207b\u00b9 ^ 12 * E.\u0394')) =\n            WeierstrassCurve.\u0394\n              (WeierstrassCurve.variableChange (WeierstrassCurve.variableChange E.toWeierstrassCurve C') C)) }\n[PROOFSTEP]\nsimp only [WeierstrassCurve.VariableChange.comp, mul_inv, mul_pow, \u2190 mul_assoc]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 \u2191(variableChange E C).\u0394' = \u2191C.u\u207b\u00b9 ^ 12 * \u2191E.\u0394'\n[PROOFSTEP]\nrw [variableChange_\u0394', Units.val_mul, Units.val_pow_eq_pow_val]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 \u2191(variableChange E C).\u0394'\u207b\u00b9 = \u2191C.u ^ 12 * \u2191E.\u0394'\u207b\u00b9\n[PROOFSTEP]\nrw [variableChange_\u0394', mul_inv, inv_pow, inv_inv, Units.val_mul, Units.val_pow_eq_pow_val]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 j (variableChange E C) = j E\n[PROOFSTEP]\nrw [j, coe_inv_variableChange_\u0394']\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 \u2191C.u ^ 12 * \u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 (variableChange E C).toWeierstrassCurve ^ 3 = j E\n[PROOFSTEP]\nhave hu : (C.u * \u2191C.u\u207b\u00b9 : R) ^ 12 = 1 := by rw [C.u.mul_inv, one_pow]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\n\u22a2 (\u2191C.u * \u2191C.u\u207b\u00b9) ^ 12 = 1\n[PROOFSTEP]\nrw [C.u.mul_inv, one_pow]\n[GOAL]\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\nhu : (\u2191C.u * \u2191C.u\u207b\u00b9) ^ 12 = 1\n\u22a2 \u2191C.u ^ 12 * \u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 (variableChange E C).toWeierstrassCurve ^ 3 = j E\n[PROOFSTEP]\nlinear_combination (norm := (rw [variableChange_toWeierstrassCurve, WeierstrassCurve.variableChange_c\u2084, j]; ring1))\n  E.j * hu\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\nhu : (\u2191C.u * \u2191C.u\u207b\u00b9) ^ 12 = 1\n\u22a2 \u2191C.u ^ 12 * \u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 (variableChange E C).toWeierstrassCurve ^ 3 - j E -\n      (j E * (\u2191C.u * \u2191C.u\u207b\u00b9) ^ 12 - j E * 1) =\n    0\n[PROOFSTEP]\nrw [variableChange_toWeierstrassCurve, WeierstrassCurve.variableChange_c\u2084, j]\n[GOAL]\ncase a\nR : Type u\ninst\u271d : CommRing R\nE : EllipticCurve R\nC : WeierstrassCurve.VariableChange R\nhu : (\u2191C.u * \u2191C.u\u207b\u00b9) ^ 12 = 1\n\u22a2 \u2191C.u ^ 12 * \u2191E.\u0394'\u207b\u00b9 * (\u2191C.u\u207b\u00b9 ^ 4 * WeierstrassCurve.c\u2084 E.toWeierstrassCurve) ^ 3 -\n        \u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 E.toWeierstrassCurve ^ 3 -\n      (\u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 E.toWeierstrassCurve ^ 3 * (\u2191C.u * \u2191C.u\u207b\u00b9) ^ 12 -\n        \u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 E.toWeierstrassCurve ^ 3 * 1) =\n    0\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\n\u22a2 \u2191(\u2191(Units.map \u2191(algebraMap R A)) E.\u0394') = WeierstrassCurve.\u0394 (WeierstrassCurve.baseChange E.toWeierstrassCurve A)\n[PROOFSTEP]\nsimp only [Units.coe_map, coe_\u0394', E.baseChange_\u0394]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\n\u22a2 \u2191\u2191(algebraMap R A) (WeierstrassCurve.\u0394 E.toWeierstrassCurve) =\n    \u2191(algebraMap R A) (WeierstrassCurve.\u0394 E.toWeierstrassCurve)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\n\u22a2 j (baseChange E A) = \u2191(algebraMap R A) (j E)\n[PROOFSTEP]\nsimp only [j, baseChange, E.baseChange_c\u2084]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\n\u22a2 \u2191(\u2191(Units.map \u2191(algebraMap R A)) E.\u0394')\u207b\u00b9 * \u2191(algebraMap R A) (WeierstrassCurve.c\u2084 E.toWeierstrassCurve) ^ 3 =\n    \u2191(algebraMap R A) (\u2191E.\u0394'\u207b\u00b9 * WeierstrassCurve.c\u2084 E.toWeierstrassCurve ^ 3)\n[PROOFSTEP]\nmap_simp\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\n\u22a2 \u2191(\u2191(Units.map \u2191(algebraMap R A)) E.\u0394')\u207b\u00b9 * \u2191(algebraMap R A) (WeierstrassCurve.c\u2084 E.toWeierstrassCurve) ^ 3 =\n    \u2191(algebraMap R A) \u2191E.\u0394'\u207b\u00b9 * \u2191(algebraMap R A) (WeierstrassCurve.c\u2084 E.toWeierstrassCurve) ^ 3\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1 : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\n\u22a2 E = E'\n[PROOFSTEP]\nrcases mk.inj h1 with \u27e8h1, h2\u27e9\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191(Units.map \u2191(algebraMap R A)) E.\u0394' = \u2191(Units.map \u2191(algebraMap R A)) E'.\u0394'\n\u22a2 E = E'\n[PROOFSTEP]\nreplace h2 := (Units.mk.inj h2).left\n[GOAL]\ncase intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\n\u22a2 E = E'\n[PROOFSTEP]\nrcases WeierstrassCurve.mk.inj h1 with \u27e8_, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 E = E'\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2081\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 E.a\u2081 = E'.a\u2081\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2082\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 E.a\u2082 = E'.a\u2082\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2083\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 E.a\u2083 = E'.a\u2083\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2084\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 E.a\u2084 = E'.a\u2084\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2086\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 E.a\u2086 = E'.a\u2086\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.intro.\u0394'.a\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191E.\u0394' = \u2191E'.\u0394'\n[PROOFSTEP]\napply_fun _ using h\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2081\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2082\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2083\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2084\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.intro.a\u2086\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.intro.intro.intro.\u0394'.a\nR : Type u\ninst\u271d\u00b2 : CommRing R\nE\u271d : EllipticCurve R\nA : Type v\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nh : Function.Injective \u2191(algebraMap R A)\nE E' : EllipticCurve R\nh1\u271d : (fun E => baseChange E A) E = (fun E => baseChange E A) E'\nh1 : WeierstrassCurve.baseChange E.toWeierstrassCurve A = WeierstrassCurve.baseChange E'.toWeierstrassCurve A\nh2 : \u2191\u2191(algebraMap R A) \u2191E.\u0394' = \u2191\u2191(algebraMap R A) \u2191E'.\u0394'\nleft\u271d\u00b3 : \u2191(algebraMap R A) E.a\u2081 = \u2191(algebraMap R A) E'.a\u2081\nleft\u271d\u00b2 : \u2191(algebraMap R A) E.a\u2082 = \u2191(algebraMap R A) E'.a\u2082\nleft\u271d\u00b9 : \u2191(algebraMap R A) E.a\u2083 = \u2191(algebraMap R A) E'.a\u2083\nleft\u271d : \u2191(algebraMap R A) E.a\u2084 = \u2191(algebraMap R A) E'.a\u2084\nright\u271d : \u2191(algebraMap R A) E.a\u2086 = \u2191(algebraMap R A) E'.a\u2086\n\u22a2 \u2191(algebraMap R A) \u2191E.\u0394' = \u2191(algebraMap R A) \u2191E'.\u0394'\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass", "llama_tokens": 130835, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.865224072151174, "lm_q2_score": 0.6723316860482763, "lm_q1q2_score": 0.5817175592389543}}
{"text": "[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\na\u271d b\u271d : M\nhm\u2081 : a\u271d \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N}\nhm\u2082 : b\u271d \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N}\nx : L\n\u22a2 \u2045x, a\u271d + b\u271d\u2046 \u2208 N\n[PROOFSTEP]\nrw [lie_add]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\na\u271d b\u271d : M\nhm\u2081 : a\u271d \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N}\nhm\u2082 : b\u271d \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N}\nx : L\n\u22a2 \u2045x, a\u271d\u2046 + \u2045x, b\u271d\u2046 \u2208 N\n[PROOFSTEP]\nexact N.add_mem' (hm\u2081 x) (hm\u2082 x)\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nx : L\n\u22a2 \u2045x, 0\u2046 \u2208 N\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nt : R\nm : M\nhm :\n  m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : M},\n                    a \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 b \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 \u2200 (x : L), \u2045x, a + b\u2046 \u2208 N) },\n          zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 \u2208 N) }.toAddSubsemigroup.carrier\nx : L\n\u22a2 \u2045x, t \u2022 m\u2046 \u2208 N\n[PROOFSTEP]\nrw [lie_smul]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nt : R\nm : M\nhm :\n  m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : M},\n                    a \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 b \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 \u2200 (x : L), \u2045x, a + b\u2046 \u2208 N) },\n          zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 \u2208 N) }.toAddSubsemigroup.carrier\nx : L\n\u22a2 t \u2022 \u2045x, m\u2046 \u2208 N\n[PROOFSTEP]\nexact N.smul_mem' t (hm x)\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nx : L\nm : M\nhm :\n  m \u2208\n    {\n            toAddSubmonoid :=\n              {\n                toAddSubsemigroup :=\n                  { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N},\n                    add_mem' :=\n                      (_ :\n                        \u2200 {a b : M},\n                          a \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192\n                            b \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 \u2200 (x : L), \u2045x, a + b\u2046 \u2208 N) },\n                zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 \u2208 N) },\n            smul_mem' :=\n              (_ :\n                \u2200 (t : R) (m : M),\n                  m \u2208\n                      {\n                            toAddSubsemigroup :=\n                              { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N},\n                                add_mem' :=\n                                  (_ :\n                                    \u2200 {a b : M},\n                                      a \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192\n                                        b \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 \u2200 (x : L), \u2045x, a + b\u2046 \u2208 N) },\n                            zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 \u2208 N) }.toAddSubsemigroup.carrier \u2192\n                    \u2200 (x : L), \u2045x, t \u2022 m\u2046 \u2208 N) }.toAddSubmonoid.toAddSubsemigroup.carrier\ny : L\n\u22a2 \u2045y, \u2045x, m\u2046\u2046 \u2208 N\n[PROOFSTEP]\nrw [leibniz_lie]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nx : L\nm : M\nhm :\n  m \u2208\n    {\n            toAddSubmonoid :=\n              {\n                toAddSubsemigroup :=\n                  { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N},\n                    add_mem' :=\n                      (_ :\n                        \u2200 {a b : M},\n                          a \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192\n                            b \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 \u2200 (x : L), \u2045x, a + b\u2046 \u2208 N) },\n                zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 \u2208 N) },\n            smul_mem' :=\n              (_ :\n                \u2200 (t : R) (m : M),\n                  m \u2208\n                      {\n                            toAddSubsemigroup :=\n                              { carrier := {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N},\n                                add_mem' :=\n                                  (_ :\n                                    \u2200 {a b : M},\n                                      a \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192\n                                        b \u2208 {m | \u2200 (x : L), \u2045x, m\u2046 \u2208 N} \u2192 \u2200 (x : L), \u2045x, a + b\u2046 \u2208 N) },\n                            zero_mem' := (_ : \u2200 (x : L), \u2045x, 0\u2046 \u2208 N) }.toAddSubsemigroup.carrier \u2192\n                    \u2200 (x : L), \u2045x, t \u2022 m\u2046 \u2208 N) }.toAddSubmonoid.toAddSubsemigroup.carrier\ny : L\n\u22a2 \u2045\u2045y, x\u2046, m\u2046 + \u2045x, \u2045y, m\u2046\u2046 \u2208 N\n[PROOFSTEP]\nexact N.add_mem' (hm \u2045y, x\u2046) (N.lie_mem (hm y))\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\n\u22a2 N \u2264 normalizer N\n[PROOFSTEP]\nintro m hm\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nm : M\nhm : m \u2208 N\n\u22a2 m \u2208 normalizer N\n[PROOFSTEP]\nrw [mem_normalizer]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nm : M\nhm : m \u2208 N\n\u22a2 \u2200 (x : L), \u2045x, m\u2046 \u2208 N\n[PROOFSTEP]\nexact fun x => N.lie_mem hm\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\n\u22a2 normalizer (N\u2081 \u2293 N\u2082) = normalizer N\u2081 \u2293 normalizer N\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nm\u271d : M\n\u22a2 m\u271d \u2208 normalizer (N\u2081 \u2293 N\u2082) \u2194 m\u271d \u2208 normalizer N\u2081 \u2293 normalizer N\u2082\n[PROOFSTEP]\nsimp [\u2190 forall_and]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\n\u22a2 Monotone normalizer\n[PROOFSTEP]\nintro N\u2081 N\u2082 h m hm\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081\u271d N\u2082\u271d N\u2081 N\u2082 : LieSubmodule R L M\nh : N\u2081 \u2264 N\u2082\nm : M\nhm : m \u2208 normalizer N\u2081\n\u22a2 m \u2208 normalizer N\u2082\n[PROOFSTEP]\nrw [mem_normalizer] at hm \u22a2\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081\u271d N\u2082\u271d N\u2081 N\u2082 : LieSubmodule R L M\nh : N\u2081 \u2264 N\u2082\nm : M\nhm : \u2200 (x : L), \u2045x, m\u2046 \u2208 N\u2081\n\u22a2 \u2200 (x : L), \u2045x, m\u2046 \u2208 N\u2082\n[PROOFSTEP]\nexact fun x => h (hm x)\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nf : M' \u2192\u2097\u2045R,L\u2046 M\n\u22a2 comap f (normalizer N) = normalizer (comap f N)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 : LieSubmodule R L M\nf : M' \u2192\u2097\u2045R,L\u2046 M\nm\u271d : M'\n\u22a2 m\u271d \u2208 comap f (normalizer N) \u2194 m\u271d \u2208 normalizer (comap f N)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 N' : LieSubmodule R L M\n\u22a2 \u2045\u22a4, N\u2046 \u2264 N' \u2194 N \u2264 normalizer N'\n[PROOFSTEP]\nrw [lie_le_iff]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 N' : LieSubmodule R L M\n\u22a2 (\u2200 (x : L), x \u2208 \u22a4 \u2192 \u2200 (m : M), m \u2208 N \u2192 \u2045x, m\u2046 \u2208 N') \u2194 N \u2264 normalizer N'\n[PROOFSTEP]\ntauto\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nsrc\u271d : LieSubmodule R { x // x \u2208 H } L := LieSubmodule.normalizer (toLieSubmodule H)\ny z : L\nhy : y \u2208 src\u271d.carrier\nhz : z \u2208 src\u271d.carrier\nx : { x // x \u2208 H }\n\u22a2 \u2045x, \u2045y, z\u2046\u2046 \u2208 toLieSubmodule H\n[PROOFSTEP]\nrw [coe_bracket_of_module, mem_toLieSubmodule, leibniz_lie, \u2190 lie_skew y, \u2190 sub_eq_add_neg]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nsrc\u271d : LieSubmodule R { x // x \u2208 H } L := LieSubmodule.normalizer (toLieSubmodule H)\ny z : L\nhy : y \u2208 src\u271d.carrier\nhz : z \u2208 src\u271d.carrier\nx : { x // x \u2208 H }\n\u22a2 \u2045\u2045\u2191x, y\u2046, z\u2046 - \u2045\u2045\u2191x, z\u2046, y\u2046 \u2208 H\n[PROOFSTEP]\nexact H.sub_mem (hz \u27e8_, hy x\u27e9) (hy \u27e8_, hz x\u27e9)\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 x \u2208 normalizer H \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045y, x\u2046 \u2208 H\n[PROOFSTEP]\nrw [Subtype.forall']\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 x \u2208 normalizer H \u2194 \u2200 (x_1 : { a // a \u2208 H }), \u2045\u2191x_1, x\u2046 \u2208 H\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 x \u2208 normalizer H \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H\n[PROOFSTEP]\nrw [mem_normalizer_iff']\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 (\u2200 (y : L), y \u2208 H \u2192 \u2045y, x\u2046 \u2208 H) \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H\n[PROOFSTEP]\nrefine' forall\u2082_congr fun y hy => _\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhy : y \u2208 H\n\u22a2 \u2045y, x\u2046 \u2208 H \u2194 \u2045x, y\u2046 \u2208 H\n[PROOFSTEP]\nrw [\u2190 lie_skew, neg_mem_iff (G := L)]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y z : L\nhx : x \u2208 normalizer H\nhy : y \u2208 Submodule.span R {x} \u2294 H.toSubmodule\nhz : z \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n\u22a2 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n[PROOFSTEP]\nrw [Submodule.mem_sup] at hy hz \n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y z : L\nhx : x \u2208 normalizer H\nhy : \u2203 y_1, y_1 \u2208 Submodule.span R {x} \u2227 \u2203 z, z \u2208 H.toSubmodule \u2227 y_1 + z = y\nhz : \u2203 y, y \u2208 Submodule.span R {x} \u2227 \u2203 z_1, z_1 \u2208 H.toSubmodule \u2227 y + z_1 = z\n\u22a2 \u2045y, z\u2046 \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n[PROOFSTEP]\nobtain \u27e8u\u2081, hu\u2081, v, hv : v \u2208 H, rfl\u27e9 := hy\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx z : L\nhx : x \u2208 normalizer H\nhz : \u2203 y, y \u2208 Submodule.span R {x} \u2227 \u2203 z_1, z_1 \u2208 H.toSubmodule \u2227 y + z_1 = z\nu\u2081 : L\nhu\u2081 : u\u2081 \u2208 Submodule.span R {x}\nv : L\nhv : v \u2208 H\n\u22a2 \u2045u\u2081 + v, z\u2046 \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n[PROOFSTEP]\nobtain \u27e8u\u2082, hu\u2082, w, hw : w \u2208 H, rfl\u27e9 := hz\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nu\u2081 : L\nhu\u2081 : u\u2081 \u2208 Submodule.span R {x}\nv : L\nhv : v \u2208 H\nu\u2082 : L\nhu\u2082 : u\u2082 \u2208 Submodule.span R {x}\nw : L\nhw : w \u2208 H\n\u22a2 \u2045u\u2081 + v, u\u2082 + w\u2046 \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n[PROOFSTEP]\nobtain \u27e8t, rfl\u27e9 := Submodule.mem_span_singleton.mp hu\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nv : L\nhv : v \u2208 H\nu\u2082 : L\nhu\u2082 : u\u2082 \u2208 Submodule.span R {x}\nw : L\nhw : w \u2208 H\nt : R\nhu\u2081 : t \u2022 x \u2208 Submodule.span R {x}\n\u22a2 \u2045t \u2022 x + v, u\u2082 + w\u2046 \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n[PROOFSTEP]\nobtain \u27e8s, rfl\u27e9 := Submodule.mem_span_singleton.mp hu\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nv : L\nhv : v \u2208 H\nw : L\nhw : w \u2208 H\nt : R\nhu\u2081 : t \u2022 x \u2208 Submodule.span R {x}\ns : R\nhu\u2082 : s \u2022 x \u2208 Submodule.span R {x}\n\u22a2 \u2045t \u2022 x + v, s \u2022 x + w\u2046 \u2208 Submodule.span R {x} \u2294 H.toSubmodule\n[PROOFSTEP]\napply Submodule.mem_sup_right\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nv : L\nhv : v \u2208 H\nw : L\nhw : w \u2208 H\nt : R\nhu\u2081 : t \u2022 x \u2208 Submodule.span R {x}\ns : R\nhu\u2082 : s \u2022 x \u2208 Submodule.span R {x}\n\u22a2 \u2045t \u2022 x + v, s \u2022 x + w\u2046 \u2208 H.toSubmodule\n[PROOFSTEP]\nsimp only [LieSubalgebra.mem_coe_submodule, smul_lie, add_lie, zero_add, lie_add, smul_zero, lie_smul, lie_self]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nv : L\nhv : v \u2208 H\nw : L\nhw : w \u2208 H\nt : R\nhu\u2081 : t \u2022 x \u2208 Submodule.span R {x}\ns : R\nhu\u2082 : s \u2022 x \u2208 Submodule.span R {x}\n\u22a2 s \u2022 \u2045v, x\u2046 + (t \u2022 \u2045x, w\u2046 + \u2045v, w\u2046) \u2208 H\n[PROOFSTEP]\nrefine' H.add_mem (H.smul_mem s _) (H.add_mem (H.smul_mem t _) (H.lie_mem hv hw))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a.refine'_1\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nv : L\nhv : v \u2208 H\nw : L\nhw : w \u2208 H\nt : R\nhu\u2081 : t \u2022 x \u2208 Submodule.span R {x}\ns : R\nhu\u2082 : s \u2022 x \u2208 Submodule.span R {x}\n\u22a2 \u2045v, x\u2046 \u2208 H\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.a.refine'_2\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\nhx : x \u2208 normalizer H\nv : L\nhv : v \u2208 H\nw : L\nhw : w \u2208 H\nt : R\nhu\u2081 : t \u2022 x \u2208 Submodule.span R {x}\ns : R\nhu\u2082 : s \u2022 x \u2208 Submodule.span R {x}\n\u22a2 \u2045x, w\u2046 \u2208 H\n[PROOFSTEP]\nexacts [(H.mem_normalizer_iff' x).mp hx v hv, (H.mem_normalizer_iff x).mp hx w hw]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhx : x \u2208 normalizer H\nhy : y \u2208 H\n\u22a2 \u2045x, y\u2046 \u2208 H\n[PROOFSTEP]\nrw [\u2190 lie_skew, neg_mem_iff (G := L)]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhx : x \u2208 normalizer H\nhy : y \u2208 H\n\u22a2 \u2045y, x\u2046 \u2208 H\n[PROOFSTEP]\nexact hx \u27e8y, hy\u27e9\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH K : LieSubalgebra R L\nh\u2081 : H \u2264 K\nh\u2082 : K \u2264 normalizer H\n\u22a2 \u2203 I, \u2191R { x // x \u2208 K } I = ofLe h\u2081\n[PROOFSTEP]\nrw [exists_nested_lieIdeal_coe_eq_iff]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH K : LieSubalgebra R L\nh\u2081 : H \u2264 K\nh\u2082 : K \u2264 normalizer H\n\u22a2 \u2200 (x y : L), x \u2208 K \u2192 y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H\n[PROOFSTEP]\nexact fun x y hx hy => ideal_in_normalizer (h\u2082 hx) hy\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\n\u22a2 normalizer H = H \u2194 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) = \u22a5\n[PROOFSTEP]\nrw [LieSubmodule.eq_bot_iff]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\n\u22a2 normalizer H = H \u2194\n    \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => le_antisymm _ H.le_normalizer\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\n\u22a2 \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\n[PROOFSTEP]\nrintro \u27e8x\u27e9 hx\n[GOAL]\ncase refine'_1.mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm\u271d : L \u29f8 toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\n\u22a2 Quot.mk Setoid.r x = 0\n[PROOFSTEP]\nsuffices x \u2208 H by\n  rwa [Submodule.Quotient.quot_mk_eq_mk, Submodule.Quotient.mk_eq_zero, coe_toLieSubmodule, mem_coe_submodule]\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm\u271d : L \u29f8 toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\nthis : x \u2208 H\n\u22a2 Quot.mk Setoid.r x = 0\n[PROOFSTEP]\nrwa [Submodule.Quotient.quot_mk_eq_mk, Submodule.Quotient.mk_eq_zero, coe_toLieSubmodule, mem_coe_submodule]\n[GOAL]\ncase refine'_1.mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm\u271d : L \u29f8 toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\n\u22a2 x \u2208 H\n[PROOFSTEP]\nrw [\u2190 h, H.mem_normalizer_iff']\n[GOAL]\ncase refine'_1.mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm\u271d : L \u29f8 toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\n\u22a2 \u2200 (y : L), y \u2208 H \u2192 \u2045y, x\u2046 \u2208 H\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase refine'_1.mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm\u271d : L \u29f8 toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\ny : L\nhy : y \u2208 H\n\u22a2 \u2045y, x\u2046 \u2208 H\n[PROOFSTEP]\nreplace hx : \u2045_, LieSubmodule.Quotient.mk' _ x\u2046 = 0 := hx \u27e8y, hy\u27e9\n[GOAL]\ncase refine'_1.mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm\u271d : L \u29f8 toLieSubmodule H\nx y : L\nhy : y \u2208 H\nhx : \u2045{ val := y, property := hy }, \u2191(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\u2046 = 0\n\u22a2 \u2045y, x\u2046 \u2208 H\n[PROOFSTEP]\nrwa [\u2190 LieModuleHom.map_lie, LieSubmodule.Quotient.mk_eq_zero] at hx \n[GOAL]\ncase refine'_2\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\n\u22a2 normalizer H \u2264 H\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_2\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\nx : L\nhx : x \u2208 normalizer H\n\u22a2 x \u2208 H\n[PROOFSTEP]\nlet y := LieSubmodule.Quotient.mk' H.toLieSubmodule x\n[GOAL]\ncase refine'_2\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\nx : L\nhx : x \u2208 normalizer H\ny : (fun x => L \u29f8 toLieSubmodule H) x := \u2191(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\n\u22a2 x \u2208 H\n[PROOFSTEP]\nhave hy : y \u2208 LieModule.maxTrivSubmodule R H (L \u29f8 H.toLieSubmodule) :=\n  by\n  rintro \u27e8z, hz\u27e9\n  rw [\u2190 LieModuleHom.map_lie, LieSubmodule.Quotient.mk_eq_zero, coe_bracket_of_module, Submodule.coe_mk,\n    mem_toLieSubmodule]\n  exact (H.mem_normalizer_iff' x).mp hx z hz\n[GOAL]\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\nx : L\nhx : x \u2208 normalizer H\ny : (fun x => L \u29f8 toLieSubmodule H) x := \u2191(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\n\u22a2 y \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\n[PROOFSTEP]\nrintro \u27e8z, hz\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\nx : L\nhx : x \u2208 normalizer H\ny : (fun x => L \u29f8 toLieSubmodule H) x := \u2191(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nz : L\nhz : z \u2208 H\n\u22a2 \u2045{ val := z, property := hz }, y\u2046 = 0\n[PROOFSTEP]\nrw [\u2190 LieModuleHom.map_lie, LieSubmodule.Quotient.mk_eq_zero, coe_bracket_of_module, Submodule.coe_mk,\n  mem_toLieSubmodule]\n[GOAL]\ncase mk\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\nx : L\nhx : x \u2208 normalizer H\ny : (fun x => L \u29f8 toLieSubmodule H) x := \u2191(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nz : L\nhz : z \u2208 H\n\u22a2 \u2045z, x\u2046 \u2208 H\n[PROOFSTEP]\nexact (H.mem_normalizer_iff' x).mp hx z hz\n[GOAL]\ncase refine'_2\nR : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\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 M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nh : \u2200 (m : L \u29f8 toLieSubmodule H), m \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H) \u2192 m = 0\nx : L\nhx : x \u2208 normalizer H\ny : (fun x => L \u29f8 toLieSubmodule H) x := \u2191(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nhy : y \u2208 LieModule.maxTrivSubmodule R { x // x \u2208 H } (L \u29f8 toLieSubmodule H)\n\u22a2 x \u2208 H\n[PROOFSTEP]\nsimpa using h y hy\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Normalizer", "llama_tokens": 16999, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835371034368, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.581461728344243}}
{"text": "[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nF : A \u2192 C \u2964 C\nzero : F 0 \u2245 \ud835\udfed C\nadd : (n m : A) \u2192 F (n + m) \u2245 F n \u22d9 F m\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 (F (m\u2081 + m\u2082 + m\u2083)).obj X = (F (m\u2081 + (m\u2082 + m\u2083))).obj X\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nF : A \u2192 C \u2964 C\nzero : F 0 \u2245 \ud835\udfed C\nadd : (n m : A) \u2192 F (n + m) \u2245 F n \u22d9 F m\nassoc_hom_app :\n  autoParam\n    (\u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n      NatTrans.app (add (m\u2081 + m\u2082) m\u2083).hom X \u226b (F m\u2083).map (NatTrans.app (add m\u2081 m\u2082).hom X) =\n        eqToHom (_ : (F (m\u2081 + m\u2082 + m\u2083)).obj X = (F (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n          NatTrans.app (add m\u2081 (m\u2082 + m\u2083)).hom X \u226b NatTrans.app (add m\u2082 m\u2083).hom ((F m\u2081).obj X))\n    _auto\u271d\nn : A\nX : C\n\u22a2 (F (0 + n)).obj X = (F n).obj ((\ud835\udfed C).obj X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nF : A \u2192 C \u2964 C\nzero : F 0 \u2245 \ud835\udfed C\nadd : (n m : A) \u2192 F (n + m) \u2245 F n \u22d9 F m\nassoc_hom_app :\n  autoParam\n    (\u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n      NatTrans.app (add (m\u2081 + m\u2082) m\u2083).hom X \u226b (F m\u2083).map (NatTrans.app (add m\u2081 m\u2082).hom X) =\n        eqToHom (_ : (F (m\u2081 + m\u2082 + m\u2083)).obj X = (F (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n          NatTrans.app (add m\u2081 (m\u2082 + m\u2083)).hom X \u226b NatTrans.app (add m\u2082 m\u2083).hom ((F m\u2081).obj X))\n    _auto\u271d\nn : A\nX : C\n\u22a2 (F (0 + n)).obj X = (F n).obj X\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nF : A \u2192 C \u2964 C\nzero : F 0 \u2245 \ud835\udfed C\nadd : (n m : A) \u2192 F (n + m) \u2245 F n \u22d9 F m\nassoc_hom_app :\n  autoParam\n    (\u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n      NatTrans.app (add (m\u2081 + m\u2082) m\u2083).hom X \u226b (F m\u2083).map (NatTrans.app (add m\u2081 m\u2082).hom X) =\n        eqToHom (_ : (F (m\u2081 + m\u2082 + m\u2083)).obj X = (F (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n          NatTrans.app (add m\u2081 (m\u2082 + m\u2083)).hom X \u226b NatTrans.app (add m\u2082 m\u2083).hom ((F m\u2081).obj X))\n    _auto\u271d\nzero_add_hom_app :\n  autoParam\n    (\u2200 (n : A) (X : C),\n      NatTrans.app (add 0 n).hom X =\n        eqToHom (_ : (F (0 + n)).obj X = (F n).obj ((\ud835\udfed C).obj X)) \u226b (F n).map (NatTrans.app zero.inv X))\n    _auto\u271d\nn : A\nX : C\n\u22a2 (F (n + 0)).obj X = (\ud835\udfed C).obj ((F n).obj X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nF : A \u2192 C \u2964 C\nzero : F 0 \u2245 \ud835\udfed C\nadd : (n m : A) \u2192 F (n + m) \u2245 F n \u22d9 F m\nassoc_hom_app :\n  autoParam\n    (\u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n      NatTrans.app (add (m\u2081 + m\u2082) m\u2083).hom X \u226b (F m\u2083).map (NatTrans.app (add m\u2081 m\u2082).hom X) =\n        eqToHom (_ : (F (m\u2081 + m\u2082 + m\u2083)).obj X = (F (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n          NatTrans.app (add m\u2081 (m\u2082 + m\u2083)).hom X \u226b NatTrans.app (add m\u2082 m\u2083).hom ((F m\u2081).obj X))\n    _auto\u271d\nzero_add_hom_app :\n  autoParam\n    (\u2200 (n : A) (X : C),\n      NatTrans.app (add 0 n).hom X =\n        eqToHom (_ : (F (0 + n)).obj X = (F n).obj ((\ud835\udfed C).obj X)) \u226b (F n).map (NatTrans.app zero.inv X))\n    _auto\u271d\nn : A\nX : C\n\u22a2 (F (n + 0)).obj X = (F n).obj X\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 (F h (m\u2081 + (m\u2082 + m\u2083))).obj X = (F h (m\u2081 + m\u2082 + m\u2083)).obj X\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 (F h m\u2083).map (NatTrans.app (add h m\u2081 m\u2082).inv X) \u226b NatTrans.app (add h (m\u2081 + m\u2082) m\u2083).inv X =\n    NatTrans.app (add h m\u2082 m\u2083).inv ((F h m\u2081).obj X) \u226b\n      NatTrans.app (add h m\u2081 (m\u2082 + m\u2083)).inv X \u226b eqToHom (_ : (F h (m\u2081 + (m\u2082 + m\u2083))).obj X = (F h (m\u2081 + m\u2082 + m\u2083)).obj X)\n[PROOFSTEP]\nrw [\u2190 cancel_mono ((h.add (m\u2081 + m\u2082) m\u2083).hom.app X \u226b (h.F m\u2083).map ((h.add m\u2081 m\u2082).hom.app X)), Category.assoc,\n  Category.assoc, Category.assoc, Iso.inv_hom_id_app_assoc, \u2190 Functor.map_comp, Iso.inv_hom_id_app, Functor.map_id,\n  h.assoc_hom_app, eqToHom_trans_assoc, eqToHom_refl, Category.id_comp, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 \ud835\udfd9 ((F h m\u2083).obj ((F h m\u2081 \u22d9 F h m\u2082).obj X)) = \ud835\udfd9 ((F h m\u2082 \u22d9 F h m\u2083).obj ((F h m\u2081).obj X))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n\u22a2 (F h n).obj ((\ud835\udfed C).obj X) = (F h (0 + n)).obj X\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n\u22a2 (F h n).obj X = (F h (0 + n)).obj X\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n\u22a2 NatTrans.app (add h 0 n).inv X =\n    (F h n).map (NatTrans.app h.zero.hom X) \u226b eqToHom (_ : (F h n).obj ((\ud835\udfed C).obj X) = (F h (0 + n)).obj X)\n[PROOFSTEP]\nrw [\u2190 cancel_epi ((h.add 0 n).hom.app X), Iso.hom_inv_id_app, h.zero_add_hom_app, Category.assoc, \u2190\n  Functor.map_comp_assoc, Iso.inv_hom_id_app, Functor.map_id, Category.id_comp, eqToHom_trans, eqToHom_refl]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n\u22a2 (\ud835\udfed C).obj ((F h n).obj X) = (F h (n + 0)).obj X\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n\u22a2 (F h n).obj X = (F h (n + 0)).obj X\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n\u22a2 NatTrans.app (add h n 0).inv X =\n    NatTrans.app h.zero.hom ((F h n).obj X) \u226b eqToHom (_ : (\ud835\udfed C).obj ((F h n).obj X) = (F h (n + 0)).obj X)\n[PROOFSTEP]\nrw [\u2190 cancel_epi ((h.add n 0).hom.app X), Iso.hom_inv_id_app, h.add_zero_hom_app, Category.assoc,\n  Iso.inv_hom_id_app_assoc, eqToHom_trans, eqToHom_refl]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\n\u22a2 \u2200 {X Y X' Y' : Discrete A} (f : X \u27f6 Y) (g : X' \u27f6 Y'),\n    MonoidalCategory.tensorHom ((Functor.mk src\u271d.toPrefunctor).map f) ((Functor.mk src\u271d.toPrefunctor).map g) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) Y Y' =\n      (fun m n => (ShiftMkCore.add h m.as n.as).inv) X X' \u226b\n        (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.tensorHom f g)\n[PROOFSTEP]\nrintro \u27e8X\u27e9 \u27e8Y\u27e9 \u27e8X'\u27e9 \u27e8Y'\u27e9 \u27e8\u27e8\u27e8rfl\u27e9\u27e9\u27e9 \u27e8\u27e8\u27e8rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase mk.mk.mk.mk.up.up.refl.up.up.refl\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nX X' : A\n\u22a2 MonoidalCategory.tensorHom\n        ((Functor.mk src\u271d.toPrefunctor).map { down := { down := (_ : { as := X }.as = { as := X }.as) } })\n        ((Functor.mk src\u271d.toPrefunctor).map { down := { down := (_ : { as := X' }.as = { as := X' }.as) } }) \u226b\n      (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := X } { as := X' } =\n    (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := X } { as := X' } \u226b\n      (Functor.mk src\u271d.toPrefunctor).map\n        (MonoidalCategory.tensorHom { down := { down := (_ : { as := X }.as = { as := X }.as) } }\n          { down := { down := (_ : { as := X' }.as = { as := X' }.as) } })\n[PROOFSTEP]\next\n[GOAL]\ncase mk.mk.mk.mk.up.up.refl.up.up.refl.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nX X' : A\nx\u271d : C\n\u22a2 NatTrans.app\n      (MonoidalCategory.tensorHom\n          ((Functor.mk src\u271d.toPrefunctor).map { down := { down := (_ : { as := X }.as = { as := X }.as) } })\n          ((Functor.mk src\u271d.toPrefunctor).map { down := { down := (_ : { as := X' }.as = { as := X' }.as) } }) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := X } { as := X' })\n      x\u271d =\n    NatTrans.app\n      ((fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := X } { as := X' } \u226b\n        (Functor.mk src\u271d.toPrefunctor).map\n          (MonoidalCategory.tensorHom { down := { down := (_ : { as := X }.as = { as := X }.as) } }\n            { down := { down := (_ : { as := X' }.as = { as := X' }.as) } }))\n      x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk.mk.mk.up.up.refl.up.up.refl.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nX X' : A\nx\u271d : C\n\u22a2 (NatTrans.app ((Discrete.functor h.F).map { down := { down := (_ : X' = X') } }) ((ShiftMkCore.F h X).obj x\u271d) \u226b\n        (ShiftMkCore.F h X').map (NatTrans.app ((Discrete.functor h.F).map { down := { down := (_ : X = X) } }) x\u271d)) \u226b\n      NatTrans.app (ShiftMkCore.add h X X').inv x\u271d =\n    NatTrans.app (ShiftMkCore.add h X X').inv x\u271d \u226b\n      NatTrans.app\n        ((Discrete.functor h.F).map\n          (MonoidalCategory.tensorHom { down := { down := (_ : X = X) } } { down := { down := (_ : X' = X') } }))\n        x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\n\u22a2 \u2200 (X Y Z : Discrete A),\n    MonoidalCategory.tensorHom ((fun m n => (ShiftMkCore.add h m.as n.as).inv) X Y)\n          (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj Z)) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) (MonoidalCategory.tensorObj X Y) Z \u226b\n          (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.associator X Y Z).hom =\n      (MonoidalCategory.associator ((Functor.mk src\u271d.toPrefunctor).obj X) ((Functor.mk src\u271d.toPrefunctor).obj Y)\n            ((Functor.mk src\u271d.toPrefunctor).obj Z)).hom \u226b\n        MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X))\n            ((fun m n => (ShiftMkCore.add h m.as n.as).inv) Y Z) \u226b\n          (fun m n => (ShiftMkCore.add h m.as n.as).inv) X (MonoidalCategory.tensorObj Y Z)\n[PROOFSTEP]\nrintro \u27e8m\u2081\u27e9 \u27e8m\u2082\u27e9 \u27e8m\u2083\u27e9\n[GOAL]\ncase mk.mk.mk\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nm\u2081 m\u2082 m\u2083 : A\n\u22a2 MonoidalCategory.tensorHom ((fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := m\u2081 } { as := m\u2082 })\n        (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2083 })) \u226b\n      (fun m n => (ShiftMkCore.add h m.as n.as).inv) (MonoidalCategory.tensorObj { as := m\u2081 } { as := m\u2082 })\n          { as := m\u2083 } \u226b\n        (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.associator { as := m\u2081 } { as := m\u2082 } { as := m\u2083 }).hom =\n    (MonoidalCategory.associator ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2081 })\n          ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2082 }) ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2083 })).hom \u226b\n      MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2081 }))\n          ((fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := m\u2082 } { as := m\u2083 }) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := m\u2081 }\n          (MonoidalCategory.tensorObj { as := m\u2082 } { as := m\u2083 })\n[PROOFSTEP]\next X\n[GOAL]\ncase mk.mk.mk.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app\n      (MonoidalCategory.tensorHom ((fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := m\u2081 } { as := m\u2082 })\n          (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2083 })) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) (MonoidalCategory.tensorObj { as := m\u2081 } { as := m\u2082 })\n            { as := m\u2083 } \u226b\n          (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.associator { as := m\u2081 } { as := m\u2082 } { as := m\u2083 }).hom)\n      X =\n    NatTrans.app\n      ((MonoidalCategory.associator ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2081 })\n            ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2082 }) ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2083 })).hom \u226b\n        MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := m\u2081 }))\n            ((fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := m\u2082 } { as := m\u2083 }) \u226b\n          (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := m\u2081 }\n            (MonoidalCategory.tensorObj { as := m\u2082 } { as := m\u2083 }))\n      X\n[PROOFSTEP]\nsimp [endofunctorMonoidalCategory, h.assoc_inv_app_assoc]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\n\u22a2 \u2200 (X : Discrete A),\n    (MonoidalCategory.leftUnitor ((Functor.mk src\u271d.toPrefunctor).obj X)).hom =\n      MonoidalCategory.tensorHom h.zero.inv (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X)) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) (MonoidalCategory.tensorUnit (Discrete A)) X \u226b\n          (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.leftUnitor X).hom\n[PROOFSTEP]\nrintro \u27e8n\u27e9\n[GOAL]\ncase mk\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nn : A\n\u22a2 (MonoidalCategory.leftUnitor ((Functor.mk src\u271d.toPrefunctor).obj { as := n })).hom =\n    MonoidalCategory.tensorHom h.zero.inv (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := n })) \u226b\n      (fun m n => (ShiftMkCore.add h m.as n.as).inv) (MonoidalCategory.tensorUnit (Discrete A)) { as := n } \u226b\n        (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.leftUnitor { as := n }).hom\n[PROOFSTEP]\next X\n[GOAL]\ncase mk.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nn : A\nX : C\n\u22a2 NatTrans.app (MonoidalCategory.leftUnitor ((Functor.mk src\u271d.toPrefunctor).obj { as := n })).hom X =\n    NatTrans.app\n      (MonoidalCategory.tensorHom h.zero.inv (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := n })) \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) (MonoidalCategory.tensorUnit (Discrete A)) { as := n } \u226b\n          (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.leftUnitor { as := n }).hom)\n      X\n[PROOFSTEP]\nsimp [endofunctorMonoidalCategory, h.zero_add_inv_app, \u2190 Functor.map_comp]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\n\u22a2 \u2200 (X : Discrete A),\n    (MonoidalCategory.rightUnitor ((Functor.mk src\u271d.toPrefunctor).obj X)).hom =\n      MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj X)) h.zero.inv \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) X (MonoidalCategory.tensorUnit (Discrete A)) \u226b\n          (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.rightUnitor X).hom\n[PROOFSTEP]\nrintro \u27e8n\u27e9\n[GOAL]\ncase mk\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nn : A\n\u22a2 (MonoidalCategory.rightUnitor ((Functor.mk src\u271d.toPrefunctor).obj { as := n })).hom =\n    MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := n })) h.zero.inv \u226b\n      (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := n } (MonoidalCategory.tensorUnit (Discrete A)) \u226b\n        (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.rightUnitor { as := n }).hom\n[PROOFSTEP]\next X\n[GOAL]\ncase mk.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : AddMonoid A\nh : ShiftMkCore C A\nsrc\u271d : Discrete A \u2964 C \u2964 C := Discrete.functor h.F\nn : A\nX : C\n\u22a2 NatTrans.app (MonoidalCategory.rightUnitor ((Functor.mk src\u271d.toPrefunctor).obj { as := n })).hom X =\n    NatTrans.app\n      (MonoidalCategory.tensorHom (\ud835\udfd9 ((Functor.mk src\u271d.toPrefunctor).obj { as := n })) h.zero.inv \u226b\n        (fun m n => (ShiftMkCore.add h m.as n.as).inv) { as := n } (MonoidalCategory.tensorUnit (Discrete A)) \u226b\n          (Functor.mk src\u271d.toPrefunctor).map (MonoidalCategory.rightUnitor { as := n }).hom)\n      X\n[PROOFSTEP]\nsimp [endofunctorMonoidalCategory, h.add_zero_inv_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\ni j k : A\nh : i + j = k\n\u22a2 shiftFunctor C k = shiftFunctor C (i + j)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 shiftFunctorAdd' C i j (i + j) (_ : i + j = i + j) = shiftFunctorAdd C i j\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 (shiftFunctorAdd' C i j (i + j) (_ : i + j = i + j)).hom = (shiftFunctorAdd C i j).hom\n[PROOFSTEP]\napply Category.id_comp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\na : A\n\u22a2 shiftFunctor C a = F h a\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\n\u22a2 shiftFunctorZero C A = h.zero\n[PROOFSTEP]\nletI := hasShiftMk C A h\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 shiftFunctorZero C A = h.zero\n[PROOFSTEP]\ndsimp [shiftFunctorZero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 (MonoidalFunctor.\u03b5Iso (shiftMonoidalFunctor C A)).symm = h.zero\n[PROOFSTEP]\nchange (shiftFunctorZero C A).symm.symm = h.zero.symm.symm\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 (shiftFunctorZero C A).symm.symm = h.zero.symm.symm\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_I\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 (shiftFunctorZero C A).symm = h.zero.symm\n[PROOFSTEP]\next\n[GOAL]\ncase e_I.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\nthis : HasShift C A := hasShiftMk C A h\nx\u271d : C\n\u22a2 NatTrans.app (shiftFunctorZero C A).symm.hom x\u271d = NatTrans.app h.zero.symm.hom x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\na b : A\n\u22a2 shiftFunctorAdd C a b = add h a b\n[PROOFSTEP]\nletI := hasShiftMk C A h\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\na b : A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 shiftFunctorAdd C a b = add h a b\n[PROOFSTEP]\nchange (shiftFunctorAdd C a b).symm.symm = (h.add a b).symm.symm\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\na b : A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 (shiftFunctorAdd C a b).symm.symm = (add h a b).symm.symm\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_I\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\na b : A\nthis : HasShift C A := hasShiftMk C A h\n\u22a2 (shiftFunctorAdd C a b).symm = (add h a b).symm\n[PROOFSTEP]\next\n[GOAL]\ncase e_I.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nh : ShiftMkCore C A\na b : A\nthis : HasShift C A := hasShiftMk C A h\nx\u271d : C\n\u22a2 NatTrans.app (shiftFunctorAdd C a b).symm.hom x\u271d = NatTrans.app (add h a b).symm.hom x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\n\u22a2 shiftFunctorAdd' C 0 a a (_ : 0 + a = a) =\n    (Functor.leftUnitor (shiftFunctor C a)).symm \u226a\u226b isoWhiskerRight (shiftFunctorZero C A).symm (shiftFunctor C a)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C 0 a a (_ : 0 + a = a)).hom X =\n    NatTrans.app\n      ((Functor.leftUnitor (shiftFunctor C a)).symm \u226a\u226b\n          isoWhiskerRight (shiftFunctorZero C A).symm (shiftFunctor C a)).hom\n      X\n[PROOFSTEP]\ndsimp [shiftFunctorAdd', shiftFunctorZero, shiftFunctor]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (eqToHom (_ : shiftFunctor C a = shiftFunctor C (0 + a))) X \u226b\n      NatTrans.app (shiftFunctorAdd C 0 a).hom X =\n    \ud835\udfd9 (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj X) \u226b\n      ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).map\n        (NatTrans.app (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.\u03b5 X)\n[PROOFSTEP]\nsimp only [eqToHom_app, obj_\u03b5_app, Discrete.addMonoidal_leftUnitor, eqToIso.inv, eqToHom_map, Category.id_comp]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 eqToHom\n        (_ :\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj X =\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := 0 + a }).obj X) \u226b\n      NatTrans.app (shiftFunctorAdd C 0 a).hom X =\n    eqToHom\n        (_ :\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj X =\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorObj MonoidalCategory.tensorUnit' { as := a })).obj\n              X) \u226b\n      NatTrans.app\n        (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) (MonoidalCategory.tensorUnit (Discrete A)) { as := a }).inv X\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\n\u22a2 shiftFunctorAdd' C a 0 a (_ : a + 0 = a) =\n    (Functor.rightUnitor (shiftFunctor C a)).symm \u226a\u226b isoWhiskerLeft (shiftFunctor C a) (shiftFunctorZero C A).symm\n[PROOFSTEP]\next\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nx\u271d : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C a 0 a (_ : a + 0 = a)).hom x\u271d =\n    NatTrans.app\n      ((Functor.rightUnitor (shiftFunctor C a)).symm \u226a\u226b\n          isoWhiskerLeft (shiftFunctor C a) (shiftFunctorZero C A).symm).hom\n      x\u271d\n[PROOFSTEP]\ndsimp [shiftFunctorAdd', shiftFunctorZero, shiftFunctor]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nx\u271d : C\n\u22a2 NatTrans.app (eqToHom (_ : shiftFunctor C a = shiftFunctor C (a + 0))) x\u271d \u226b\n      NatTrans.app (shiftFunctorAdd C a 0).hom x\u271d =\n    \ud835\udfd9 (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj x\u271d) \u226b\n      NatTrans.app (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.\u03b5\n        (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj x\u271d)\n[PROOFSTEP]\nsimp only [eqToHom_app, \u03b5_app_obj, Discrete.addMonoidal_rightUnitor, eqToIso.inv, eqToHom_map, Category.id_comp]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nx\u271d : C\n\u22a2 eqToHom\n        (_ :\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj x\u271d =\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a + 0 }).obj x\u271d) \u226b\n      NatTrans.app (shiftFunctorAdd C a 0).hom x\u271d =\n    eqToHom\n        (_ :\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a }).obj x\u271d =\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorObj { as := a } MonoidalCategory.tensorUnit')).obj\n              x\u271d) \u226b\n      NatTrans.app\n        (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a } (MonoidalCategory.tensorUnit (Discrete A))).inv x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\n\u22a2 a\u2081\u2082 + a\u2083 = a\u2081\u2082\u2083\n[PROOFSTEP]\nrw [\u2190 h\u2081\u2082, h\u2081\u2082\u2083]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\n\u22a2 a\u2081 + a\u2082\u2083 = a\u2081\u2082\u2083\n[PROOFSTEP]\nrw [\u2190 h\u2082\u2083, \u2190 add_assoc, h\u2081\u2082\u2083]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\n\u22a2 shiftFunctorAdd' C a\u2081\u2082 a\u2083 a\u2081\u2082\u2083 (_ : a\u2081\u2082 + a\u2083 = a\u2081\u2082\u2083) \u226a\u226b\n      isoWhiskerRight (shiftFunctorAdd' C a\u2081 a\u2082 a\u2081\u2082 h\u2081\u2082) (shiftFunctor C a\u2083) \u226a\u226b\n        Functor.associator (shiftFunctor C a\u2081) (shiftFunctor C a\u2082) (shiftFunctor C a\u2083) =\n    shiftFunctorAdd' C a\u2081 a\u2082\u2083 a\u2081\u2082\u2083 (_ : a\u2081 + a\u2082\u2083 = a\u2081\u2082\u2083) \u226a\u226b\n      isoWhiskerLeft (shiftFunctor C a\u2081) (shiftFunctorAdd' C a\u2082 a\u2083 a\u2082\u2083 h\u2082\u2083)\n[PROOFSTEP]\nsubst h\u2081\u2082 h\u2082\u2083 h\u2081\u2082\u2083\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\n\u22a2 shiftFunctorAdd' C (a\u2081 + a\u2082) a\u2083 (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + a\u2082 + a\u2083 = a\u2081 + a\u2082 + a\u2083) \u226a\u226b\n      isoWhiskerRight (shiftFunctorAdd' C a\u2081 a\u2082 (a\u2081 + a\u2082) (_ : a\u2081 + a\u2082 = a\u2081 + a\u2082)) (shiftFunctor C a\u2083) \u226a\u226b\n        Functor.associator (shiftFunctor C a\u2081) (shiftFunctor C a\u2082) (shiftFunctor C a\u2083) =\n    shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083) \u226a\u226b\n      isoWhiskerLeft (shiftFunctor C a\u2081) (shiftFunctorAdd' C a\u2082 a\u2083 (a\u2082 + a\u2083) (_ : a\u2082 + a\u2083 = a\u2082 + a\u2083))\n[PROOFSTEP]\next X\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app\n      (shiftFunctorAdd' C (a\u2081 + a\u2082) a\u2083 (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + a\u2082 + a\u2083 = a\u2081 + a\u2082 + a\u2083) \u226a\u226b\n          isoWhiskerRight (shiftFunctorAdd' C a\u2081 a\u2082 (a\u2081 + a\u2082) (_ : a\u2081 + a\u2082 = a\u2081 + a\u2082)) (shiftFunctor C a\u2083) \u226a\u226b\n            Functor.associator (shiftFunctor C a\u2081) (shiftFunctor C a\u2082) (shiftFunctor C a\u2083)).hom\n      X =\n    NatTrans.app\n      (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083) \u226a\u226b\n          isoWhiskerLeft (shiftFunctor C a\u2081) (shiftFunctorAdd' C a\u2082 a\u2083 (a\u2082 + a\u2083) (_ : a\u2082 + a\u2083 = a\u2082 + a\u2083))).hom\n      X\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C (a\u2081 + a\u2082) a\u2083 (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + a\u2082 + a\u2083 = a\u2081 + a\u2082 + a\u2083)).hom X \u226b\n      (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd' C a\u2081 a\u2082 (a\u2081 + a\u2082) (_ : a\u2081 + a\u2082 = a\u2081 + a\u2082)).hom X) \u226b\n        \ud835\udfd9 ((shiftFunctor C a\u2083).obj ((shiftFunctor C a\u2082).obj ((shiftFunctor C a\u2081).obj X))) =\n    NatTrans.app (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd' C a\u2082 a\u2083 (a\u2082 + a\u2083) (_ : a\u2082 + a\u2083 = a\u2082 + a\u2083)).hom ((shiftFunctor C a\u2081).obj X)\n[PROOFSTEP]\nsimp only [shiftFunctorAdd'_eq_shiftFunctorAdd, Category.comp_id]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083).hom X \u226b\n      (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd C a\u2081 a\u2082).hom X) =\n    NatTrans.app (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd C a\u2082 a\u2083).hom ((shiftFunctor C a\u2081).obj X)\n[PROOFSTEP]\ndsimp [shiftFunctorAdd']\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083).hom X \u226b\n      (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd C a\u2081 a\u2082).hom X) =\n    (NatTrans.app (eqToHom (_ : shiftFunctor C (a\u2081 + a\u2082 + a\u2083) = shiftFunctor C (a\u2081 + (a\u2082 + a\u2083)))) X \u226b\n        NatTrans.app (shiftFunctorAdd C a\u2081 (a\u2082 + a\u2083)).hom X) \u226b\n      NatTrans.app (shiftFunctorAdd C a\u2082 a\u2083).hom ((shiftFunctor C a\u2081).obj X)\n[PROOFSTEP]\nsimp only [eqToHom_app]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083).hom X \u226b\n      (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd C a\u2081 a\u2082).hom X) =\n    (eqToHom (_ : (shiftFunctor C (a\u2081 + a\u2082 + a\u2083)).obj X = (shiftFunctor C (a\u2081 + (a\u2082 + a\u2083))).obj X) \u226b\n        NatTrans.app (shiftFunctorAdd C a\u2081 (a\u2082 + a\u2083)).hom X) \u226b\n      NatTrans.app (shiftFunctorAdd C a\u2082 a\u2083).hom ((shiftFunctor C a\u2081).obj X)\n[PROOFSTEP]\ndsimp [shiftFunctorAdd, shiftFunctor]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 + a\u2082 } { as := a\u2083 }).inv X \u226b\n      ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2083 }).map\n        (NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 } { as := a\u2082 }).inv X) =\n    (eqToHom\n          (_ :\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 + a\u2082 + a\u2083 }).obj X =\n              ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 + (a\u2082 + a\u2083) }).obj X) \u226b\n        NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 } { as := a\u2082 + a\u2083 }).inv X) \u226b\n      NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2082 } { as := a\u2083 }).inv\n        (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 }).obj X)\n[PROOFSTEP]\nsimp only [obj_\u03bc_inv_app, Discrete.addMonoidal_associator, eqToIso.hom, eqToHom_map, eqToHom_app]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 + a\u2082 } { as := a\u2083 }).inv X \u226b\n      NatTrans.app\n          (LaxMonoidalFunctor.\u03bc (shiftMonoidalFunctor C A).toLaxMonoidalFunctor\n            (MonoidalCategory.tensorObj { as := a\u2081 } { as := a\u2082 }) { as := a\u2083 })\n          X \u226b\n        eqToHom\n            (_ :\n              ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                      (MonoidalCategory.tensorObj (MonoidalCategory.tensorObj { as := a\u2081 } { as := a\u2082 })\n                        { as := a\u2083 })).obj\n                  X =\n                ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                      (MonoidalCategory.tensorObj { as := a\u2081 }\n                        (MonoidalCategory.tensorObj { as := a\u2082 } { as := a\u2083 }))).obj\n                  X) \u226b\n          NatTrans.app\n              (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 }\n                  (MonoidalCategory.tensorObj { as := a\u2082 } { as := a\u2083 })).inv\n              X \u226b\n            NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2082 } { as := a\u2083 }).inv\n              (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 }).obj X) =\n    (eqToHom\n          (_ :\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 + a\u2082 + a\u2083 }).obj X =\n              ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 + (a\u2082 + a\u2083) }).obj X) \u226b\n        NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 } { as := a\u2082 + a\u2083 }).inv X) \u226b\n      NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2082 } { as := a\u2083 }).inv\n        (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 }).obj X)\n[PROOFSTEP]\nerw [Iso.inv_hom_id_app_assoc, Category.assoc]\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 eqToHom\n        (_ :\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorObj (MonoidalCategory.tensorObj { as := a\u2081 } { as := a\u2082 }) { as := a\u2083 })).obj\n              X =\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorObj { as := a\u2081 } (MonoidalCategory.tensorObj { as := a\u2082 } { as := a\u2083 }))).obj\n              X) \u226b\n      NatTrans.app\n          (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 }\n              (MonoidalCategory.tensorObj { as := a\u2082 } { as := a\u2083 })).inv\n          X \u226b\n        NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2082 } { as := a\u2083 }).inv\n          (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 }).obj X) =\n    eqToHom\n        (_ :\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 + a\u2082 + a\u2083 }).obj X =\n            ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 + (a\u2082 + a\u2083) }).obj X) \u226b\n      NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2081 } { as := a\u2082 + a\u2083 }).inv X \u226b\n        NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := a\u2082 } { as := a\u2083 }).inv\n          (((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj { as := a\u2081 }).obj X)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\n\u22a2 shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083 \u226a\u226b\n      isoWhiskerRight (shiftFunctorAdd C a\u2081 a\u2082) (shiftFunctor C a\u2083) \u226a\u226b\n        Functor.associator (shiftFunctor C a\u2081) (shiftFunctor C a\u2082) (shiftFunctor C a\u2083) =\n    shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083) \u226a\u226b\n      isoWhiskerLeft (shiftFunctor C a\u2081) (shiftFunctorAdd C a\u2082 a\u2083)\n[PROOFSTEP]\next X\n[GOAL]\ncase w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app\n      (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083 \u226a\u226b\n          isoWhiskerRight (shiftFunctorAdd C a\u2081 a\u2082) (shiftFunctor C a\u2083) \u226a\u226b\n            Functor.associator (shiftFunctor C a\u2081) (shiftFunctor C a\u2082) (shiftFunctor C a\u2083)).hom\n      X =\n    NatTrans.app\n      (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083) \u226a\u226b\n          isoWhiskerLeft (shiftFunctor C a\u2081) (shiftFunctorAdd C a\u2082 a\u2083)).hom\n      X\n[PROOFSTEP]\nsimpa [shiftFunctorAdd'_eq_shiftFunctorAdd] using\n  NatTrans.congr_app (congr_arg Iso.hom (shiftFunctorAdd'_assoc C a\u2081 a\u2082 a\u2083 _ _ _ rfl rfl rfl)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C 0 a a (_ : 0 + a = a)).hom X =\n    (shiftFunctor C a).map (NatTrans.app (shiftFunctorZero C A).inv X)\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.hom (shiftFunctorAdd'_zero_add C a)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C (0 + a)).obj X = (shiftFunctor C a).obj ((\ud835\udfed C).obj X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C (0 + a)).obj X = (shiftFunctor C a).obj X\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C 0 a).hom X =\n    eqToHom (_ : (shiftFunctor C (0 + a)).obj X = (shiftFunctor C a).obj ((\ud835\udfed C).obj X)) \u226b\n      (shiftFunctor C a).map (NatTrans.app (shiftFunctorZero C A).inv X)\n[PROOFSTEP]\nsimp [\u2190 shiftFunctorAdd'_zero_add_hom_app, shiftFunctorAdd']\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C 0 a a (_ : 0 + a = a)).inv X =\n    (shiftFunctor C a).map (NatTrans.app (shiftFunctorZero C A).hom X)\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd'_zero_add C a)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C a).obj ((\ud835\udfed C).obj X) = (shiftFunctor C (0 + a)).obj X\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C a).obj X = (shiftFunctor C (0 + a)).obj X\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C 0 a).inv X =\n    (shiftFunctor C a).map (NatTrans.app (shiftFunctorZero C A).hom X) \u226b\n      eqToHom (_ : (shiftFunctor C a).obj ((\ud835\udfed C).obj X) = (shiftFunctor C (0 + a)).obj X)\n[PROOFSTEP]\nsimp [\u2190 shiftFunctorAdd'_zero_add_inv_app, shiftFunctorAdd']\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C a 0 a (_ : a + 0 = a)).hom X =\n    NatTrans.app (shiftFunctorZero C A).inv ((shiftFunctor C a).obj X)\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.hom (shiftFunctorAdd'_add_zero C a)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C (a + 0)).obj X = (\ud835\udfed C).obj ((shiftFunctor C a).obj X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C (a + 0)).obj X = (shiftFunctor C a).obj X\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C a 0).hom X =\n    eqToHom (_ : (shiftFunctor C (a + 0)).obj X = (\ud835\udfed C).obj ((shiftFunctor C a).obj X)) \u226b\n      NatTrans.app (shiftFunctorZero C A).inv ((shiftFunctor C a).obj X)\n[PROOFSTEP]\nsimp [\u2190 shiftFunctorAdd'_add_zero_hom_app, shiftFunctorAdd']\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C a 0 a (_ : a + 0 = a)).inv X =\n    NatTrans.app (shiftFunctorZero C A).hom ((shiftFunctor C a).obj X)\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd'_add_zero C a)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (\ud835\udfed C).obj ((shiftFunctor C a).obj X) = (shiftFunctor C (a + 0)).obj X\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 (shiftFunctor C a).obj X = (shiftFunctor C (a + 0)).obj X\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C a 0).inv X =\n    NatTrans.app (shiftFunctorZero C A).hom ((shiftFunctor C a).obj X) \u226b\n      eqToHom (_ : (\ud835\udfed C).obj ((shiftFunctor C a).obj X) = (shiftFunctor C (a + 0)).obj X)\n[PROOFSTEP]\nsimp [\u2190 shiftFunctorAdd'_add_zero_inv_app, shiftFunctorAdd']\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\nX : C\n\u22a2 a\u2081\u2082 + a\u2083 = a\u2081\u2082\u2083\n[PROOFSTEP]\nrw [\u2190 h\u2081\u2082, h\u2081\u2082\u2083]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\nX : C\n\u22a2 a\u2081 + a\u2082\u2083 = a\u2081\u2082\u2083\n[PROOFSTEP]\nrw [\u2190 h\u2082\u2083, \u2190 add_assoc, h\u2081\u2082\u2083]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd' C a\u2081\u2082 a\u2083 a\u2081\u2082\u2083 (_ : a\u2081\u2082 + a\u2083 = a\u2081\u2082\u2083)).hom X \u226b\n      (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd' C a\u2081 a\u2082 a\u2081\u2082 h\u2081\u2082).hom X) =\n    NatTrans.app (shiftFunctorAdd' C a\u2081 a\u2082\u2083 a\u2081\u2082\u2083 (_ : a\u2081 + a\u2082\u2083 = a\u2081\u2082\u2083)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd' C a\u2082 a\u2083 a\u2082\u2083 h\u2082\u2083).hom ((shiftFunctor C a\u2081).obj X)\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.hom (shiftFunctorAdd'_assoc C _ _ _ _ _ _ h\u2081\u2082 h\u2082\u2083 h\u2081\u2082\u2083)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\nX : C\n\u22a2 a\u2081\u2082 + a\u2083 = a\u2081\u2082\u2083\n[PROOFSTEP]\nrw [\u2190 h\u2081\u2082, h\u2081\u2082\u2083]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\nX : C\n\u22a2 a\u2081 + a\u2082\u2083 = a\u2081\u2082\u2083\n[PROOFSTEP]\nrw [\u2190 h\u2082\u2083, \u2190 add_assoc, h\u2081\u2082\u2083]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 a\u2081\u2082 a\u2082\u2083 a\u2081\u2082\u2083 : A\nh\u2081\u2082 : a\u2081 + a\u2082 = a\u2081\u2082\nh\u2082\u2083 : a\u2082 + a\u2083 = a\u2082\u2083\nh\u2081\u2082\u2083 : a\u2081 + a\u2082 + a\u2083 = a\u2081\u2082\u2083\nX : C\n\u22a2 (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd' C a\u2081 a\u2082 a\u2081\u2082 h\u2081\u2082).inv X) \u226b\n      NatTrans.app (shiftFunctorAdd' C a\u2081\u2082 a\u2083 a\u2081\u2082\u2083 (_ : a\u2081\u2082 + a\u2083 = a\u2081\u2082\u2083)).inv X =\n    NatTrans.app (shiftFunctorAdd' C a\u2082 a\u2083 a\u2082\u2083 h\u2082\u2083).inv ((shiftFunctor C a\u2081).obj X) \u226b\n      NatTrans.app (shiftFunctorAdd' C a\u2081 a\u2082\u2083 a\u2081\u2082\u2083 (_ : a\u2081 + a\u2082\u2083 = a\u2081\u2082\u2083)).inv X\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd'_assoc C _ _ _ _ _ _ h\u2081\u2082 h\u2082\u2083 h\u2081\u2082\u2083)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083).hom X \u226b\n      (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd C a\u2081 a\u2082).hom X) =\n    NatTrans.app (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd C a\u2082 a\u2083).hom ((shiftFunctor C a\u2081).obj X)\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.hom (shiftFunctorAdd_assoc C a\u2081 a\u2082 a\u2083)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\na\u2081 a\u2082 a\u2083 : A\nX : C\n\u22a2 (shiftFunctor C a\u2083).map (NatTrans.app (shiftFunctorAdd C a\u2081 a\u2082).inv X) \u226b\n      NatTrans.app (shiftFunctorAdd C (a\u2081 + a\u2082) a\u2083).inv X =\n    NatTrans.app (shiftFunctorAdd C a\u2082 a\u2083).inv ((shiftFunctor C a\u2081).obj X) \u226b\n      NatTrans.app (shiftFunctorAdd' C a\u2081 (a\u2082 + a\u2083) (a\u2081 + a\u2082 + a\u2083) (_ : a\u2081 + (a\u2082 + a\u2083) = a\u2081 + a\u2082 + a\u2083)).inv X\n[PROOFSTEP]\nsimpa using NatTrans.congr_app (congr_arg Iso.inv (shiftFunctorAdd_assoc C a\u2081 a\u2082 a\u2083)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftFunctor C j).map ((shiftFunctor C i).map f) =\n    (shiftAdd X i j).inv \u226b (shiftFunctor C (i + j)).map f \u226b (shiftAdd Y i j).hom\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftAdd X i j).inv \u226b (shiftFunctor C (i + j)).map f \u226b (shiftAdd Y i j).hom =\n    (shiftFunctor C j).map ((shiftFunctor C i).map f)\n[PROOFSTEP]\napply NatIso.naturality_1\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\n\u22a2 (shiftFunctor C 0).map f = (shiftZero A X).hom \u226b f \u226b (shiftZero A Y).inv\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\n\u22a2 (shiftZero A X).hom \u226b f \u226b (shiftZero A Y).inv = (shiftFunctor C 0).map f\n[PROOFSTEP]\napply NatIso.naturality_2\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\n\u22a2 j + i = 0\n[PROOFSTEP]\nrw [\u2190 add_left_inj j, add_assoc, h, zero_add, add_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\n\u22a2 (shiftFunctor C i).map (NatTrans.app (shiftFunctorCompIsoId C i j h).symm.hom X) \u226b\n      NatTrans.app (shiftFunctorCompIsoId C j i (_ : j + i = 0)).hom ((shiftFunctor C i).obj X) =\n    \ud835\udfd9 ((shiftFunctor C i).obj X)\n[PROOFSTEP]\nconvert\n  (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) \u27e8i\u27e9 \u27e8j\u27e9 (Discrete.eqToIso h)\n        (Discrete.eqToIso (by dsimp; rw [\u2190 add_left_inj j, add_assoc, h, zero_add, add_zero]))\n        (Subsingleton.elim _ _)).functor_unitIso_comp\n    X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\n\u22a2 (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\n\u22a2 j + i = 0\n[PROOFSTEP]\nrw [\u2190 add_left_inj j, add_assoc, h, zero_add, add_zero]\n[GOAL]\ncase h.e'_2.h.h.e'_6.h.h.e'_8.h.h.e'_7.h.h.e'_5.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X))\ne_3\u271d :\n  (shiftFunctor C i).obj ((\ud835\udfed C).obj X) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      ((\ud835\udfed C).obj X)\ne_4\u271d\u00b9 :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X)\ne_7\u271d :\n  (shiftFunctor C i \u22d9 shiftFunctor C j).obj X =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n          (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n      X\ne_6\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\ne_4\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\n\u22a2 (shiftFunctorCompIsoId C i j h).symm =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n        (Discrete.eqToIso\n          (_ : (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n        (_ :\n          MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n              (MonoidalCategory.leftUnitor { as := i }).hom =\n            (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n              MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                  (Discrete.eqToIso\n                      (_ :\n                        (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                          (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                (MonoidalCategory.rightUnitor { as := i }).hom)).unitIso\ncase h.e'_2.h.h.e'_7.h.h.e'_7.h.h.e'_5.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X))\ne_4\u271d :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X)\ne_5\u271d\u00b9 :\n  (\ud835\udfed C).obj ((shiftFunctor C i).obj X) =\n    (\ud835\udfed C).obj\n      ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        X)\ne_5\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\ne_3\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\n\u22a2 shiftFunctorCompIsoId C j i (_ : j + i = 0) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n        (Discrete.eqToIso\n          (_ : (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n        (_ :\n          MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n              (MonoidalCategory.leftUnitor { as := i }).hom =\n            (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n              MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                  (Discrete.eqToIso\n                      (_ :\n                        (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                          (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                (MonoidalCategory.rightUnitor { as := i }).hom)).counitIso\n[PROOFSTEP]\nall_goals\n  ext X\n  dsimp [shiftFunctorCompIsoId, unitOfTensorIsoUnit, shiftFunctorAdd']\n  simp only [Category.assoc, eqToHom_map]\n  rfl\n[GOAL]\ncase h.e'_2.h.h.e'_6.h.h.e'_8.h.h.e'_7.h.h.e'_5.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X))\ne_3\u271d :\n  (shiftFunctor C i).obj ((\ud835\udfed C).obj X) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      ((\ud835\udfed C).obj X)\ne_4\u271d\u00b9 :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X)\ne_7\u271d :\n  (shiftFunctor C i \u22d9 shiftFunctor C j).obj X =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n          (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n      X\ne_6\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\ne_4\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\n\u22a2 (shiftFunctorCompIsoId C i j h).symm =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n        (Discrete.eqToIso\n          (_ : (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n        (_ :\n          MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n              (MonoidalCategory.leftUnitor { as := i }).hom =\n            (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n              MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                  (Discrete.eqToIso\n                      (_ :\n                        (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                          (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                (MonoidalCategory.rightUnitor { as := i }).hom)).unitIso\n[PROOFSTEP]\next X\n[GOAL]\ncase h.e'_2.h.h.e'_6.h.h.e'_8.h.h.e'_7.h.h.e'_5.h.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX\u271d : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X\u271d) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X\u271d))\ne_3\u271d :\n  (shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      ((\ud835\udfed C).obj X\u271d)\ne_4\u271d\u00b9 :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X\u271d)\ne_7\u271d :\n  (shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n          (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n      X\u271d\ne_6\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\ne_4\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\nX : C\n\u22a2 NatTrans.app (shiftFunctorCompIsoId C i j h).symm.hom X =\n    NatTrans.app\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).unitIso.hom\n      X\n[PROOFSTEP]\ndsimp [shiftFunctorCompIsoId, unitOfTensorIsoUnit, shiftFunctorAdd']\n[GOAL]\ncase h.e'_2.h.h.e'_6.h.h.e'_8.h.h.e'_7.h.h.e'_5.h.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX\u271d : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X\u271d) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X\u271d))\ne_3\u271d :\n  (shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      ((\ud835\udfed C).obj X\u271d)\ne_4\u271d\u00b9 :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X\u271d)\ne_7\u271d :\n  (shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n          (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n      X\u271d\ne_6\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\ne_4\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\nX : C\n\u22a2 NatTrans.app (shiftFunctorZero C A).inv X \u226b\n      NatTrans.app (eqToHom (_ : shiftFunctor C 0 = shiftFunctor C (i + j))) X \u226b\n        NatTrans.app (shiftFunctorAdd C i j).hom X =\n    (NatTrans.app (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.\u03b5 X \u226b\n        NatTrans.app\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.map\n            (eqToHom\n              (_ : MonoidalCategory.tensorUnit (Discrete A) = MonoidalCategory.tensorObj { as := i } { as := j })))\n          X) \u226b\n      NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := i } { as := j }).inv X\n[PROOFSTEP]\nsimp only [Category.assoc, eqToHom_map]\n[GOAL]\ncase h.e'_2.h.h.e'_6.h.h.e'_8.h.h.e'_7.h.h.e'_5.h.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX\u271d : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X\u271d) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X\u271d))\ne_3\u271d :\n  (shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      ((\ud835\udfed C).obj X\u271d)\ne_4\u271d\u00b9 :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X\u271d)\ne_7\u271d :\n  (shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n          (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n      X\u271d\ne_6\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\ne_4\u271d :\n  shiftFunctor C i \u22d9 shiftFunctor C j =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse\nX : C\n\u22a2 NatTrans.app (shiftFunctorZero C A).inv X \u226b\n      NatTrans.app (eqToHom (_ : shiftFunctor C 0 = shiftFunctor C (i + j))) X \u226b\n        NatTrans.app (shiftFunctorAdd C i j).hom X =\n    NatTrans.app (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.\u03b5 X \u226b\n      NatTrans.app\n          (eqToHom\n            (_ :\n              (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj (MonoidalCategory.tensorUnit (Discrete A)) =\n                (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorObj { as := i } { as := j })))\n          X \u226b\n        NatTrans.app (MonoidalFunctor.\u03bcIso (shiftMonoidalFunctor C A) { as := i } { as := j }).inv X\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.h.h.e'_7.h.h.e'_7.h.h.e'_5.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X))\ne_4\u271d :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X)\ne_5\u271d\u00b9 :\n  (\ud835\udfed C).obj ((shiftFunctor C i).obj X) =\n    (\ud835\udfed C).obj\n      ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        X)\ne_5\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\ne_3\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\n\u22a2 shiftFunctorCompIsoId C j i (_ : j + i = 0) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n        (Discrete.eqToIso\n          (_ : (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n        (_ :\n          MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n              (MonoidalCategory.leftUnitor { as := i }).hom =\n            (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n              MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                  (Discrete.eqToIso\n                      (_ :\n                        (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                          (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                (MonoidalCategory.rightUnitor { as := i }).hom)).counitIso\n[PROOFSTEP]\next X\n[GOAL]\ncase h.e'_2.h.h.e'_7.h.h.e'_7.h.h.e'_5.h.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX\u271d : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X\u271d) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X\u271d))\ne_4\u271d :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X\u271d)\ne_5\u271d\u00b9 :\n  (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d) =\n    (\ud835\udfed C).obj\n      ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        X\u271d)\ne_5\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\ne_3\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\nX : C\n\u22a2 NatTrans.app (shiftFunctorCompIsoId C j i (_ : j + i = 0)).hom X =\n    NatTrans.app\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).counitIso.hom\n      X\n[PROOFSTEP]\ndsimp [shiftFunctorCompIsoId, unitOfTensorIsoUnit, shiftFunctorAdd']\n[GOAL]\ncase h.e'_2.h.h.e'_7.h.h.e'_7.h.h.e'_5.h.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX\u271d : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X\u271d) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X\u271d))\ne_4\u271d :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X\u271d)\ne_5\u271d\u00b9 :\n  (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d) =\n    (\ud835\udfed C).obj\n      ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        X\u271d)\ne_5\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\ne_3\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\nX : C\n\u22a2 (NatTrans.app (shiftFunctorAdd C j i).inv X \u226b\n        NatTrans.app (eqToHom (_ : shiftFunctor C (j + i) = shiftFunctor C 0)) X) \u226b\n      NatTrans.app (shiftFunctorZero C A).hom X =\n    NatTrans.app (LaxMonoidalFunctor.\u03bc (shiftMonoidalFunctor C A).toLaxMonoidalFunctor { as := j } { as := i }) X \u226b\n      NatTrans.app\n          ((shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.map\n            (eqToHom\n              (_ : MonoidalCategory.tensorObj { as := j } { as := i } = MonoidalCategory.tensorUnit (Discrete A))))\n          X \u226b\n        NatTrans.app (MonoidalFunctor.\u03b5Iso (shiftMonoidalFunctor C A)).inv X\n[PROOFSTEP]\nsimp only [Category.assoc, eqToHom_map]\n[GOAL]\ncase h.e'_2.h.h.e'_7.h.h.e'_7.h.h.e'_5.h.w.w.h\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\ni j : A\nh : i + j = 0\nX\u271d : C\ne_1\u271d :\n  ((shiftFunctor C i).obj ((\ud835\udfed C).obj X\u271d) \u27f6 (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d)) =\n    ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        ((\ud835\udfed C).obj X\u271d) \u27f6\n      (\ud835\udfed C).obj\n        ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n          X\u271d))\ne_4\u271d :\n  (shiftFunctor C i).obj ((shiftFunctor C i \u22d9 shiftFunctor C j).obj X\u271d) =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n            (Discrete.eqToIso\n              (_ :\n                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                  (MonoidalCategory.tensorUnit (Discrete A)).as))\n            (_ :\n              MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                  (MonoidalCategory.leftUnitor { as := i }).hom =\n                (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                  MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                      (Discrete.eqToIso\n                          (_ :\n                            (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                              (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                    (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n      (((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).functor \u22d9\n            (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n                (Discrete.eqToIso\n                  (_ :\n                    (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                      (MonoidalCategory.tensorUnit (Discrete A)).as))\n                (_ :\n                  MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                      (MonoidalCategory.leftUnitor { as := i }).hom =\n                    (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                      MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                          (Discrete.eqToIso\n                              (_ :\n                                (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                  (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                        (MonoidalCategory.rightUnitor { as := i }).hom)).inverse).obj\n        X\u271d)\ne_5\u271d\u00b9 :\n  (\ud835\udfed C).obj ((shiftFunctor C i).obj X\u271d) =\n    (\ud835\udfed C).obj\n      ((equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n              (Discrete.eqToIso\n                (_ :\n                  (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                    (MonoidalCategory.tensorUnit (Discrete A)).as))\n              (_ :\n                MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                    (MonoidalCategory.leftUnitor { as := i }).hom =\n                  (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                    MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                        (Discrete.eqToIso\n                            (_ :\n                              (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                                (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                      (MonoidalCategory.rightUnitor { as := i }).hom)).functor.obj\n        X\u271d)\ne_5\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\ne_3\u271d :\n  shiftFunctor C j \u22d9 shiftFunctor C i =\n    (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).inverse \u22d9\n      (equivOfTensorIsoUnit (shiftMonoidalFunctor C A) { as := i } { as := j } (Discrete.eqToIso h)\n          (Discrete.eqToIso\n            (_ :\n              (MonoidalCategory.tensorObj { as := j } { as := i }).as = (MonoidalCategory.tensorUnit (Discrete A)).as))\n          (_ :\n            MonoidalCategory.tensorHom (Discrete.eqToIso h).hom (\ud835\udfd9 { as := i }) \u226b\n                (MonoidalCategory.leftUnitor { as := i }).hom =\n              (MonoidalCategory.associator { as := i } { as := j } { as := i }).hom \u226b\n                MonoidalCategory.tensorHom (\ud835\udfd9 { as := i })\n                    (Discrete.eqToIso\n                        (_ :\n                          (MonoidalCategory.tensorObj { as := j } { as := i }).as =\n                            (MonoidalCategory.tensorUnit (Discrete A)).as)).hom \u226b\n                  (MonoidalCategory.rightUnitor { as := i }).hom)).functor\nX : C\n\u22a2 NatTrans.app (shiftFunctorAdd C j i).inv X \u226b\n      NatTrans.app (eqToHom (_ : shiftFunctor C (j + i) = shiftFunctor C 0)) X \u226b\n        NatTrans.app (shiftFunctorZero C A).hom X =\n    NatTrans.app (LaxMonoidalFunctor.\u03bc (shiftMonoidalFunctor C A).toLaxMonoidalFunctor { as := j } { as := i }) X \u226b\n      NatTrans.app\n          (eqToHom\n            (_ :\n              (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorObj { as := j } { as := i }) =\n                (shiftMonoidalFunctor C A).toLaxMonoidalFunctor.toFunctor.obj\n                  (MonoidalCategory.tensorUnit (Discrete A))))\n          X \u226b\n        NatTrans.app (MonoidalFunctor.\u03b5Iso (shiftMonoidalFunctor C A)).inv X\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni : A\n\u22a2 IsEquivalence (shiftFunctor C i)\n[PROOFSTEP]\nchange IsEquivalence (shiftEquiv C i).functor\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni : A\n\u22a2 IsEquivalence (shiftEquiv C i).functor\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 m + n = 0\n[PROOFSTEP]\nrw [\u2190 neg_eq_of_add_eq_zero_left h, add_right_neg]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 (shiftFunctor C n).map (NatTrans.app (shiftFunctorCompIsoId C n m h).hom X) =\n    NatTrans.app (shiftFunctorCompIsoId C m n (_ : m + n = 0)).hom ((shiftFunctor C n).obj X)\n[PROOFSTEP]\ndsimp [shiftFunctorCompIsoId]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 (shiftFunctor C n).map (NatTrans.app (shiftFunctorAdd' C n m 0 h).inv X \u226b NatTrans.app (shiftFunctorZero C A).hom X) =\n    NatTrans.app (shiftFunctorAdd' C m n 0 (_ : m + n = 0)).inv ((shiftFunctor C n).obj X) \u226b\n      NatTrans.app (shiftFunctorZero C A).hom ((shiftFunctor C n).obj X)\n[PROOFSTEP]\nsimpa only [Functor.map_comp, \u2190 shiftFunctorAdd'_zero_add_inv_app n X, \u2190 shiftFunctorAdd'_add_zero_inv_app n X] using\n  shiftFunctorAdd'_assoc_inv_app n m n 0 0 n h (by rw [\u2190 neg_eq_of_add_eq_zero_left h, add_right_neg])\n    (by rw [h, zero_add]) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 m + n = 0\n[PROOFSTEP]\nrw [\u2190 neg_eq_of_add_eq_zero_left h, add_right_neg]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 n + m + n = n\n[PROOFSTEP]\nrw [h, zero_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 m + n = 0\n[PROOFSTEP]\nrw [\u2190 neg_eq_of_add_eq_zero_left h, add_right_neg]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 (shiftFunctor C n).map (NatTrans.app (shiftFunctorCompIsoId C n m h).inv X) =\n    NatTrans.app (shiftFunctorCompIsoId C m n (_ : m + n = 0)).inv ((shiftFunctor C n).obj X)\n[PROOFSTEP]\nrw [\u2190 cancel_mono (((shiftFunctorCompIsoId C n m h).hom.app X)\u27e6n\u27e7'), \u2190 Functor.map_comp, Iso.inv_hom_id_app,\n  Functor.map_id, shift_shiftFunctorCompIsoId_hom_app, Iso.inv_hom_id_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn m : A\nh : n + m = 0\nX : C\n\u22a2 \ud835\udfd9 ((shiftFunctor C n).obj ((\ud835\udfed C).obj X)) = \ud835\udfd9 ((\ud835\udfed C).obj ((shiftFunctor C n).obj X))\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn : A\nX : C\n\u22a2 (shiftFunctor C n).map (NatTrans.app (shiftFunctorCompIsoId C n (-n) (_ : n + -n = 0)).hom X) =\n    NatTrans.app (shiftFunctorCompIsoId C (-n) n (_ : -n + n = 0)).hom ((shiftFunctor C n).obj X)\n[PROOFSTEP]\napply shift_shiftFunctorCompIsoId_hom_app\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn : A\nX : C\n\u22a2 (shiftFunctor C n).map (NatTrans.app (shiftFunctorCompIsoId C n (-n) (_ : n + -n = 0)).inv X) =\n    NatTrans.app (shiftFunctorCompIsoId C (-n) n (_ : -n + n = 0)).inv ((shiftFunctor C n).obj X)\n[PROOFSTEP]\napply shift_shiftFunctorCompIsoId_inv_app\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn : A\nX : C\n\u22a2 (shiftFunctor C (-n)).map (NatTrans.app (shiftFunctorCompIsoId C (-n) n (_ : -n + n = 0)).hom X) =\n    NatTrans.app (shiftFunctorCompIsoId C n (-n) (_ : n + -n = 0)).hom ((shiftFunctor C (-n)).obj X)\n[PROOFSTEP]\napply shift_shiftFunctorCompIsoId_hom_app\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nn : A\nX : C\n\u22a2 (shiftFunctor C (-n)).map (NatTrans.app (shiftFunctorCompIsoId C (-n) n (_ : -n + n = 0)).inv X) =\n    NatTrans.app (shiftFunctorCompIsoId C n (-n) (_ : n + -n = 0)).inv ((shiftFunctor C (-n)).obj X)\n[PROOFSTEP]\napply shift_shiftFunctorCompIsoId_inv_app\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j k : A\nh : i + j = k\n\u22a2 j + i = k\n[PROOFSTEP]\nrw [add_comm j i, h]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j k : A\nh : i + j = k\n\u22a2 shiftFunctorComm C i j = (shiftFunctorAdd' C i j k h).symm \u226a\u226b shiftFunctorAdd' C j i k (_ : j + i = k)\n[PROOFSTEP]\nsubst h\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 shiftFunctorComm C i j =\n    (shiftFunctorAdd' C i j (i + j) (_ : i + j = i + j)).symm \u226a\u226b shiftFunctorAdd' C j i (i + j) (_ : j + i = i + j)\n[PROOFSTEP]\nrw [shiftFunctorAdd'_eq_shiftFunctorAdd]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 shiftFunctorComm C i j = (shiftFunctorAdd C i j).symm \u226a\u226b shiftFunctorAdd' C j i (i + j) (_ : j + i = i + j)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 (shiftFunctorComm C i j).symm = shiftFunctorComm C j i\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 (shiftFunctorComm C i j).symm.hom = (shiftFunctorComm C j i).hom\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 (shiftFunctorComm C i j).inv = (shiftFunctorComm C j i).hom\n[PROOFSTEP]\nrw [shiftFunctorComm_eq C i j (i + j) rfl, shiftFunctorComm_eq C j i (i + j) (add_comm j i)]\n[GOAL]\ncase w\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\ni j : A\n\u22a2 ((shiftFunctorAdd' C i j (i + j) (_ : i + j = i + j)).symm \u226a\u226b\n        shiftFunctorAdd' C j i (i + j) (_ : j + i = i + j)).inv =\n    ((shiftFunctorAdd' C j i (i + j) (_ : j + i = i + j)).symm \u226a\u226b\n        shiftFunctorAdd' C i j (i + j) (_ : i + j = i + j)).hom\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftComm X i j).symm = shiftComm X j i\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftComm X i j).symm.hom = (shiftComm X j i).hom\n[PROOFSTEP]\nexact NatTrans.congr_app (congr_arg Iso.hom (shiftFunctorComm_symm C i j)) X\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftFunctor C j).map ((shiftFunctor C i).map f) =\n    (shiftComm X i j).hom \u226b (shiftFunctor C i).map ((shiftFunctor C j).map f) \u226b (shiftComm Y j i).hom\n[PROOFSTEP]\nerw [\u2190 shiftComm_symm Y i j, \u2190 ((shiftFunctorComm C i j).hom.naturality_assoc f)]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftFunctor C j).map ((shiftFunctor C i).map f) =\n    (shiftFunctor C i \u22d9 shiftFunctor C j).map f \u226b\n      NatTrans.app (shiftFunctorComm C i j).hom Y \u226b (shiftComm Y i j).symm.hom\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftFunctor C j).map ((shiftFunctor C i).map f) =\n    (shiftFunctor C j).map ((shiftFunctor C i).map f) \u226b\n      NatTrans.app (shiftFunctorComm C i j).hom Y \u226b NatTrans.app (shiftFunctorComm C i j).inv Y\n[PROOFSTEP]\nsimp only [Iso.hom_inv_id_app, Functor.comp_obj, Category.comp_id]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\ni j : A\n\u22a2 (shiftComm X i j).hom \u226b (shiftFunctor C i).map ((shiftFunctor C j).map f) =\n    (shiftFunctor C j).map ((shiftFunctor C i).map f) \u226b (shiftComm Y i j).hom\n[PROOFSTEP]\nrw [shiftComm', \u2190 shiftComm_symm, Iso.symm_hom, Iso.inv_hom_id_assoc]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\nn : A\n\u22a2 NatTrans.app (shiftFunctorZero C A).hom ((shiftFunctor C n).obj X) =\n    NatTrans.app (shiftFunctorComm C n 0).hom X \u226b (shiftFunctor C n).map (NatTrans.app (shiftFunctorZero C A).hom X)\n[PROOFSTEP]\nrw [\u2190 shiftFunctorAdd'_zero_add_inv_app n X, shiftFunctorComm_eq C n 0 n (add_zero n)]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\nn : A\n\u22a2 NatTrans.app (shiftFunctorZero C A).hom ((shiftFunctor C n).obj X) =\n    NatTrans.app ((shiftFunctorAdd' C n 0 n (_ : n + 0 = n)).symm \u226a\u226b shiftFunctorAdd' C 0 n n (_ : 0 + n = n)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd' C 0 n n (_ : 0 + n = n)).inv X\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\nn : A\n\u22a2 NatTrans.app (shiftFunctorZero C A).hom ((shiftFunctor C n).obj X) =\n    (NatTrans.app (shiftFunctorAdd' C n 0 n (_ : n + 0 = n)).inv X \u226b\n        NatTrans.app (shiftFunctorAdd' C 0 n n (_ : 0 + n = n)).hom X) \u226b\n      NatTrans.app (shiftFunctorAdd' C 0 n n (_ : 0 + n = n)).inv X\n[PROOFSTEP]\nrw [Category.assoc, Iso.hom_inv_id_app, Category.comp_id, shiftFunctorAdd'_add_zero_inv_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\nn : A\n\u22a2 NatTrans.app (shiftFunctorZero C A).inv ((shiftFunctor C n).obj X) =\n    (shiftFunctor C n).map (NatTrans.app (shiftFunctorZero C A).inv X) \u226b NatTrans.app (shiftFunctorComm C n 0).inv X\n[PROOFSTEP]\nrw [\u2190 cancel_mono ((shiftFunctorZero C A).hom.app (X\u27e6n\u27e7)), Category.assoc, Iso.inv_hom_id_app,\n  shiftFunctorZero_hom_app_shift, Iso.inv_hom_id_app_assoc, \u2190 Functor.map_comp, Iso.inv_hom_id_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\nn : A\n\u22a2 \ud835\udfd9 ((\ud835\udfed C).obj ((shiftFunctor C n).obj X)) = (shiftFunctor C n).map (\ud835\udfd9 ((\ud835\udfed C).obj X))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX Y : C\nf : X \u27f6 Y\nn : A\n\u22a2 \ud835\udfd9 ((shiftFunctor C n).obj X) = (shiftFunctor C n).map (\ud835\udfd9 X)\n[PROOFSTEP]\nrw [Functor.map_id]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorComm C m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom X) =\n    NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom ((shiftFunctor C m\u2081).obj X) \u226b\n      (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).hom X) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).hom ((shiftFunctor C m\u2082).obj X)\n[PROOFSTEP]\nrw [\u2190 cancel_mono ((shiftFunctorComm C m\u2081 m\u2083).inv.app (X\u27e6m\u2082\u27e7)), \u2190\n  cancel_mono (((shiftFunctorComm C m\u2081 m\u2082).inv.app X)\u27e6m\u2083\u27e7')]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 ((NatTrans.app (shiftFunctorComm C m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n          (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom X)) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).inv ((shiftFunctor C m\u2082).obj X)) \u226b\n      (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X) =\n    ((NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom ((shiftFunctor C m\u2081).obj X) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).hom X) \u226b\n            NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).hom ((shiftFunctor C m\u2082).obj X)) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).inv ((shiftFunctor C m\u2082).obj X)) \u226b\n      (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X)\n[PROOFSTEP]\nsimp only [Category.assoc, Iso.hom_inv_id_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorComm C m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom X) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).inv ((shiftFunctor C m\u2082).obj X) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X) =\n    NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom ((shiftFunctor C m\u2081).obj X) \u226b\n      (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).hom X) \u226b\n        \ud835\udfd9 ((shiftFunctor C m\u2081 \u22d9 shiftFunctor C m\u2083).obj ((shiftFunctor C m\u2082).obj X)) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorComm C m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom X) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).inv ((shiftFunctor C m\u2082).obj X) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X) =\n    NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom ((shiftFunctor C m\u2081).obj X) \u226b\n      (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).hom X) \u226b\n        \ud835\udfd9 ((shiftFunctor C m\u2083).obj ((shiftFunctor C m\u2081).obj ((shiftFunctor C m\u2082).obj X))) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X)\n[PROOFSTEP]\nsimp only [Category.id_comp, \u2190 Functor.map_comp, Iso.hom_inv_id_app]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorComm C m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom X) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).inv ((shiftFunctor C m\u2082).obj X) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X) =\n    NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom ((shiftFunctor C m\u2081).obj X) \u226b\n      (shiftFunctor C m\u2083).map (\ud835\udfd9 ((shiftFunctor C m\u2081 \u22d9 shiftFunctor C m\u2082).obj X))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app (shiftFunctorComm C m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom X) \u226b\n        NatTrans.app (shiftFunctorComm C m\u2081 m\u2083).inv ((shiftFunctor C m\u2082).obj X) \u226b\n          (shiftFunctor C m\u2083).map (NatTrans.app (shiftFunctorComm C m\u2081 m\u2082).inv X) =\n    NatTrans.app (shiftFunctorAdd C m\u2082 m\u2083).hom ((shiftFunctor C m\u2081).obj X) \u226b\n      (shiftFunctor C m\u2083).map (\ud835\udfd9 ((shiftFunctor C m\u2082).obj ((shiftFunctor C m\u2081).obj X)))\n[PROOFSTEP]\nsimp only [Functor.map_id, Category.comp_id, shiftFunctorComm_eq C _ _ _ rfl, \u2190 shiftFunctorAdd'_eq_shiftFunctorAdd]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 NatTrans.app\n        ((shiftFunctorAdd' C m\u2081 (m\u2082 + m\u2083) (m\u2081 + (m\u2082 + m\u2083)) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + (m\u2082 + m\u2083))).symm \u226a\u226b\n            shiftFunctorAdd' C (m\u2082 + m\u2083) m\u2081 (m\u2081 + (m\u2082 + m\u2083)) (_ : m\u2082 + m\u2083 + m\u2081 = m\u2081 + (m\u2082 + m\u2083))).hom\n        X \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd' C m\u2082 m\u2083 (m\u2082 + m\u2083) (_ : m\u2082 + m\u2083 = m\u2082 + m\u2083)).hom X) \u226b\n        NatTrans.app\n            ((shiftFunctorAdd' C m\u2081 m\u2083 (m\u2081 + m\u2083) (_ : m\u2081 + m\u2083 = m\u2081 + m\u2083)).symm \u226a\u226b\n                shiftFunctorAdd' C m\u2083 m\u2081 (m\u2081 + m\u2083) (_ : m\u2083 + m\u2081 = m\u2081 + m\u2083)).inv\n            ((shiftFunctor C m\u2082).obj X) \u226b\n          (shiftFunctor C m\u2083).map\n            (NatTrans.app\n              ((shiftFunctorAdd' C m\u2081 m\u2082 (m\u2081 + m\u2082) (_ : m\u2081 + m\u2082 = m\u2081 + m\u2082)).symm \u226a\u226b\n                  shiftFunctorAdd' C m\u2082 m\u2081 (m\u2081 + m\u2082) (_ : m\u2082 + m\u2081 = m\u2081 + m\u2082)).inv\n              X) =\n    NatTrans.app (shiftFunctorAdd' C m\u2082 m\u2083 (m\u2082 + m\u2083) (_ : m\u2082 + m\u2083 = m\u2082 + m\u2083)).hom ((shiftFunctor C m\u2081).obj X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 (NatTrans.app (shiftFunctorAdd' C m\u2081 (m\u2082 + m\u2083) (m\u2081 + (m\u2082 + m\u2083)) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + (m\u2082 + m\u2083))).inv X \u226b\n        NatTrans.app (shiftFunctorAdd' C (m\u2082 + m\u2083) m\u2081 (m\u2081 + (m\u2082 + m\u2083)) (_ : m\u2082 + m\u2083 + m\u2081 = m\u2081 + (m\u2082 + m\u2083))).hom X) \u226b\n      (shiftFunctor C m\u2081).map (NatTrans.app (shiftFunctorAdd' C m\u2082 m\u2083 (m\u2082 + m\u2083) (_ : m\u2082 + m\u2083 = m\u2082 + m\u2083)).hom X) \u226b\n        (NatTrans.app (shiftFunctorAdd' C m\u2083 m\u2081 (m\u2081 + m\u2083) (_ : m\u2083 + m\u2081 = m\u2081 + m\u2083)).inv ((shiftFunctor C m\u2082).obj X) \u226b\n            NatTrans.app (shiftFunctorAdd' C m\u2081 m\u2083 (m\u2081 + m\u2083) (_ : m\u2081 + m\u2083 = m\u2081 + m\u2083)).hom ((shiftFunctor C m\u2082).obj X)) \u226b\n          (shiftFunctor C m\u2083).map\n            (NatTrans.app (shiftFunctorAdd' C m\u2082 m\u2081 (m\u2081 + m\u2082) (_ : m\u2082 + m\u2081 = m\u2081 + m\u2082)).inv X \u226b\n              NatTrans.app (shiftFunctorAdd' C m\u2081 m\u2082 (m\u2081 + m\u2082) (_ : m\u2081 + m\u2082 = m\u2081 + m\u2082)).hom X) =\n    NatTrans.app (shiftFunctorAdd' C m\u2082 m\u2083 (m\u2082 + m\u2083) (_ : m\u2082 + m\u2083 = m\u2082 + m\u2083)).hom ((shiftFunctor C m\u2081).obj X)\n[PROOFSTEP]\nsimp only [Category.assoc, Iso.hom_inv_id_app_assoc, Iso.inv_hom_id_app_assoc, \u2190 Functor.map_comp,\n  shiftFunctorAdd'_assoc_hom_app_assoc m\u2082 m\u2083 m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2083) (m\u2081 + (m\u2082 + m\u2083)) rfl (add_comm m\u2083 m\u2081)\n    (add_comm _ m\u2081) X,\n  \u2190\n  shiftFunctorAdd'_assoc_hom_app_assoc m\u2082 m\u2081 m\u2083 (m\u2081 + m\u2082) (m\u2081 + m\u2083) (m\u2081 + (m\u2082 + m\u2083)) (add_comm _ _) rfl\n    (by rw [add_comm m\u2082 m\u2081, add_assoc]) X,\n  shiftFunctorAdd'_assoc_hom_app m\u2081 m\u2082 m\u2083 (m\u2081 + m\u2082) (m\u2082 + m\u2083) (m\u2081 + (m\u2082 + m\u2083)) rfl rfl (add_assoc _ _ _) X]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : HasShift C A\nX\u271d Y : C\nf : X\u271d \u27f6 Y\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 m\u2082 + m\u2081 + m\u2083 = m\u2081 + (m\u2082 + m\u2083)\n[PROOFSTEP]\nrw [add_comm m\u2082 m\u2081, add_assoc]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nX : C\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).hom X) =\n    NatTrans.app (i 0).hom X \u226b NatTrans.app (shiftFunctorZero D A).hom (F.obj X)\n[PROOFSTEP]\nsimp [hasShiftOfFullyFaithful_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nX : C\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X) =\n    NatTrans.app (shiftFunctorZero D A).inv (F.obj X) \u226b NatTrans.app (i 0).inv X\n[PROOFSTEP]\nsimp [hasShiftOfFullyFaithful_zero]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\na b : A\nX : C\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i a b).hom X) =\n    NatTrans.app (i (a + b)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd D a b).hom (F.obj X) \u226b\n        (shiftFunctor D b).map (NatTrans.app (i a).inv X) \u226b NatTrans.app (i b).inv ((s a).obj X)\n[PROOFSTEP]\ndsimp [hasShiftOfFullyFaithful_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\na b : A\nX : C\n\u22a2 F.map\n      (F.preimage\n        (NatTrans.app (i (a + b)).hom X \u226b\n          NatTrans.app (shiftFunctorAdd D a b).hom (F.obj X) \u226b\n            \ud835\udfd9 ((shiftFunctor D b).obj ((shiftFunctor D a).obj (F.obj X))) \u226b\n              (shiftFunctor D b).map (NatTrans.app (i a).inv X) \u226b\n                \ud835\udfd9 ((shiftFunctor D b).obj (F.obj ((s a).obj X))) \u226b\n                  NatTrans.app (i b).inv ((s a).obj X) \u226b \ud835\udfd9 (F.obj ((s b).obj ((s a).obj X))))) =\n    NatTrans.app (i (a + b)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd D a b).hom (F.obj X) \u226b\n        (shiftFunctor D b).map (NatTrans.app (i a).inv X) \u226b NatTrans.app (i b).inv ((s a).obj X)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\na b : A\nX : C\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i a b).inv X) =\n    NatTrans.app (i b).hom ((s a).obj X) \u226b\n      (shiftFunctor D b).map (NatTrans.app (i a).hom X) \u226b\n        NatTrans.app (shiftFunctorAdd D a b).inv (F.obj X) \u226b NatTrans.app (i (a + b)).inv X\n[PROOFSTEP]\ndsimp [hasShiftOfFullyFaithful_add]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{u_3, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\na b : A\nX : C\n\u22a2 F.map\n      (F.preimage\n        ((((((\ud835\udfd9 (F.obj ((s b).obj ((s a).obj X))) \u226b NatTrans.app (i b).hom ((s a).obj X)) \u226b\n                  \ud835\udfd9 ((shiftFunctor D b).obj (F.obj ((s a).obj X)))) \u226b\n                (shiftFunctor D b).map (NatTrans.app (i a).hom X)) \u226b\n              \ud835\udfd9 ((shiftFunctor D b).obj ((shiftFunctor D a).obj (F.obj X)))) \u226b\n            NatTrans.app (shiftFunctorAdd D a b).inv (F.obj X)) \u226b\n          NatTrans.app (i (a + b)).inv X)) =\n    NatTrans.app (i b).hom ((s a).obj X) \u226b\n      (shiftFunctor D b).map (NatTrans.app (i a).hom X) \u226b\n        NatTrans.app (shiftFunctorAdd D a b).inv (F.obj X) \u226b NatTrans.app (i (a + b)).inv X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\n\u22a2 F.map\n      (NatTrans.app (hasShiftOfFullyFaithful_add F s i (m\u2081 + m\u2082) m\u2083).hom X \u226b\n        (s m\u2083).map (NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 m\u2082).hom X)) =\n    F.map\n      (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n        NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n          NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2082 m\u2083).hom ((s m\u2081).obj X))\n[PROOFSTEP]\nhave h := shiftFunctorAdd'_assoc_hom_app m\u2081 m\u2082 m\u2083 _ _ (m\u2081 + m\u2082 + m\u2083) rfl rfl rfl (F.obj X)\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\nh :\n  NatTrans.app (shiftFunctorAdd' D (m\u2081 + m\u2082) m\u2083 (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + m\u2082 + m\u2083 = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      (shiftFunctor D m\u2083).map\n        (NatTrans.app (shiftFunctorAdd' D m\u2081 m\u2082 (m\u2081 + m\u2082) (_ : m\u2081 + m\u2082 = m\u2081 + m\u2082)).hom (F.obj X)) =\n    NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      NatTrans.app (shiftFunctorAdd' D m\u2082 m\u2083 (m\u2082 + m\u2083) (_ : m\u2082 + m\u2083 = m\u2082 + m\u2083)).hom ((shiftFunctor D m\u2081).obj (F.obj X))\n\u22a2 F.map\n      (NatTrans.app (hasShiftOfFullyFaithful_add F s i (m\u2081 + m\u2082) m\u2083).hom X \u226b\n        (s m\u2083).map (NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 m\u2082).hom X)) =\n    F.map\n      (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n        NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n          NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2082 m\u2083).hom ((s m\u2081).obj X))\n[PROOFSTEP]\nsimp only [shiftFunctorAdd'_eq_shiftFunctorAdd] at h \n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\nh :\n  NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n      (shiftFunctor D m\u2083).map (NatTrans.app (shiftFunctorAdd D m\u2081 m\u2082).hom (F.obj X)) =\n    NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X))\n\u22a2 F.map\n      (NatTrans.app (hasShiftOfFullyFaithful_add F s i (m\u2081 + m\u2082) m\u2083).hom X \u226b\n        (s m\u2083).map (NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 m\u2082).hom X)) =\n    F.map\n      (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n        NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n          NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2082 m\u2083).hom ((s m\u2081).obj X))\n[PROOFSTEP]\nrw [\u2190 cancel_mono ((i m\u2083).hom.app ((s m\u2082).obj ((s m\u2081).obj X)))]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\nh :\n  NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n      (shiftFunctor D m\u2083).map (NatTrans.app (shiftFunctorAdd D m\u2081 m\u2082).hom (F.obj X)) =\n    NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X))\n\u22a2 F.map\n        (NatTrans.app (hasShiftOfFullyFaithful_add F s i (m\u2081 + m\u2082) m\u2083).hom X \u226b\n          (s m\u2083).map (NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 m\u2082).hom X)) \u226b\n      NatTrans.app (i m\u2083).hom ((s m\u2082).obj ((s m\u2081).obj X)) =\n    F.map\n        (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X) \u226b\n          NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 (m\u2082 + m\u2083)).hom X \u226b\n            NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2082 m\u2083).hom ((s m\u2081).obj X)) \u226b\n      NatTrans.app (i m\u2083).hom ((s m\u2082).obj ((s m\u2081).obj X))\n[PROOFSTEP]\nsimp only [Functor.comp_obj, Functor.map_comp, map_hasShiftOfFullyFaithful_add_hom_app, Category.assoc,\n  Iso.inv_hom_id_app_assoc, NatTrans.naturality_assoc, Functor.comp_map, Iso.inv_hom_id_app, Category.comp_id]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\nh :\n  NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n      (shiftFunctor D m\u2083).map (NatTrans.app (shiftFunctorAdd D m\u2081 m\u2082).hom (F.obj X)) =\n    NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X))\n\u22a2 NatTrans.app (i (m\u2081 + m\u2082 + m\u2083)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n        (shiftFunctor D m\u2083).map (NatTrans.app (i (m\u2081 + m\u2082)).inv X) \u226b\n          NatTrans.app (i m\u2083).inv ((s (m\u2081 + m\u2082)).obj X) \u226b\n            F.map ((s m\u2083).map (NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 m\u2082).hom X)) \u226b\n              NatTrans.app (i m\u2083).hom ((s m\u2082).obj ((s m\u2081).obj X)) =\n    F.map (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X)) \u226b\n      NatTrans.app (i (m\u2081 + (m\u2082 + m\u2083))).hom X \u226b\n        NatTrans.app (shiftFunctorAdd D m\u2081 (m\u2082 + m\u2083)).hom (F.obj X) \u226b\n          NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X)) \u226b\n            (shiftFunctor D m\u2083).map ((shiftFunctor D m\u2082).map (NatTrans.app (i m\u2081).inv X)) \u226b\n              (shiftFunctor D m\u2083).map (NatTrans.app (i m\u2082).inv ((s m\u2081).obj X))\n[PROOFSTEP]\nerw [(i m\u2083).hom.naturality]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\nh :\n  NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n      (shiftFunctor D m\u2083).map (NatTrans.app (shiftFunctorAdd D m\u2081 m\u2082).hom (F.obj X)) =\n    NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X))\n\u22a2 NatTrans.app (i (m\u2081 + m\u2082 + m\u2083)).hom X \u226b\n      NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n        (shiftFunctor D m\u2083).map (NatTrans.app (i (m\u2081 + m\u2082)).inv X) \u226b\n          NatTrans.app (i m\u2083).inv ((s (m\u2081 + m\u2082)).obj X) \u226b\n            NatTrans.app (i m\u2083).hom ((s (m\u2081 + m\u2082)).obj X) \u226b\n              (F \u22d9 shiftFunctor D m\u2083).map (NatTrans.app (hasShiftOfFullyFaithful_add F s i m\u2081 m\u2082).hom X) =\n    F.map (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X)) \u226b\n      NatTrans.app (i (m\u2081 + (m\u2082 + m\u2083))).hom X \u226b\n        NatTrans.app (shiftFunctorAdd D m\u2081 (m\u2082 + m\u2083)).hom (F.obj X) \u226b\n          NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X)) \u226b\n            (shiftFunctor D m\u2083).map ((shiftFunctor D m\u2082).map (NatTrans.app (i m\u2081).inv X)) \u226b\n              (shiftFunctor D m\u2083).map (NatTrans.app (i m\u2082).inv ((s m\u2081).obj X))\n[PROOFSTEP]\nrw [Functor.comp_map, map_hasShiftOfFullyFaithful_add_hom_app, Functor.map_comp, Functor.map_comp,\n  Iso.inv_hom_id_app_assoc, \u2190 Functor.map_comp_assoc _ ((i (m\u2081 + m\u2082)).inv.app X), Iso.inv_hom_id_app, Functor.map_id,\n  Category.id_comp, reassoc_of% h, dcongr_arg (fun a => (i a).hom.app X) (add_assoc m\u2081 m\u2082 m\u2083)]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nm\u2081 m\u2082 m\u2083 : A\nX : C\nh :\n  NatTrans.app (shiftFunctorAdd D (m\u2081 + m\u2082) m\u2083).hom (F.obj X) \u226b\n      (shiftFunctor D m\u2083).map (NatTrans.app (shiftFunctorAdd D m\u2081 m\u2082).hom (F.obj X)) =\n    NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n      NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X))\n\u22a2 (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083) \u22d9 F).obj X = (s (m\u2081 + (m\u2082 + m\u2083)) \u22d9 F).obj X) \u226b\n        NatTrans.app (i (m\u2081 + (m\u2082 + m\u2083))).hom X \u226b\n          eqToHom (_ : (F \u22d9 shiftFunctor D (m\u2081 + (m\u2082 + m\u2083))).obj X = (F \u22d9 shiftFunctor D (m\u2081 + m\u2082 + m\u2083)).obj X)) \u226b\n      NatTrans.app (shiftFunctorAdd' D m\u2081 (m\u2082 + m\u2083) (m\u2081 + m\u2082 + m\u2083) (_ : m\u2081 + (m\u2082 + m\u2083) = m\u2081 + m\u2082 + m\u2083)).hom (F.obj X) \u226b\n        NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X)) \u226b\n          (shiftFunctor D m\u2083).map\n            ((shiftFunctor D m\u2082).map (NatTrans.app (i m\u2081).inv X) \u226b NatTrans.app (i m\u2082).inv ((s m\u2081).obj X)) =\n    F.map (eqToHom (_ : (s (m\u2081 + m\u2082 + m\u2083)).obj X = (s (m\u2081 + (m\u2082 + m\u2083))).obj X)) \u226b\n      NatTrans.app (i (m\u2081 + (m\u2082 + m\u2083))).hom X \u226b\n        NatTrans.app (shiftFunctorAdd D m\u2081 (m\u2082 + m\u2083)).hom (F.obj X) \u226b\n          NatTrans.app (shiftFunctorAdd D m\u2082 m\u2083).hom ((shiftFunctor D m\u2081).obj (F.obj X)) \u226b\n            (shiftFunctor D m\u2083).map ((shiftFunctor D m\u2082).map (NatTrans.app (i m\u2081).inv X)) \u226b\n              (shiftFunctor D m\u2083).map (NatTrans.app (i m\u2082).inv ((s m\u2081).obj X))\n[PROOFSTEP]\nsimp [shiftFunctorAdd', eqToHom_map]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i 0 n).hom X) =\n    F.map\n      (eqToHom (_ : (s (0 + n)).obj X = (s n).obj ((\ud835\udfed C).obj X)) \u226b\n        (s n).map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X))\n[PROOFSTEP]\nhave this := dcongr_arg (fun a => (i a).hom.app X) (zero_add n)\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (0 + n)).hom X =\n    eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (0 + n)).obj X)\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i 0 n).hom X) =\n    F.map\n      (eqToHom (_ : (s (0 + n)).obj X = (s n).obj ((\ud835\udfed C).obj X)) \u226b\n        (s n).map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X))\n[PROOFSTEP]\nrw [\u2190 cancel_mono ((i n).hom.app ((s 0).obj X))]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (0 + n)).hom X =\n    eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (0 + n)).obj X)\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i 0 n).hom X) \u226b NatTrans.app (i n).hom ((s 0).obj X) =\n    F.map\n        (eqToHom (_ : (s (0 + n)).obj X = (s n).obj ((\ud835\udfed C).obj X)) \u226b\n          (s n).map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X)) \u226b\n      NatTrans.app (i n).hom ((s 0).obj X)\n[PROOFSTEP]\nsimp [this, map_hasShiftOfFullyFaithful_add_hom_app, shiftFunctorAdd_zero_add_hom_app, eqToHom_map]\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (0 + n)).hom X =\n    eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (0 + n)).obj X)\n\u22a2 eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b\n        (shiftFunctor D n).map (NatTrans.app (shiftFunctorZero D A).inv (F.obj X)) \u226b\n          (shiftFunctor D n).map (NatTrans.app (i 0).inv X) =\n    eqToHom (_ : F.obj ((s (0 + n)).obj X) = F.obj ((s n).obj X)) \u226b\n      F.map ((s n).map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X)) \u226b NatTrans.app (i n).hom ((s 0).obj X)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (0 + n)).hom X =\n    eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (0 + n)).obj X)\n\u22a2 NatTrans.app (i n).hom X \u226b\n      (shiftFunctor D n).map (NatTrans.app (shiftFunctorZero D A).inv (F.obj X)) \u226b\n        (shiftFunctor D n).map (NatTrans.app (i 0).inv X) =\n    F.map ((s n).map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X)) \u226b NatTrans.app (i n).hom ((s 0).obj X)\n[PROOFSTEP]\nerw [(i n).hom.naturality]\n[GOAL]\ncase e_a\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (0 + n)).hom X =\n    eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (0 + n)).obj X)\n\u22a2 NatTrans.app (i n).hom X \u226b\n      (shiftFunctor D n).map (NatTrans.app (shiftFunctorZero D A).inv (F.obj X)) \u226b\n        (shiftFunctor D n).map (NatTrans.app (i 0).inv X) =\n    NatTrans.app (i n).hom X \u226b (F \u22d9 shiftFunctor D n).map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase e_a\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (0 + n)).hom X =\n    eqToHom (_ : (s (0 + n) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (0 + n)).obj X)\n\u22a2 NatTrans.app (i n).hom X \u226b\n      (shiftFunctor D n).map (NatTrans.app (shiftFunctorZero D A).inv (F.obj X)) \u226b\n        (shiftFunctor D n).map (NatTrans.app (i 0).inv X) =\n    NatTrans.app (i n).hom X \u226b (shiftFunctor D n).map (F.map (NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv X))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i n 0).hom X) =\n    F.map\n      (eqToHom (_ : (s (n + 0)).obj X = (\ud835\udfed C).obj ((s n).obj X)) \u226b\n        NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv ((s n).obj X))\n[PROOFSTEP]\nhave := dcongr_arg (fun a => (i a).hom.app X) (add_zero n)\n[GOAL]\nC : Type u\nA : Type u_1\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.291249, u_2} D\ninst\u271d\u00b3 : AddMonoid A\ninst\u271d\u00b2 : HasShift D A\nF : C \u2964 D\ninst\u271d\u00b9 : Full F\ninst\u271d : Faithful F\ns : A \u2192 C \u2964 C\ni : (i : A) \u2192 s i \u22d9 F \u2245 F \u22d9 shiftFunctor D i\nn : A\nX : C\nthis :\n  NatTrans.app (i (n + 0)).hom X =\n    eqToHom (_ : (s (n + 0) \u22d9 F).obj X = (s n \u22d9 F).obj X) \u226b\n      NatTrans.app (i n).hom X \u226b eqToHom (_ : (F \u22d9 shiftFunctor D n).obj X = (F \u22d9 shiftFunctor D (n + 0)).obj X)\n\u22a2 F.map (NatTrans.app (hasShiftOfFullyFaithful_add F s i n 0).hom X) =\n    F.map\n      (eqToHom (_ : (s (n + 0)).obj X = (\ud835\udfed C).obj ((s n).obj X)) \u226b\n        NatTrans.app (hasShiftOfFullyFaithful_zero F s i).inv ((s n).obj X))\n[PROOFSTEP]\nsimp [this, \u2190 NatTrans.naturality_assoc, eqToHom_map, shiftFunctorAdd_add_zero_hom_app]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Shift.Basic", "llama_tokens": 77349, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.6791786991753929, "lm_q1q2_score": 0.581275863690703}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\nh : 0 = 1\n\u22a2 False\n[PROOFSTEP]\nsimpa [\u2190 h, C.pos_iff] using C.one_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nchange C.nonneg (1 - 0)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\n\u22a2 AddCommGroup.PositiveCone.nonneg C.toPositiveCone (1 - 0)\n[PROOFSTEP]\nconvert C.one_nonneg\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\n\u22a2 1 - 0 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\nx y : \u03b1\nxp : 0 < x\nyp : 0 < y\n\u22a2 0 < x * y\n[PROOFSTEP]\nchange\n  C.pos\n    (x * y - 0)\n      -- porting note: used to be convert, but it relied on unfolding definitions\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\nx y : \u03b1\nxp : 0 < x\nyp : 0 < y\n\u22a2 AddCommGroup.PositiveCone.pos C.toPositiveCone (x * y - 0)\n[PROOFSTEP]\nrw [sub_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\nx y : \u03b1\nxp : 0 < x\nyp : 0 < y\n\u22a2 AddCommGroup.PositiveCone.pos C.toPositiveCone (x * y)\n[PROOFSTEP]\nexact C.mul_pos x y (by rwa [\u2190 sub_zero x]) (by rwa [\u2190 sub_zero y])\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\nx y : \u03b1\nxp : 0 < x\nyp : 0 < y\n\u22a2 AddCommGroup.PositiveCone.pos C.toPositiveCone x\n[PROOFSTEP]\nrwa [\u2190 sub_zero x]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : Nontrivial \u03b1\nC : PositiveCone \u03b1\nsrc\u271d\u00b9 : Ring \u03b1 := inst\u271d\u00b9\nsrc\u271d : OrderedAddCommGroup \u03b1 := OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone\nx y : \u03b1\nxp : 0 < x\nyp : 0 < y\n\u22a2 AddCommGroup.PositiveCone.pos C.toPositiveCone y\n[PROOFSTEP]\nrwa [\u2190 sub_zero y]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Ring.Cone", "llama_tokens": 1233, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.7401743735019595, "lm_q1q2_score": 0.5812652868643092}}
{"text": "[GOAL]\ne : \u2124\u02e3\n\u22a2 \u2016\u2191e\u2016\u208a = 1\n[PROOFSTEP]\nobtain rfl | rfl := units_eq_one_or e\n[GOAL]\ncase inl\n\u22a2 \u2016\u21911\u2016\u208a = 1\n[PROOFSTEP]\nsimp only [Units.coe_neg_one, Units.val_one, nnnorm_neg, nnnorm_one]\n[GOAL]\ncase inr\n\u22a2 \u2016\u2191(-1)\u2016\u208a = 1\n[PROOFSTEP]\nsimp only [Units.coe_neg_one, Units.val_one, nnnorm_neg, nnnorm_one]\n[GOAL]\ne : \u2124\u02e3\n\u22a2 \u2016\u2191e\u2016 = 1\n[PROOFSTEP]\nrw [\u2190 coe_nnnorm, nnnorm_coe_units, NNReal.coe_one]\n[GOAL]\nn : \u2124\n\u22a2 \u2191(toNat n) + \u2191(toNat (-n)) = \u2016n\u2016\u208a\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add, toNat_add_toNat_neg_eq_natAbs, NNReal.coe_natAbs]\n[GOAL]\nn : \u2124\n\u22a2 \u2191(toNat n) + \u2191(toNat (-n)) = \u2016n\u2016\n[PROOFSTEP]\nsimpa only [NNReal.coe_nat_cast, NNReal.coe_add] using congrArg NNReal.toReal (toNat_add_toNat_neg_eq_nnnorm n)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Int", "llama_tokens": 410, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7981867777396211, "lm_q2_score": 0.7279754430043072, "lm_q1q2_score": 0.5810603731251811}}
{"text": "[GOAL]\nsrc\u271d\u00b9 : CommRing \u2124 := inferInstanceAs (CommRing \u2124)\nsrc\u271d : Nontrivial \u2124 := inferInstanceAs (Nontrivial \u2124)\na b : \u2124\nb0 : b \u2260 0\n\u22a2 \u2191(natAbs ((fun x x_1 => x % x_1) a b)) < \u2191(natAbs b)\n[PROOFSTEP]\nrw [Int.natAbs_of_nonneg (Int.emod_nonneg _ b0), \u2190 Int.abs_eq_natAbs]\n[GOAL]\nsrc\u271d\u00b9 : CommRing \u2124 := inferInstanceAs (CommRing \u2124)\nsrc\u271d : Nontrivial \u2124 := inferInstanceAs (Nontrivial \u2124)\na b : \u2124\nb0 : b \u2260 0\n\u22a2 a % b < |b|\n[PROOFSTEP]\nexact Int.emod_lt _ b0\n[GOAL]\nsrc\u271d\u00b9 : CommRing \u2124 := inferInstanceAs (CommRing \u2124)\nsrc\u271d : Nontrivial \u2124 := inferInstanceAs (Nontrivial \u2124)\na b : \u2124\nb0 : b \u2260 0\n\u22a2 natAbs (a * b) \u2265 natAbs a\n[PROOFSTEP]\nrw [\u2190 mul_one a.natAbs, Int.natAbs_mul]\n[GOAL]\nsrc\u271d\u00b9 : CommRing \u2124 := inferInstanceAs (CommRing \u2124)\nsrc\u271d : Nontrivial \u2124 := inferInstanceAs (Nontrivial \u2124)\na b : \u2124\nb0 : b \u2260 0\n\u22a2 natAbs a * natAbs b \u2265 natAbs a * 1\n[PROOFSTEP]\nrw [\u2190 Int.natAbs_pos] at b0 \n[GOAL]\nsrc\u271d\u00b9 : CommRing \u2124 := inferInstanceAs (CommRing \u2124)\nsrc\u271d : Nontrivial \u2124 := inferInstanceAs (Nontrivial \u2124)\na b : \u2124\nb0 : 0 < natAbs b\n\u22a2 natAbs a * natAbs b \u2265 natAbs a * 1\n[PROOFSTEP]\nexact Nat.mul_le_mul_of_nonneg_left b0\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\na b : K\n\u22a2 b * (fun x x_1 => x / x_1) a b + (fun a b => a - a * b / b) a b = a\n[PROOFSTEP]\nby_cases h : b = 0\n[GOAL]\ncase pos\nK : Type u_1\ninst\u271d : Field K\na b : K\nh : b = 0\n\u22a2 b * (fun x x_1 => x / x_1) a b + (fun a b => a - a * b / b) a b = a\n[PROOFSTEP]\nsimp [h, mul_div_cancel']\n[GOAL]\ncase neg\nK : Type u_1\ninst\u271d : Field K\na b : K\nh : \u00acb = 0\n\u22a2 b * (fun x x_1 => x / x_1) a b + (fun a b => a - a * b / b) a b = a\n[PROOFSTEP]\nsimp [h, mul_div_cancel']\n[GOAL]\nK : Type u_1\ninst\u271d : Field K\na b : K\nhnb : b \u2260 0\n\u22a2 (fun a b => a = 0 \u2227 b \u2260 0) ((fun a b => a - a * b / b) a b) b\n[PROOFSTEP]\nsimp [hnb]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.EuclideanDomain.Instances", "llama_tokens": 912, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506635289836, "lm_q2_score": 0.7634837581726991, "lm_q1q2_score": 0.5807444272476656}}
{"text": "[GOAL]\nn : \u2115\nh : n \u2260 0\n\u22a2 filter (fun x => x \u2223 n) (range (succ n)) = divisors n\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : \u2115\nh : n \u2260 0\na\u271d : \u2115\n\u22a2 a\u271d \u2208 filter (fun x => x \u2223 n) (range (succ n)) \u2194 a\u271d \u2208 divisors n\n[PROOFSTEP]\nsimp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]\n[GOAL]\ncase a\nn : \u2115\nh : n \u2260 0\na\u271d : \u2115\n\u22a2 a\u271d \u2223 n \u2192 a\u271d < succ n \u2192 1 \u2264 a\u271d\n[PROOFSTEP]\nexact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)\n[GOAL]\nn : \u2115\nh : n \u2260 0\n\u22a2 filter (fun x => x \u2223 n) (range n) = properDivisors n\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : \u2115\nh : n \u2260 0\na\u271d : \u2115\n\u22a2 a\u271d \u2208 filter (fun x => x \u2223 n) (range n) \u2194 a\u271d \u2208 properDivisors n\n[PROOFSTEP]\nsimp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]\n[GOAL]\ncase a\nn : \u2115\nh : n \u2260 0\na\u271d : \u2115\n\u22a2 a\u271d \u2223 n \u2192 a\u271d < n \u2192 1 \u2264 a\u271d\n[PROOFSTEP]\nexact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)\n[GOAL]\nn : \u2115\n\u22a2 \u00acn \u2208 properDivisors n\n[PROOFSTEP]\nsimp [properDivisors]\n[GOAL]\nn m : \u2115\n\u22a2 n \u2208 properDivisors m \u2194 n \u2223 m \u2227 n < m\n[PROOFSTEP]\nrcases eq_or_ne m 0 with (rfl | hm)\n[GOAL]\ncase inl\nn : \u2115\n\u22a2 n \u2208 properDivisors 0 \u2194 n \u2223 0 \u2227 n < 0\n[PROOFSTEP]\nsimp [properDivisors]\n[GOAL]\ncase inr\nn m : \u2115\nhm : m \u2260 0\n\u22a2 n \u2208 properDivisors m \u2194 n \u2223 m \u2227 n < m\n[PROOFSTEP]\nsimp only [and_comm, \u2190 filter_dvd_eq_properDivisors hm, mem_filter, mem_range]\n[GOAL]\nn : \u2115\nh : n \u2260 0\n\u22a2 insert n (properDivisors n) = divisors n\n[PROOFSTEP]\nrw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h), Finset.filter_insert,\n  if_pos (dvd_refl n)]\n[GOAL]\nn : \u2115\nh : n \u2260 0\n\u22a2 cons n (properDivisors n) (_ : \u00acn \u2208 properDivisors n) = divisors n\n[PROOFSTEP]\nrw [cons_eq_insert, insert_self_properDivisors h]\n[GOAL]\nn m : \u2115\n\u22a2 n \u2208 divisors m \u2194 n \u2223 m \u2227 m \u2260 0\n[PROOFSTEP]\nrcases eq_or_ne m 0 with (rfl | hm)\n[GOAL]\ncase inl\nn : \u2115\n\u22a2 n \u2208 divisors 0 \u2194 n \u2223 0 \u2227 0 \u2260 0\n[PROOFSTEP]\nsimp [divisors]\n[GOAL]\ncase inr\nn m : \u2115\nhm : m \u2260 0\n\u22a2 n \u2208 divisors m \u2194 n \u2223 m \u2227 m \u2260 0\n[PROOFSTEP]\nsimp only [hm, Ne.def, not_false_iff, and_true_iff, \u2190 filter_dvd_eq_divisors hm, mem_filter, mem_range,\n  and_iff_right_iff_imp, lt_succ_iff]\n[GOAL]\ncase inr\nn m : \u2115\nhm : m \u2260 0\n\u22a2 n \u2223 m \u2192 n \u2264 m\n[PROOFSTEP]\nexact le_of_dvd hm.bot_lt\n[GOAL]\nn : \u2115\n\u22a2 1 \u2208 divisors n \u2194 n \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn m : \u2115\nh : n \u2208 divisors m\n\u22a2 n \u2223 m\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\nn : \u2115\nh : n \u2208 divisors zero\n\u22a2 n \u2223 zero\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase succ\nn n\u271d : \u2115\nh : n \u2208 divisors (succ n\u271d)\n\u22a2 n \u2223 succ n\u271d\n[PROOFSTEP]\nsimp [mem_divisors.1 h]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x \u2208 divisorsAntidiagonal n \u2194 x.fst * x.snd = n \u2227 n \u2260 0\n[PROOFSTEP]\nsimp only [divisorsAntidiagonal, Finset.mem_Ico, Ne.def, Finset.mem_filter, Finset.mem_product]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 ((1 \u2264 x.fst \u2227 x.fst < n + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < n + 1) \u2227 x.fst * x.snd = n \u2194 x.fst * x.snd = n \u2227 \u00acn = 0\n[PROOFSTEP]\nrw [and_comm]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.fst * x.snd = n \u2227 (1 \u2264 x.fst \u2227 x.fst < n + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < n + 1 \u2194 x.fst * x.snd = n \u2227 \u00acn = 0\n[PROOFSTEP]\napply and_congr_right\n[GOAL]\ncase h\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.fst * x.snd = n \u2192 ((1 \u2264 x.fst \u2227 x.fst < n + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < n + 1 \u2194 \u00acn = 0)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase h\nx : \u2115 \u00d7 \u2115\n\u22a2 (1 \u2264 x.fst \u2227 x.fst < x.fst * x.snd + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < x.fst * x.snd + 1 \u2194 \u00acx.fst * x.snd = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nx : \u2115 \u00d7 \u2115\n\u22a2 (1 \u2264 x.fst \u2227 x.fst < x.fst * x.snd + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < x.fst * x.snd + 1 \u2192 \u00acx.fst * x.snd = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nx : \u2115 \u00d7 \u2115\n\u22a2 \u00acx.fst * x.snd = 0 \u2192 (1 \u2264 x.fst \u2227 x.fst < x.fst * x.snd + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < x.fst * x.snd + 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mp\nx : \u2115 \u00d7 \u2115\nh : (1 \u2264 x.fst \u2227 x.fst < x.fst * x.snd + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < x.fst * x.snd + 1\n\u22a2 \u00acx.fst * x.snd = 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase h.mp\nx : \u2115 \u00d7 \u2115\nh : x.fst * x.snd = 0\n\u22a2 1 \u2264 x.fst \u2227 x.fst < x.fst * x.snd + 1 \u2192 1 \u2264 x.snd \u2192 x.fst * x.snd + 1 \u2264 x.snd\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.fst * x.snd = 0\n\u22a2 (1 \u2264 x.fst \u2227 x.fst < x.fst * x.snd + 1) \u2227 1 \u2264 x.snd \u2227 x.snd < x.fst * x.snd + 1\n[PROOFSTEP]\nrw [Nat.lt_add_one_iff, Nat.lt_add_one_iff]\n[GOAL]\ncase h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.fst * x.snd = 0\n\u22a2 (1 \u2264 x.fst \u2227 x.fst \u2264 x.fst * x.snd) \u2227 1 \u2264 x.snd \u2227 x.snd \u2264 x.fst * x.snd\n[PROOFSTEP]\nrw [mul_eq_zero, not_or] at h \n[GOAL]\ncase h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.fst = 0 \u2227 \u00acx.snd = 0\n\u22a2 (1 \u2264 x.fst \u2227 x.fst \u2264 x.fst * x.snd) \u2227 1 \u2264 x.snd \u2227 x.snd \u2264 x.fst * x.snd\n[PROOFSTEP]\nsimp only [succ_le_of_lt (Nat.pos_of_ne_zero h.1), succ_le_of_lt (Nat.pos_of_ne_zero h.2), true_and_iff]\n[GOAL]\ncase h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.fst = 0 \u2227 \u00acx.snd = 0\n\u22a2 x.fst \u2264 x.fst * x.snd \u2227 x.snd \u2264 x.fst * x.snd\n[PROOFSTEP]\nexact \u27e8le_mul_of_pos_right (Nat.pos_of_ne_zero h.2), le_mul_of_pos_left (Nat.pos_of_ne_zero h.1)\u27e9\n[GOAL]\nn m : \u2115\n\u22a2 n \u2208 divisors m \u2192 n \u2264 m\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 n \u2208 divisors zero \u2192 n \u2264 zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn m : \u2115\n\u22a2 n \u2208 divisors (succ m) \u2192 n \u2264 succ m\n[PROOFSTEP]\nsimp only [mem_divisors, Nat.succ_ne_zero m, and_true_iff, Ne.def, not_false_iff]\n[GOAL]\ncase succ\nn m : \u2115\n\u22a2 n \u2223 succ m \u2192 n \u2264 succ m\n[PROOFSTEP]\nexact Nat.le_of_dvd (Nat.succ_pos m)\n[GOAL]\nn m : \u2115\nhzero : n \u2260 0\nh : m \u2223 n\nhdiff : m \u2260 n\n\u22a2 divisors m \u2286 properDivisors n\n[PROOFSTEP]\napply Finset.subset_iff.2\n[GOAL]\nn m : \u2115\nhzero : n \u2260 0\nh : m \u2223 n\nhdiff : m \u2260 n\n\u22a2 \u2200 \u2983x : \u2115\u2984, x \u2208 divisors m \u2192 x \u2208 properDivisors n\n[PROOFSTEP]\nintro x hx\n[GOAL]\nn m : \u2115\nhzero : n \u2260 0\nh : m \u2223 n\nhdiff : m \u2260 n\nx : \u2115\nhx : x \u2208 divisors m\n\u22a2 x \u2208 properDivisors n\n[PROOFSTEP]\nexact\n  Nat.mem_properDivisors.2\n    \u27e8(Nat.mem_divisors.1 hx).1.trans h,\n      lt_of_le_of_lt (divisor_le hx) (lt_of_le_of_ne (divisor_le (Nat.mem_divisors.2 \u27e8h, hzero\u27e9)) hdiff)\u27e9\n[GOAL]\nn : \u2115\n\u22a2 divisors 0 = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn a\u271d : \u2115\n\u22a2 a\u271d \u2208 divisors 0 \u2194 a\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 properDivisors 0 = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn a\u271d : \u2115\n\u22a2 a\u271d \u2208 properDivisors 0 \u2194 a\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 divisors 1 = {1}\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn a\u271d : \u2115\n\u22a2 a\u271d \u2208 divisors 1 \u2194 a\u271d \u2208 {1}\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 properDivisors 1 = \u2205\n[PROOFSTEP]\nrw [properDivisors, Ico_self, filter_empty]\n[GOAL]\nn m : \u2115\nh : m \u2208 divisors n\n\u22a2 0 < m\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\nn : \u2115\nh : zero \u2208 divisors n\n\u22a2 0 < zero\n[PROOFSTEP]\nrw [mem_divisors, zero_eq, zero_dvd_iff (a := n)] at h \n[GOAL]\ncase zero\nn : \u2115\nh : n = 0 \u2227 n \u2260 0\n\u22a2 0 < zero\n[PROOFSTEP]\ncases h.2 h.1\n[GOAL]\ncase succ\nn n\u271d : \u2115\nh : succ n\u271d \u2208 divisors n\n\u22a2 0 < succ n\u271d\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\nn : \u2115\n\u22a2 1 \u2208 properDivisors n \u2194 1 < n\n[PROOFSTEP]\nrw [mem_properDivisors, and_iff_right (one_dvd _)]\n[GOAL]\nn : \u2115\n\u22a2 divisorsAntidiagonal 0 = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : \u2115\na\u271d : \u2115 \u00d7 \u2115\n\u22a2 a\u271d \u2208 divisorsAntidiagonal 0 \u2194 a\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 divisorsAntidiagonal 1 = {(1, 1)}\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : \u2115\na\u271d : \u2115 \u00d7 \u2115\n\u22a2 a\u271d \u2208 divisorsAntidiagonal 1 \u2194 a\u271d \u2208 {(1, 1)}\n[PROOFSTEP]\nsimp [mul_eq_one, Prod.ext_iff]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 Prod.swap x \u2208 divisorsAntidiagonal n \u2194 x \u2208 divisorsAntidiagonal n\n[PROOFSTEP]\nrw [mem_divisorsAntidiagonal, mem_divisorsAntidiagonal, mul_comm, Prod.swap]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.snd * x.fst = n \u2227 \u00acn = 0 \u2194 x \u2208 divisorsAntidiagonal n\n[PROOFSTEP]\nrw [mem_divisorsAntidiagonal, mul_comm]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\nh : x \u2208 divisorsAntidiagonal n\n\u22a2 x.fst \u2208 divisors n\n[PROOFSTEP]\nrw [mem_divisorsAntidiagonal] at h \n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\nh : x.fst * x.snd = n \u2227 n \u2260 0\n\u22a2 x.fst \u2208 divisors n\n[PROOFSTEP]\nsimp [Dvd.intro _ h.1, h.2]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\nh : x \u2208 divisorsAntidiagonal n\n\u22a2 x.snd \u2208 divisors n\n[PROOFSTEP]\nrw [mem_divisorsAntidiagonal] at h \n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\nh : x.fst * x.snd = n \u2227 n \u2260 0\n\u22a2 x.snd \u2208 divisors n\n[PROOFSTEP]\nsimp [Dvd.intro_left _ h.1, h.2]\n[GOAL]\nn : \u2115\n\u22a2 map (Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115)) (divisorsAntidiagonal n) = divisorsAntidiagonal n\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, Equiv.coe_toEmbedding, Equiv.coe_prodComm, Set.image_swap_eq_preimage_swap]\n[GOAL]\nn : \u2115\n\u22a2 Prod.swap \u207b\u00b9' \u2191(divisorsAntidiagonal n) = \u2191(divisorsAntidiagonal n)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nn : \u2115\nx\u271d : \u2115 \u00d7 \u2115\n\u22a2 x\u271d \u2208 Prod.swap \u207b\u00b9' \u2191(divisorsAntidiagonal n) \u2194 x\u271d \u2208 \u2191(divisorsAntidiagonal n)\n[PROOFSTEP]\nexact swap_mem_divisorsAntidiagonal\n[GOAL]\nn : \u2115\n\u22a2 image Prod.fst (divisorsAntidiagonal n) = divisors n\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn a\u271d : \u2115\n\u22a2 a\u271d \u2208 image Prod.fst (divisorsAntidiagonal n) \u2194 a\u271d \u2208 divisors n\n[PROOFSTEP]\nsimp [Dvd.dvd, @eq_comm _ n (_ * _)]\n[GOAL]\nn : \u2115\n\u22a2 image Prod.snd (divisorsAntidiagonal n) = divisors n\n[PROOFSTEP]\nrw [\u2190 map_swap_divisorsAntidiagonal, map_eq_image, image_image]\n[GOAL]\nn : \u2115\n\u22a2 image (Prod.snd \u2218 \u2191(Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115))) (divisorsAntidiagonal n) = divisors n\n[PROOFSTEP]\nexact image_fst_divisorsAntidiagonal\n[GOAL]\nn : \u2115\n\u22a2 map\n      { toFun := fun d => (d, n / d),\n        inj' :=\n          (_ :\n            \u2200 (p\u2081 p\u2082 : \u2115),\n              (fun d => (d, n / d)) p\u2081 = (fun d => (d, n / d)) p\u2082 \u2192\n                ((fun d => (d, n / d)) p\u2081).fst = ((fun d => (d, n / d)) p\u2082).fst) }\n      (divisors n) =\n    divisorsAntidiagonal n\n[PROOFSTEP]\next \u27e8d, nd\u27e9\n[GOAL]\ncase a.mk\nn d nd : \u2115\n\u22a2 (d, nd) \u2208\n      map\n        { toFun := fun d => (d, n / d),\n          inj' :=\n            (_ :\n              \u2200 (p\u2081 p\u2082 : \u2115),\n                (fun d => (d, n / d)) p\u2081 = (fun d => (d, n / d)) p\u2082 \u2192\n                  ((fun d => (d, n / d)) p\u2081).fst = ((fun d => (d, n / d)) p\u2082).fst) }\n        (divisors n) \u2194\n    (d, nd) \u2208 divisorsAntidiagonal n\n[PROOFSTEP]\nsimp only [mem_map, mem_divisorsAntidiagonal, Function.Embedding.coeFn_mk, mem_divisors, Prod.ext_iff, exists_prop,\n  and_left_comm, exists_eq_left]\n[GOAL]\ncase a.mk\nn d nd : \u2115\n\u22a2 (d \u2223 n \u2227 n \u2260 0) \u2227 n / d = nd \u2194 d * nd = n \u2227 n \u2260 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mk.mp\nn d nd : \u2115\n\u22a2 (d \u2223 n \u2227 n \u2260 0) \u2227 n / d = nd \u2192 d * nd = n \u2227 n \u2260 0\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8k, rfl\u27e9, hn\u27e9, rfl\u27e9\n[GOAL]\ncase a.mk.mp.intro.intro.intro\nd k : \u2115\nhn : d * k \u2260 0\n\u22a2 d * (d * k / d) = d * k \u2227 d * k \u2260 0\n[PROOFSTEP]\nrw [Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt]\n[GOAL]\ncase a.mk.mp.intro.intro.intro\nd k : \u2115\nhn : d * k \u2260 0\n\u22a2 d * k = d * k \u2227 d * k \u2260 0\n[PROOFSTEP]\nexact \u27e8rfl, hn\u27e9\n[GOAL]\ncase a.mk.mpr\nn d nd : \u2115\n\u22a2 d * nd = n \u2227 n \u2260 0 \u2192 (d \u2223 n \u2227 n \u2260 0) \u2227 n / d = nd\n[PROOFSTEP]\nrintro \u27e8rfl, hn\u27e9\n[GOAL]\ncase a.mk.mpr.intro\nd nd : \u2115\nhn : d * nd \u2260 0\n\u22a2 (d \u2223 d * nd \u2227 d * nd \u2260 0) \u2227 d * nd / d = nd\n[PROOFSTEP]\nexact \u27e8\u27e8dvd_mul_right _ _, hn\u27e9, Nat.mul_div_cancel_left _ (left_ne_zero_of_mul hn).bot_lt\u27e9\n[GOAL]\nn : \u2115\n\u22a2 map\n      { toFun := fun d => (n / d, d),\n        inj' :=\n          (_ :\n            \u2200 (p\u2081 p\u2082 : \u2115),\n              (fun d => (n / d, d)) p\u2081 = (fun d => (n / d, d)) p\u2082 \u2192\n                ((fun d => (n / d, d)) p\u2081).snd = ((fun d => (n / d, d)) p\u2082).snd) }\n      (divisors n) =\n    divisorsAntidiagonal n\n[PROOFSTEP]\napply Finset.map_injective (Equiv.prodComm _ _).toEmbedding\n[GOAL]\ncase a\nn : \u2115\n\u22a2 map (Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115))\n      (map\n        { toFun := fun d => (n / d, d),\n          inj' :=\n            (_ :\n              \u2200 (p\u2081 p\u2082 : \u2115),\n                (fun d => (n / d, d)) p\u2081 = (fun d => (n / d, d)) p\u2082 \u2192\n                  ((fun d => (n / d, d)) p\u2081).snd = ((fun d => (n / d, d)) p\u2082).snd) }\n        (divisors n)) =\n    map (Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115)) (divisorsAntidiagonal n)\n[PROOFSTEP]\nrw [map_swap_divisorsAntidiagonal, \u2190 map_div_right_divisors, Finset.map_map]\n[GOAL]\ncase a\nn : \u2115\n\u22a2 map\n      (Function.Embedding.trans\n        { toFun := fun d => (n / d, d),\n          inj' :=\n            (_ :\n              \u2200 (p\u2081 p\u2082 : \u2115),\n                (fun d => (n / d, d)) p\u2081 = (fun d => (n / d, d)) p\u2082 \u2192\n                  ((fun d => (n / d, d)) p\u2081).snd = ((fun d => (n / d, d)) p\u2082).snd) }\n        (Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115)))\n      (divisors n) =\n    map\n      { toFun := fun d => (d, n / d),\n        inj' :=\n          (_ :\n            \u2200 (p\u2081 p\u2082 : \u2115),\n              (fun d => (d, n / d)) p\u2081 = (fun d => (d, n / d)) p\u2082 \u2192\n                ((fun d => (d, n / d)) p\u2081).fst = ((fun d => (d, n / d)) p\u2082).fst) }\n      (divisors n)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 \u2211 i in divisors n, i = \u2211 i in properDivisors n, i + n\n[PROOFSTEP]\nrcases Decidable.eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\n\u22a2 \u2211 i in divisors 0, i = \u2211 i in properDivisors 0, i + 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2211 i in divisors n, i = \u2211 i in properDivisors n, i + n\n[PROOFSTEP]\nrw [\u2190 cons_self_properDivisors hn, Finset.sum_cons, add_comm]\n[GOAL]\nn : \u2115\nh : 0 < n\n\u22a2 Perfect n \u2194 \u2211 i in divisors n, i = 2 * n\n[PROOFSTEP]\nrw [perfect_iff_sum_properDivisors h, sum_divisors_eq_sum_properDivisors_add_self, two_mul]\n[GOAL]\nn : \u2115\nh : 0 < n\n\u22a2 \u2211 i in properDivisors n, i = n \u2194 \u2211 i in properDivisors n, i + n = n + n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115\nh : 0 < n\n\u22a2 \u2211 i in properDivisors n, i = n \u2192 \u2211 i in properDivisors n, i + n = n + n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nn : \u2115\nh : 0 < n\n\u22a2 \u2211 i in properDivisors n, i + n = n + n \u2192 \u2211 i in properDivisors n, i = n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nn : \u2115\nh\u271d : 0 < n\nh : \u2211 i in properDivisors n, i = n\n\u22a2 \u2211 i in properDivisors n, i + n = n + n\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nn : \u2115\nh\u271d : 0 < n\nh : \u2211 i in properDivisors n, i + n = n + n\n\u22a2 \u2211 i in properDivisors n, i = n\n[PROOFSTEP]\napply add_right_cancel h\n[GOAL]\nn p : \u2115\npp : Prime p\nk x : \u2115\n\u22a2 x \u2208 divisors (p ^ k) \u2194 \u2203 j x_1, x = p ^ j\n[PROOFSTEP]\nrw [mem_divisors, Nat.dvd_prime_pow pp, and_iff_left (ne_of_gt (pow_pos pp.pos k))]\n[GOAL]\nn p : \u2115\npp : Prime p\nk x : \u2115\n\u22a2 (\u2203 k_1, k_1 \u2264 k \u2227 x = p ^ k_1) \u2194 \u2203 j x_1, x = p ^ j\n[PROOFSTEP]\nsimp\n[GOAL]\nn p : \u2115\npp : Prime p\n\u22a2 Nat.divisors p = {1, p}\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\na\u271d : \u2115\n\u22a2 a\u271d \u2208 Nat.divisors p \u2194 a\u271d \u2208 {1, p}\n[PROOFSTEP]\nrw [mem_divisors, dvd_prime pp, and_iff_left pp.ne_zero, Finset.mem_insert, Finset.mem_singleton]\n[GOAL]\nn p : \u2115\npp : Prime p\n\u22a2 Nat.properDivisors p = {1}\n[PROOFSTEP]\nrw [\u2190 erase_insert properDivisors.not_self_mem, insert_self_properDivisors pp.ne_zero, pp.divisors, pair_comm,\n  erase_insert fun con => pp.ne_one (mem_singleton.1 con)]\n[GOAL]\nn p : \u2115\npp : Prime p\nk : \u2115\n\u22a2 divisors (p ^ k) = map { toFun := Nat.pow p, inj' := (_ : Function.Injective fun n => p ^ n) } (range (k + 1))\n[PROOFSTEP]\next a\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\n\u22a2 a \u2208 divisors (p ^ k) \u2194 a \u2208 map { toFun := Nat.pow p, inj' := (_ : Function.Injective fun n => p ^ n) } (range (k + 1))\n[PROOFSTEP]\nsimp only [mem_divisors, mem_map, mem_range, lt_succ_iff, Function.Embedding.coeFn_mk, Nat.pow_eq,\n  mem_divisors_prime_pow pp k]\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\n\u22a2 a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nhave := mem_divisors_prime_pow pp k (x := a)\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2208 divisors (p ^ k) \u2194 \u2203 j x, a = p ^ j\n\u22a2 a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrw [mem_divisors] at this \n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\n\u22a2 a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\n\u22a2 (\u2203 j x, a = p ^ j) \u2194 \u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrefine \u27e8?_, ?_\u27e9\n[GOAL]\ncase a.refine_1\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\n\u22a2 (\u2203 j x, a = p ^ j) \u2192 \u2203 a_2, a_2 \u2264 k \u2227 p ^ a_2 = a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase a.refine_1\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\nh : \u2203 j x, a = p ^ j\n\u22a2 \u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrcases h with \u27e8x, hx, hap\u27e9\n[GOAL]\ncase a.refine_1.intro.intro\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\nx : \u2115\nhx : x \u2264 k\nhap : a = p ^ x\n\u22a2 \u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\nx : \u2115\nhx : x \u2264 k\nhap : a = p ^ x\n\u22a2 x \u2264 k \u2227 p ^ x = a\n[PROOFSTEP]\ntauto\n[GOAL]\ncase a.refine_2\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 p ^ k \u2260 0 \u2194 \u2203 j x, a = p ^ j\n\u22a2 (\u2203 a_1, a_1 \u2264 k \u2227 p ^ a_1 = a) \u2192 \u2203 j x, a = p ^ j\n[PROOFSTEP]\ntauto\n[GOAL]\nn : \u2115\ns : Finset \u2115\nhsub : s \u2286 properDivisors n\n\u22a2 \u2211 x in s, x = \u2211 x in properDivisors n, x \u2192 s = properDivisors n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\ns : Finset \u2115\nhsub : s \u2286 properDivisors zero\n\u22a2 \u2211 x in s, x = \u2211 x in properDivisors zero, x \u2192 s = properDivisors zero\n[PROOFSTEP]\nrw [properDivisors_zero, subset_empty] at hsub \n[GOAL]\ncase zero\ns : Finset \u2115\nhsub : s = \u2205\n\u22a2 \u2211 x in s, x = \u2211 x in properDivisors zero, x \u2192 s = properDivisors zero\n[PROOFSTEP]\nsimp [hsub]\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\n\u22a2 \u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d), x \u2192 s = properDivisors (succ n\u271d)\n[PROOFSTEP]\nclassical\nrw [\u2190 sum_sdiff hsub]\nintro h\napply Subset.antisymm hsub\nrw [\u2190 sdiff_eq_empty_iff_subset]\ncontrapose h\nrw [\u2190 Ne.def, \u2190 nonempty_iff_ne_empty] at h \napply ne_of_lt\nrw [\u2190 zero_add (\u2211 x in s, x), \u2190 add_assoc, add_zero]\napply add_lt_add_right\nhave hlt := sum_lt_sum_of_nonempty h fun x hx => pos_of_mem_properDivisors (sdiff_subset _ _ hx)\nsimp only [sum_const_zero] at hlt \napply hlt\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\n\u22a2 \u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d), x \u2192 s = properDivisors (succ n\u271d)\n[PROOFSTEP]\nrw [\u2190 sum_sdiff hsub]\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\n\u22a2 \u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x \u2192 s = properDivisors (succ n\u271d)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : \u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n\u22a2 s = properDivisors (succ n\u271d)\n[PROOFSTEP]\napply Subset.antisymm hsub\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : \u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n\u22a2 properDivisors (succ n\u271d) \u2286 s\n[PROOFSTEP]\nrw [\u2190 sdiff_eq_empty_iff_subset]\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : \u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n\u22a2 properDivisors (succ n\u271d) \\ s = \u2205\n[PROOFSTEP]\ncontrapose h\n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : \u00acproperDivisors (succ n\u271d) \\ s = \u2205\n\u22a2 \u00ac\u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n[PROOFSTEP]\nrw [\u2190 Ne.def, \u2190 nonempty_iff_ne_empty] at h \n[GOAL]\ncase succ\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : Finset.Nonempty (properDivisors (succ n\u271d) \\ s)\n\u22a2 \u00ac\u2211 x in s, x = \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase succ.h\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : Finset.Nonempty (properDivisors (succ n\u271d) \\ s)\n\u22a2 \u2211 x in s, x < \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n[PROOFSTEP]\nrw [\u2190 zero_add (\u2211 x in s, x), \u2190 add_assoc, add_zero]\n[GOAL]\ncase succ.h\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : Finset.Nonempty (properDivisors (succ n\u271d) \\ s)\n\u22a2 0 + \u2211 x in s, x < \u2211 x in properDivisors (succ n\u271d) \\ s, x + \u2211 x in s, x\n[PROOFSTEP]\napply add_lt_add_right\n[GOAL]\ncase succ.h.bc\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : Finset.Nonempty (properDivisors (succ n\u271d) \\ s)\n\u22a2 0 < \u2211 x in properDivisors (succ n\u271d) \\ s, x\n[PROOFSTEP]\nhave hlt := sum_lt_sum_of_nonempty h fun x hx => pos_of_mem_properDivisors (sdiff_subset _ _ hx)\n[GOAL]\ncase succ.h.bc\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : Finset.Nonempty (properDivisors (succ n\u271d) \\ s)\nhlt : \u2211 i in properDivisors (succ n\u271d) \\ s, 0 < \u2211 i in properDivisors (succ n\u271d) \\ s, i\n\u22a2 0 < \u2211 x in properDivisors (succ n\u271d) \\ s, x\n[PROOFSTEP]\nsimp only [sum_const_zero] at hlt \n[GOAL]\ncase succ.h.bc\ns : Finset \u2115\nn\u271d : \u2115\nhsub : s \u2286 properDivisors (succ n\u271d)\nh : Finset.Nonempty (properDivisors (succ n\u271d) \\ s)\nhlt : 0 < \u2211 i in properDivisors (succ n\u271d) \\ s, i\n\u22a2 0 < \u2211 x in properDivisors (succ n\u271d) \\ s, x\n[PROOFSTEP]\napply hlt\n[GOAL]\nn : \u2115\nh : \u2211 x in properDivisors n, x \u2223 n\n\u22a2 \u2211 x in properDivisors n, x = 1 \u2228 \u2211 x in properDivisors n, x = n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nh : \u2211 x in properDivisors zero, x \u2223 zero\n\u22a2 \u2211 x in properDivisors zero, x = 1 \u2228 \u2211 x in properDivisors zero, x = zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\nh : \u2211 x in properDivisors (succ n), x \u2223 succ n\n\u22a2 \u2211 x in properDivisors (succ n), x = 1 \u2228 \u2211 x in properDivisors (succ n), x = succ n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.zero\nh : \u2211 x in properDivisors (succ zero), x \u2223 succ zero\n\u22a2 \u2211 x in properDivisors (succ zero), x = 1 \u2228 \u2211 x in properDivisors (succ zero), x = succ zero\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase succ.zero\nh : \u2211 x in properDivisors (succ zero), x \u2260 1 \u2227 \u2211 x in properDivisors (succ zero), x \u2260 succ zero\n\u22a2 \u00ac\u2211 x in properDivisors (succ zero), x \u2223 succ zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.succ\nn : \u2115\nh : \u2211 x in properDivisors (succ (succ n)), x \u2223 succ (succ n)\n\u22a2 \u2211 x in properDivisors (succ (succ n)), x = 1 \u2228 \u2211 x in properDivisors (succ (succ n)), x = succ (succ n)\n[PROOFSTEP]\nrw [or_iff_not_imp_right]\n[GOAL]\ncase succ.succ\nn : \u2115\nh : \u2211 x in properDivisors (succ (succ n)), x \u2223 succ (succ n)\n\u22a2 \u00ac\u2211 x in properDivisors (succ (succ n)), x = succ (succ n) \u2192 \u2211 x in properDivisors (succ (succ n)), x = 1\n[PROOFSTEP]\nintro ne_n\n[GOAL]\ncase succ.succ\nn : \u2115\nh : \u2211 x in properDivisors (succ (succ n)), x \u2223 succ (succ n)\nne_n : \u00ac\u2211 x in properDivisors (succ (succ n)), x = succ (succ n)\n\u22a2 \u2211 x in properDivisors (succ (succ n)), x = 1\n[PROOFSTEP]\nhave hlt : \u2211 x in n.succ.succ.properDivisors, x < n.succ.succ := lt_of_le_of_ne (Nat.le_of_dvd (Nat.succ_pos _) h) ne_n\n[GOAL]\ncase succ.succ\nn : \u2115\nh : \u2211 x in properDivisors (succ (succ n)), x \u2223 succ (succ n)\nne_n : \u00ac\u2211 x in properDivisors (succ (succ n)), x = succ (succ n)\nhlt : \u2211 x in properDivisors (succ (succ n)), x < succ (succ n)\n\u22a2 \u2211 x in properDivisors (succ (succ n)), x = 1\n[PROOFSTEP]\nsymm\n[GOAL]\ncase succ.succ\nn : \u2115\nh : \u2211 x in properDivisors (succ (succ n)), x \u2223 succ (succ n)\nne_n : \u00ac\u2211 x in properDivisors (succ (succ n)), x = succ (succ n)\nhlt : \u2211 x in properDivisors (succ (succ n)), x < succ (succ n)\n\u22a2 1 = \u2211 x in properDivisors (succ (succ n)), x\n[PROOFSTEP]\nrw [\u2190 mem_singleton,\n  eq_properDivisors_of_subset_of_sum_eq_sum (singleton_subset_iff.2 (mem_properDivisors.2 \u27e8h, hlt\u27e9)) sum_singleton,\n  mem_properDivisors]\n[GOAL]\ncase succ.succ\nn : \u2115\nh : \u2211 x in properDivisors (succ (succ n)), x \u2223 succ (succ n)\nne_n : \u00ac\u2211 x in properDivisors (succ (succ n)), x = succ (succ n)\nhlt : \u2211 x in properDivisors (succ (succ n)), x < succ (succ n)\n\u22a2 1 \u2223 succ (succ n) \u2227 1 < succ (succ n)\n[PROOFSTEP]\nrefine' \u27e8one_dvd _, Nat.succ_lt_succ (Nat.succ_pos _)\u27e9\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\np : \u2115\nf : \u2115 \u2192 \u03b1\nh : Prime p\n\u22a2 \u220f x in Nat.properDivisors p, f x = f 1\n[PROOFSTEP]\nsimp [h.properDivisors]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\np : \u2115\nf : \u2115 \u2192 \u03b1\nh : Prime p\n\u22a2 \u220f x in Nat.divisors p, f x = f p * f 1\n[PROOFSTEP]\nrw [\u2190 cons_self_properDivisors h.ne_zero, prod_cons, h.prod_properDivisors]\n[GOAL]\nn : \u2115\n\u22a2 properDivisors n = {1} \u2194 Prime n\n[PROOFSTEP]\nrefine \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_1\nn : \u2115\n\u22a2 properDivisors n = {1} \u2192 Prime n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine_1\nn : \u2115\nh : properDivisors n = {1}\n\u22a2 Prime n\n[PROOFSTEP]\nrefine' Nat.prime_def_lt''.mpr \u27e8_, fun m hdvd => _\u27e9\n[GOAL]\ncase refine_1.refine'_1\nn : \u2115\nh : properDivisors n = {1}\n\u22a2 2 \u2264 n\n[PROOFSTEP]\nmatch n with\n| 0 => contradiction\n| 1 => contradiction\n| Nat.succ (Nat.succ n) => simp [succ_le_succ]\n[GOAL]\nn : \u2115\nh : properDivisors 0 = {1}\n\u22a2 2 \u2264 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\nh : properDivisors 1 = {1}\n\u22a2 2 \u2264 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn\u271d n : \u2115\nh : properDivisors (succ (succ n)) = {1}\n\u22a2 2 \u2264 succ (succ n)\n[PROOFSTEP]\nsimp [succ_le_succ]\n[GOAL]\ncase refine_1.refine'_2\nn : \u2115\nh : properDivisors n = {1}\nm : \u2115\nhdvd : m \u2223 n\n\u22a2 m = 1 \u2228 m = n\n[PROOFSTEP]\nrw [\u2190 mem_singleton, \u2190 h, mem_properDivisors]\n[GOAL]\ncase refine_1.refine'_2\nn : \u2115\nh : properDivisors n = {1}\nm : \u2115\nhdvd : m \u2223 n\n\u22a2 m \u2223 n \u2227 m < n \u2228 m = n\n[PROOFSTEP]\nhave := Nat.le_of_dvd ?_ hdvd\n[GOAL]\ncase refine_1.refine'_2.refine_2\nn : \u2115\nh : properDivisors n = {1}\nm : \u2115\nhdvd : m \u2223 n\nthis : m \u2264 n\n\u22a2 m \u2223 n \u2227 m < n \u2228 m = n\n[PROOFSTEP]\nsimp [hdvd, this]\n[GOAL]\ncase refine_1.refine'_2.refine_2\nn : \u2115\nh : properDivisors n = {1}\nm : \u2115\nhdvd : m \u2223 n\nthis : m \u2264 n\n\u22a2 m < n \u2228 m = n\n[PROOFSTEP]\nexact (le_iff_eq_or_lt.mp this).symm\n[GOAL]\ncase refine_1.refine'_2.refine_1\nn : \u2115\nh : properDivisors n = {1}\nm : \u2115\nhdvd : m \u2223 n\n\u22a2 0 < n\n[PROOFSTEP]\nby_contra'\n[GOAL]\ncase refine_1.refine'_2.refine_1\nn : \u2115\nh : properDivisors n = {1}\nm : \u2115\nhdvd : m \u2223 n\nthis : n \u2264 0\n\u22a2 False\n[PROOFSTEP]\nsimp [nonpos_iff_eq_zero.mp this, this] at h \n[GOAL]\ncase refine_2\nn : \u2115\n\u22a2 Prime n \u2192 properDivisors n = {1}\n[PROOFSTEP]\nexact fun h => Prime.properDivisors h\n[GOAL]\nn : \u2115\n\u22a2 \u2211 x in properDivisors n, x = 1 \u2194 Prime n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u22a2 \u2211 x in properDivisors zero, x = 1 \u2194 Prime zero\n[PROOFSTEP]\nsimp [Nat.not_prime_zero]\n[GOAL]\ncase succ\nn : \u2115\n\u22a2 \u2211 x in properDivisors (succ n), x = 1 \u2194 Prime (succ n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase succ.zero\n\u22a2 \u2211 x in properDivisors (succ zero), x = 1 \u2194 Prime (succ zero)\n[PROOFSTEP]\nsimp [Nat.not_prime_one]\n[GOAL]\ncase succ.succ\nn\u271d : \u2115\n\u22a2 \u2211 x in properDivisors (succ (succ n\u271d)), x = 1 \u2194 Prime (succ (succ n\u271d))\n[PROOFSTEP]\nrw [\u2190 properDivisors_eq_singleton_one_iff_prime]\n[GOAL]\ncase succ.succ\nn\u271d : \u2115\n\u22a2 \u2211 x in properDivisors (succ (succ n\u271d)), x = 1 \u2194 properDivisors (succ (succ n\u271d)) = {1}\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.symm \u25b8 sum_singleton\u27e9\n[GOAL]\ncase succ.succ\nn\u271d : \u2115\nh : \u2211 x in properDivisors (succ (succ n\u271d)), x = 1\n\u22a2 properDivisors (succ (succ n\u271d)) = {1}\n[PROOFSTEP]\nrw [@eq_comm (Finset \u2115) _ _]\n[GOAL]\ncase succ.succ\nn\u271d : \u2115\nh : \u2211 x in properDivisors (succ (succ n\u271d)), x = 1\n\u22a2 {1} = properDivisors (succ (succ n\u271d))\n[PROOFSTEP]\napply\n  eq_properDivisors_of_subset_of_sum_eq_sum\n    (singleton_subset_iff.2 (one_mem_properDivisors_iff_one_lt.2 (succ_lt_succ (Nat.succ_pos _))))\n    (Eq.trans sum_singleton h.symm)\n[GOAL]\nn p : \u2115\npp : Prime p\nk x : \u2115\n\u22a2 x \u2208 properDivisors (p ^ k) \u2194 \u2203 j x_1, x = p ^ j\n[PROOFSTEP]\nrw [mem_properDivisors, Nat.dvd_prime_pow pp, \u2190 exists_and_right]\n[GOAL]\nn p : \u2115\npp : Prime p\nk x : \u2115\n\u22a2 (\u2203 x_1, (x_1 \u2264 k \u2227 x = p ^ x_1) \u2227 x < p ^ k) \u2194 \u2203 j x_1, x = p ^ j\n[PROOFSTEP]\nsimp only [exists_prop, and_assoc]\n[GOAL]\nn p : \u2115\npp : Prime p\nk x : \u2115\n\u22a2 (\u2203 x_1, x_1 \u2264 k \u2227 x = p ^ x_1 \u2227 x < p ^ k) \u2194 \u2203 j, j < k \u2227 x = p ^ j\n[PROOFSTEP]\napply exists_congr\n[GOAL]\ncase h\nn p : \u2115\npp : Prime p\nk x : \u2115\n\u22a2 \u2200 (a : \u2115), a \u2264 k \u2227 x = p ^ a \u2227 x < p ^ k \u2194 a < k \u2227 x = p ^ a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nn p : \u2115\npp : Prime p\nk x a : \u2115\n\u22a2 a \u2264 k \u2227 x = p ^ a \u2227 x < p ^ k \u2194 a < k \u2227 x = p ^ a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nn p : \u2115\npp : Prime p\nk x a : \u2115\n\u22a2 a \u2264 k \u2227 x = p ^ a \u2227 x < p ^ k \u2192 a < k \u2227 x = p ^ a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nn p : \u2115\npp : Prime p\nk x a : \u2115\n\u22a2 a < k \u2227 x = p ^ a \u2192 a \u2264 k \u2227 x = p ^ a \u2227 x < p ^ k\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mp\nn p : \u2115\npp : Prime p\nk x a : \u2115\nh : a \u2264 k \u2227 x = p ^ a \u2227 x < p ^ k\n\u22a2 a < k \u2227 x = p ^ a\n[PROOFSTEP]\nrcases h with \u27e8_h_left, rfl, h_right\u27e9\n[GOAL]\ncase h.mp.intro.intro\nn p : \u2115\npp : Prime p\nk a : \u2115\n_h_left : a \u2264 k\nh_right : p ^ a < p ^ k\n\u22a2 a < k \u2227 p ^ a = p ^ a\n[PROOFSTEP]\nrw [pow_lt_pow_iff pp.one_lt] at h_right \n[GOAL]\ncase h.mp.intro.intro\nn p : \u2115\npp : Prime p\nk a : \u2115\n_h_left : a \u2264 k\nh_right : a < k\n\u22a2 a < k \u2227 p ^ a = p ^ a\n[PROOFSTEP]\nexact \u27e8h_right, by rfl\u27e9\n[GOAL]\nn p : \u2115\npp : Prime p\nk a : \u2115\n_h_left : a \u2264 k\nh_right : a < k\n\u22a2 p ^ a = p ^ a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.mpr\nn p : \u2115\npp : Prime p\nk x a : \u2115\nh : a < k \u2227 x = p ^ a\n\u22a2 a \u2264 k \u2227 x = p ^ a \u2227 x < p ^ k\n[PROOFSTEP]\nrcases h with \u27e8h_left, rfl\u27e9\n[GOAL]\ncase h.mpr.intro\nn p : \u2115\npp : Prime p\nk a : \u2115\nh_left : a < k\n\u22a2 a \u2264 k \u2227 p ^ a = p ^ a \u2227 p ^ a < p ^ k\n[PROOFSTEP]\nrw [pow_lt_pow_iff pp.one_lt]\n[GOAL]\ncase h.mpr.intro\nn p : \u2115\npp : Prime p\nk a : \u2115\nh_left : a < k\n\u22a2 a \u2264 k \u2227 p ^ a = p ^ a \u2227 a < k\n[PROOFSTEP]\nsimp [h_left, le_of_lt]\n[GOAL]\nn p : \u2115\npp : Prime p\nk : \u2115\n\u22a2 properDivisors (p ^ k) = map { toFun := Nat.pow p, inj' := (_ : Function.Injective fun n => p ^ n) } (range k)\n[PROOFSTEP]\next a\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\n\u22a2 a \u2208 properDivisors (p ^ k) \u2194 a \u2208 map { toFun := Nat.pow p, inj' := (_ : Function.Injective fun n => p ^ n) } (range k)\n[PROOFSTEP]\nsimp only [mem_properDivisors, Nat.isUnit_iff, mem_map, mem_range, Function.Embedding.coeFn_mk, pow_eq]\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\n\u22a2 a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 a_1, a_1 < k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nhave := mem_properDivisors_prime_pow pp k (x := a)\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2208 properDivisors (p ^ k) \u2194 \u2203 j x, a = p ^ j\n\u22a2 a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 a_1, a_1 < k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrw [mem_properDivisors] at this \n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\n\u22a2 a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 a_1, a_1 < k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\n\u22a2 (\u2203 j x, a = p ^ j) \u2194 \u2203 a_1, a_1 < k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrefine \u27e8?_, ?_\u27e9\n[GOAL]\ncase a.refine_1\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\n\u22a2 (\u2203 j x, a = p ^ j) \u2192 \u2203 a_2, a_2 < k \u2227 p ^ a_2 = a\n[PROOFSTEP]\nintro h\n[GOAL]\ncase a.refine_1\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\nh : \u2203 j x, a = p ^ j\n\u22a2 \u2203 a_1, a_1 < k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nrcases h with \u27e8j, hj, hap\u27e9\n[GOAL]\ncase a.refine_1.intro.intro\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\nj : \u2115\nhj : j < k\nhap : a = p ^ j\n\u22a2 \u2203 a_1, a_1 < k \u2227 p ^ a_1 = a\n[PROOFSTEP]\nuse j\n[GOAL]\ncase h\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\nj : \u2115\nhj : j < k\nhap : a = p ^ j\n\u22a2 j < k \u2227 p ^ j = a\n[PROOFSTEP]\ntauto\n[GOAL]\ncase a.refine_2\nn p : \u2115\npp : Prime p\nk a : \u2115\nthis : a \u2223 p ^ k \u2227 a < p ^ k \u2194 \u2203 j x, a = p ^ j\n\u22a2 (\u2203 a_1, a_1 < k \u2227 p ^ a_1 = a) \u2192 \u2203 j x, a = p ^ j\n[PROOFSTEP]\ntauto\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nk p : \u2115\nf : \u2115 \u2192 \u03b1\nh : Prime p\n\u22a2 \u220f x in properDivisors (p ^ k), f x = \u220f x in range k, f (p ^ x)\n[PROOFSTEP]\nsimp [h, properDivisors_prime_pow]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nk p : \u2115\nf : \u2115 \u2192 \u03b1\nh : Prime p\n\u22a2 \u220f x in divisors (p ^ k), f x = \u220f x in range (k + 1), f (p ^ x)\n[PROOFSTEP]\nsimp [h, divisors_prime_pow]\n[GOAL]\nn\u271d : \u2115\nM : Type u_1\ninst\u271d : CommMonoid M\nf : \u2115 \u2192 \u2115 \u2192 M\nn : \u2115\n\u22a2 \u220f i in divisorsAntidiagonal n, f i.fst i.snd = \u220f i in divisors n, f i (n / i)\n[PROOFSTEP]\nrw [\u2190 map_div_right_divisors, Finset.prod_map]\n[GOAL]\nn\u271d : \u2115\nM : Type u_1\ninst\u271d : CommMonoid M\nf : \u2115 \u2192 \u2115 \u2192 M\nn : \u2115\n\u22a2 \u220f x in divisors n,\n      f\n        (\u2191{ toFun := fun d => (d, n / d),\n                inj' :=\n                  (_ :\n                    \u2200 (p\u2081 p\u2082 : \u2115),\n                      (fun d => (d, n / d)) p\u2081 = (fun d => (d, n / d)) p\u2082 \u2192\n                        ((fun d => (d, n / d)) p\u2081).fst = ((fun d => (d, n / d)) p\u2082).fst) }\n            x).fst\n        (\u2191{ toFun := fun d => (d, n / d),\n                inj' :=\n                  (_ :\n                    \u2200 (p\u2081 p\u2082 : \u2115),\n                      (fun d => (d, n / d)) p\u2081 = (fun d => (d, n / d)) p\u2082 \u2192\n                        ((fun d => (d, n / d)) p\u2081).fst = ((fun d => (d, n / d)) p\u2082).fst) }\n            x).snd =\n    \u220f i in divisors n, f i (n / i)\n[PROOFSTEP]\nrfl\n[GOAL]\nn\u271d : \u2115\nM : Type u_1\ninst\u271d : CommMonoid M\nf : \u2115 \u2192 \u2115 \u2192 M\nn : \u2115\n\u22a2 \u220f i in divisorsAntidiagonal n, f i.fst i.snd = \u220f i in divisors n, f (n / i) i\n[PROOFSTEP]\nrw [\u2190 map_swap_divisorsAntidiagonal, Finset.prod_map]\n[GOAL]\nn\u271d : \u2115\nM : Type u_1\ninst\u271d : CommMonoid M\nf : \u2115 \u2192 \u2115 \u2192 M\nn : \u2115\n\u22a2 \u220f x in divisorsAntidiagonal n,\n      f (\u2191(Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115)) x).fst (\u2191(Equiv.toEmbedding (Equiv.prodComm \u2115 \u2115)) x).snd =\n    \u220f i in divisors n, f (n / i) i\n[PROOFSTEP]\nexact prod_divisorsAntidiagonal fun i j => f j i\n[GOAL]\nn\u271d n : \u2115\n\u22a2 List.toFinset (factors n) = filter Prime (divisors n)\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\nn : \u2115\n\u22a2 List.toFinset (factors 0) = filter Prime (divisors 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn\u271d n : \u2115\nhn : n > 0\n\u22a2 List.toFinset (factors n) = filter Prime (divisors n)\n[PROOFSTEP]\next q\n[GOAL]\ncase inr.a\nn\u271d n : \u2115\nhn : n > 0\nq : \u2115\n\u22a2 q \u2208 List.toFinset (factors n) \u2194 q \u2208 filter Prime (divisors n)\n[PROOFSTEP]\nsimpa [hn, hn.ne', mem_factors] using and_comm\n[GOAL]\nn\u271d n : \u2115\n\u22a2 image (fun x => n / x) (divisors n) = divisors n\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nn\u271d n : \u2115\nhn : n = 0\n\u22a2 image (fun x => n / x) (divisors n) = divisors n\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase neg\nn\u271d n : \u2115\nhn : \u00acn = 0\n\u22a2 image (fun x => n / x) (divisors n) = divisors n\n[PROOFSTEP]\next a\n[GOAL]\ncase neg.a\nn\u271d n : \u2115\nhn : \u00acn = 0\na : \u2115\n\u22a2 a \u2208 image (fun x => n / x) (divisors n) \u2194 a \u2208 divisors n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase neg.a.mp\nn\u271d n : \u2115\nhn : \u00acn = 0\na : \u2115\n\u22a2 a \u2208 image (fun x => n / x) (divisors n) \u2192 a \u2208 divisors n\n[PROOFSTEP]\nrw [mem_image]\n[GOAL]\ncase neg.a.mp\nn\u271d n : \u2115\nhn : \u00acn = 0\na : \u2115\n\u22a2 (\u2203 a_1, a_1 \u2208 divisors n \u2227 n / a_1 = a) \u2192 a \u2208 divisors n\n[PROOFSTEP]\nrintro \u27e8x, hx1, hx2\u27e9\n[GOAL]\ncase neg.a.mp.intro.intro\nn\u271d n : \u2115\nhn : \u00acn = 0\na x : \u2115\nhx1 : x \u2208 divisors n\nhx2 : n / x = a\n\u22a2 a \u2208 divisors n\n[PROOFSTEP]\nrw [mem_divisors] at *\n[GOAL]\ncase neg.a.mp.intro.intro\nn\u271d n : \u2115\nhn : \u00acn = 0\na x : \u2115\nhx1 : x \u2223 n \u2227 n \u2260 0\nhx2 : n / x = a\n\u22a2 a \u2223 n \u2227 n \u2260 0\n[PROOFSTEP]\nrefine' \u27e8_, hn\u27e9\n[GOAL]\ncase neg.a.mp.intro.intro\nn\u271d n : \u2115\nhn : \u00acn = 0\na x : \u2115\nhx1 : x \u2223 n \u2227 n \u2260 0\nhx2 : n / x = a\n\u22a2 a \u2223 n\n[PROOFSTEP]\nrw [\u2190 hx2]\n[GOAL]\ncase neg.a.mp.intro.intro\nn\u271d n : \u2115\nhn : \u00acn = 0\na x : \u2115\nhx1 : x \u2223 n \u2227 n \u2260 0\nhx2 : n / x = a\n\u22a2 n / x \u2223 n\n[PROOFSTEP]\nexact div_dvd_of_dvd hx1.1\n[GOAL]\ncase neg.a.mpr\nn\u271d n : \u2115\nhn : \u00acn = 0\na : \u2115\n\u22a2 a \u2208 divisors n \u2192 a \u2208 image (fun x => n / x) (divisors n)\n[PROOFSTEP]\nrw [mem_divisors, mem_image]\n[GOAL]\ncase neg.a.mpr\nn\u271d n : \u2115\nhn : \u00acn = 0\na : \u2115\n\u22a2 a \u2223 n \u2227 n \u2260 0 \u2192 \u2203 a_2, a_2 \u2208 divisors n \u2227 n / a_2 = a\n[PROOFSTEP]\nrintro \u27e8h1, -\u27e9\n[GOAL]\ncase neg.a.mpr.intro\nn\u271d n : \u2115\nhn : \u00acn = 0\na : \u2115\nh1 : a \u2223 n\n\u22a2 \u2203 a_1, a_1 \u2208 divisors n \u2227 n / a_1 = a\n[PROOFSTEP]\nexact \u27e8n / a, mem_divisors.mpr \u27e8div_dvd_of_dvd h1, hn\u27e9, Nat.div_div_self h1 hn\u27e9\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\n\u22a2 \u220f d in divisors n, f (n / d) = Finset.prod (divisors n) f\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : n = 0\n\u22a2 \u220f d in divisors n, f (n / d) = Finset.prod (divisors n) f\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase neg\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : \u00acn = 0\n\u22a2 \u220f d in divisors n, f (n / d) = Finset.prod (divisors n) f\n[PROOFSTEP]\nrw [\u2190 prod_image]\n[GOAL]\ncase neg\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : \u00acn = 0\n\u22a2 \u220f x in image (fun d => n / d) (divisors n), f x = Finset.prod (divisors n) f\n[PROOFSTEP]\nexact prod_congr (image_div_divisors_eq_divisors n) (by simp)\n[GOAL]\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : \u00acn = 0\n\u22a2 \u2200 (x : \u2115), x \u2208 divisors n \u2192 f x = f x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : \u00acn = 0\n\u22a2 \u2200 (x : \u2115), x \u2208 divisors n \u2192 \u2200 (y : \u2115), y \u2208 divisors n \u2192 n / x = n / y \u2192 x = y\n[PROOFSTEP]\nintro x hx y hy h\n[GOAL]\ncase neg\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : \u00acn = 0\nx : \u2115\nhx : x \u2208 divisors n\ny : \u2115\nhy : y \u2208 divisors n\nh : n / x = n / y\n\u22a2 x = y\n[PROOFSTEP]\nrw [mem_divisors] at hx hy \n[GOAL]\ncase neg\nn\u271d : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : \u2115 \u2192 \u03b1\nhn : \u00acn = 0\nx : \u2115\nhx : x \u2223 n \u2227 n \u2260 0\ny : \u2115\nhy : y \u2223 n \u2227 n \u2260 0\nh : n / x = n / y\n\u22a2 x = y\n[PROOFSTEP]\nexact (div_eq_iff_eq_of_dvd_dvd hn hx.1 hy.1).mp h\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Divisors", "llama_tokens": 19185, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424217727027, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.5802737323212265}}
{"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 : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh : \u2200 (b : R), b \u2208 l \u2192 Commute a b\n\u22a2 Commute a (sum l)\n[PROOFSTEP]\ninduction' l with x xs 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 : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nh : \u2200 (b : R), b \u2208 [] \u2192 Commute a b\n\u22a2 Commute a (sum [])\n[PROOFSTEP]\nexact Commute.zero_right _\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 : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nx : R\nxs : List R\nih : (\u2200 (b : R), b \u2208 xs \u2192 Commute a b) \u2192 Commute a (sum xs)\nh : \u2200 (b : R), b \u2208 x :: xs \u2192 Commute a b\n\u22a2 Commute a (sum (x :: xs))\n[PROOFSTEP]\nrw [List.sum_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 : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nx : R\nxs : List R\nih : (\u2200 (b : R), b \u2208 xs \u2192 Commute a b) \u2192 Commute a (sum xs)\nh : \u2200 (b : R), b \u2208 x :: xs \u2192 Commute a b\n\u22a2 Commute a (x + sum xs)\n[PROOFSTEP]\nexact (h _ <| mem_cons_self _ _).add_right (ih fun j hj => h _ <| mem_cons_of_mem _ hj)\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\nh : \u2200 (x : M), x \u2208 l \u2192 x = 1\n\u22a2 prod l = 1\n[PROOFSTEP]\nrw [List.eq_replicate.2 \u27e8_, h\u27e9, prod_replicate, one_pow]\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\nh : \u2200 (x : M), x \u2208 l \u2192 x = 1\n\u22a2 \u2115\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\nh : \u2200 (x : M), x \u2208 l \u2192 x = 1\n\u22a2 length l = ?m.7726\n[PROOFSTEP]\nexact (length l)\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\nh : \u2200 (x : M), x \u2208 l \u2192 x = 1\n\u22a2 length l = length l\n[PROOFSTEP]\nrfl\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\nh : prod l = -1\n\u22a2 -1 \u2208 l\n[PROOFSTEP]\nobtain \u27e8x, h\u2081, h\u2082\u27e9 := exists_mem_ne_one_of_prod_ne_one (ne_of_eq_of_ne h (by decide))\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\nh : prod l = -1\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase intro.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\nl : List \u2124\nh : prod l = -1\nx : \u2124\nh\u2081 : x \u2208 l\nh\u2082 : x \u2260 1\n\u22a2 -1 \u2208 l\n[PROOFSTEP]\nexact\n  Or.resolve_left (Int.isUnit_iff.mp (prod_isUnit_iff.mp (h.symm \u25b8 IsUnit.neg isUnit_one : IsUnit l.prod) x h\u2081)) h\u2082 \u25b8 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\nL : List \u2115\nh : \u2200 (i : \u2115), i \u2208 L \u2192 1 \u2264 i\n\u22a2 length L \u2264 sum L\n[PROOFSTEP]\ninduction' L with j L IH 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\nL : List \u2115\nh\u271d : \u2200 (i : \u2115), i \u2208 L \u2192 1 \u2264 i\nh : \u2200 (i : \u2115), i \u2208 [] \u2192 1 \u2264 i\n\u22a2 length [] \u2264 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\nL\u271d : List \u2115\nh\u271d : \u2200 (i : \u2115), i \u2208 L\u271d \u2192 1 \u2264 i\nj : \u2115\nL : List \u2115\nIH : (\u2200 (i : \u2115), i \u2208 L \u2192 1 \u2264 i) \u2192 length L \u2264 sum L\nh : \u2200 (i : \u2115), i \u2208 j :: L \u2192 1 \u2264 i\n\u22a2 length (j :: L) \u2264 sum (j :: L)\n[PROOFSTEP]\nrw [sum_cons, length, add_comm]\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\nL\u271d : List \u2115\nh\u271d : \u2200 (i : \u2115), i \u2208 L\u271d \u2192 1 \u2264 i\nj : \u2115\nL : List \u2115\nIH : (\u2200 (i : \u2115), i \u2208 L \u2192 1 \u2264 i) \u2192 length L \u2264 sum L\nh : \u2200 (i : \u2115), i \u2208 j :: L \u2192 1 \u2264 i\n\u22a2 1 + length L \u2264 j + sum L\n[PROOFSTEP]\nexact add_le_add (h _ (mem_cons_self _ _)) (IH fun i hi => h i (mem_cons.2 (Or.inr hi)))\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 : CommMonoid M\na : M\nl : List M\nha : a \u2208 l\n\u22a2 a \u2223 prod l\n[PROOFSTEP]\nlet \u27e8s, t, h\u27e9 := mem_split ha\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 : CommMonoid M\na : M\nl : List M\nha : a \u2208 l\ns t : List M\nh : l = s ++ a :: t\n\u22a2 a \u2223 prod l\n[PROOFSTEP]\nrw [h, prod_append, prod_cons, mul_left_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 : CommMonoid M\na : M\nl : List M\nha : a \u2208 l\ns t : List M\nh : l = s ++ a :: t\n\u22a2 a \u2223 a * (prod s * prod t)\n[PROOFSTEP]\nexact dvd_mul_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 : Semiring R\na : R\nl : List R\nh : \u2200 (x : R), x \u2208 l \u2192 a \u2223 x\n\u22a2 a \u2223 sum l\n[PROOFSTEP]\ninduction' l with x 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 : Semiring R\na : R\nl : List R\nh\u271d : \u2200 (x : R), x \u2208 l \u2192 a \u2223 x\nh : \u2200 (x : R), x \u2208 [] \u2192 a \u2223 x\n\u22a2 a \u2223 sum []\n[PROOFSTEP]\nexact dvd_zero _\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 : Semiring R\na : R\nl\u271d : List R\nh\u271d : \u2200 (x : R), x \u2208 l\u271d \u2192 a \u2223 x\nx : R\nl : List R\nih : (\u2200 (x : R), x \u2208 l \u2192 a \u2223 x) \u2192 a \u2223 sum l\nh : \u2200 (x_1 : R), x_1 \u2208 x :: l \u2192 a \u2223 x_1\n\u22a2 a \u2223 sum (x :: l)\n[PROOFSTEP]\nrw [List.sum_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 : Semiring R\na : R\nl\u271d : List R\nh\u271d : \u2200 (x : R), x \u2208 l\u271d \u2192 a \u2223 x\nx : R\nl : List R\nih : (\u2200 (x : R), x \u2208 l \u2192 a \u2223 x) \u2192 a \u2223 sum l\nh : \u2200 (x_1 : R), x_1 \u2208 x :: l \u2192 a \u2223 x_1\n\u22a2 a \u2223 x + sum l\n[PROOFSTEP]\nexact dvd_add (h _ (mem_cons_self _ _)) (ih fun x hx => h x (mem_cons_of_mem _ 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 : CommGroup \u03b1\nl\u2082 : List \u03b1\n\u22a2 alternatingProd ([] ++ l\u2082) = alternatingProd [] * alternatingProd l\u2082 ^ (-1) ^ length []\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 \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 alternatingProd (a :: l\u2081 ++ l\u2082) = alternatingProd (a :: l\u2081) * alternatingProd l\u2082 ^ (-1) ^ length (a :: l\u2081)\n[PROOFSTEP]\nsimp_rw [cons_append, alternatingProd_cons, alternatingProd_append, length_cons, pow_succ, neg_mul, one_mul, zpow_neg, \u2190\n  div_eq_mul_inv, div_div]\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\n\u22a2 alternatingProd (reverse []) = alternatingProd [] ^ (-1) ^ (length [] + 1)\n[PROOFSTEP]\nsimp only [alternatingProd_nil, one_zpow, reverse_nil]\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 (reverse (a :: l)) = alternatingProd (a :: l) ^ (-1) ^ (length (a :: l) + 1)\n[PROOFSTEP]\nsimp_rw [reverse_cons, alternatingProd_append, alternatingProd_reverse, alternatingProd_singleton, alternatingProd_cons,\n  length_reverse, length, pow_succ, neg_mul, one_mul, zpow_neg, inv_inv]\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 l ^ (-1) ^ length l)\u207b\u00b9 * a ^ (-1) ^ length l = (a / alternatingProd l) ^ (-1) ^ length l\n[PROOFSTEP]\nrw [mul_comm, \u2190 div_eq_mul_inv, div_zpow]\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\n\u22a2 \u2200 (l : List M), op (prod l) = prod (reverse (map op l))\n[PROOFSTEP]\nintro l\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\n\u22a2 op (prod l) = prod (reverse (map op l))\n[PROOFSTEP]\ninduction l with\n| nil => rfl\n| cons x xs ih => rw [List.prod_cons, List.map_cons, List.reverse_cons', List.prod_concat, op_mul, 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 : Monoid M\nl : List M\n\u22a2 op (prod l) = prod (reverse (map op l))\n[PROOFSTEP]\ninduction l with\n| nil => rfl\n| cons x xs ih => rw [List.prod_cons, List.map_cons, List.reverse_cons', List.prod_concat, op_mul, 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 : Monoid M\n\u22a2 op (prod []) = prod (reverse (map op []))\n[PROOFSTEP]\n\n| nil => rfl\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\n\u22a2 op (prod []) = prod (reverse (map op []))\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\nx : M\nxs : List M\nih : op (prod xs) = prod (reverse (map op xs))\n\u22a2 op (prod (x :: xs)) = prod (reverse (map op (x :: xs)))\n[PROOFSTEP]\n\n| cons x xs ih => rw [List.prod_cons, List.map_cons, List.reverse_cons', List.prod_concat, op_mul, ih]\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\nx : M\nxs : List M\nih : op (prod xs) = prod (reverse (map op xs))\n\u22a2 op (prod (x :: xs)) = prod (reverse (map op (x :: xs)))\n[PROOFSTEP]\nrw [List.prod_cons, List.map_cons, List.reverse_cons', List.prod_concat, op_mul, 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 : Monoid M\nl : List M\u1d50\u1d52\u1d56\n\u22a2 unop (prod l) = prod (reverse (map unop l))\n[PROOFSTEP]\nrw [\u2190 op_inj, op_unop, MulOpposite.op_list_prod, map_reverse, map_map, reverse_reverse, op_comp_unop, map_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 : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\nF : Type u_9\ninst\u271d : MonoidHomClass F M N\u1d50\u1d52\u1d56\nf : F\nl : List M\n\u22a2 unop (\u2191f (prod l)) = prod (reverse (map (unop \u2218 \u2191f) l))\n[PROOFSTEP]\nrw [map_list_prod f l, MulOpposite.unop_list_prod, List.map_map]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.BigOperators.Lemmas", "llama_tokens": 5440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.7490872131147275, "lm_q1q2_score": 0.5802305085690768}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\np : Palindrome l\n\u22a2 reverse l = l\n[PROOFSTEP]\ninduction p\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\n\u22a2 reverse [] = []\n[PROOFSTEP]\ntry (exact rfl)\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\n\u22a2 reverse [] = []\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase singleton\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\nx\u271d : \u03b1\n\u22a2 reverse [x\u271d] = [x\u271d]\n[PROOFSTEP]\ntry (exact rfl)\n[GOAL]\ncase singleton\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\nx\u271d : \u03b1\n\u22a2 reverse [x\u271d] = [x\u271d]\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase cons_concat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l : List \u03b1\nx\u271d : \u03b1\nl\u271d : List \u03b1\na\u271d : Palindrome l\u271d\na_ih\u271d : reverse l\u271d = l\u271d\n\u22a2 reverse (x\u271d :: (l\u271d ++ [x\u271d])) = x\u271d :: (l\u271d ++ [x\u271d])\n[PROOFSTEP]\ntry (exact rfl)\n[GOAL]\ncase cons_concat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l : List \u03b1\nx\u271d : \u03b1\nl\u271d : List \u03b1\na\u271d : Palindrome l\u271d\na_ih\u271d : reverse l\u271d = l\u271d\n\u22a2 reverse (x\u271d :: (l\u271d ++ [x\u271d])) = x\u271d :: (l\u271d ++ [x\u271d])\n[PROOFSTEP]\nexact rfl\n[GOAL]\ncase cons_concat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l : List \u03b1\nx\u271d : \u03b1\nl\u271d : List \u03b1\na\u271d : Palindrome l\u271d\na_ih\u271d : reverse l\u271d = l\u271d\n\u22a2 reverse (x\u271d :: (l\u271d ++ [x\u271d])) = x\u271d :: (l\u271d ++ [x\u271d])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons_concat\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l : List \u03b1\nx\u271d : \u03b1\nl\u271d : List \u03b1\na\u271d : Palindrome l\u271d\na_ih\u271d : reverse l\u271d = l\u271d\n\u22a2 reverse l\u271d = l\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\n\u22a2 reverse l = l \u2192 Palindrome l\n[PROOFSTEP]\nrefine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\n\u22a2 \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1),\n    (reverse l = l \u2192 Palindrome l) \u2192 reverse (a :: (l ++ [b])) = a :: (l ++ [b]) \u2192 Palindrome (a :: (l ++ [b]))\n[PROOFSTEP]\nintro x l y hp hr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse (x :: (l ++ [y])) = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [y]))\n[PROOFSTEP]\nrw [reverse_cons, reverse_append] at hr \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [y]))\n[PROOFSTEP]\nrw [head_eq_of_cons_eq hr]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [x]))\n[PROOFSTEP]\nhave : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\nthis : Palindrome l\n\u22a2 Palindrome (x :: (l ++ [x]))\n[PROOFSTEP]\nexact Palindrome.cons_concat x this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\n\u22a2 Palindrome (l ++ reverse l)\n[PROOFSTEP]\napply of_reverse_eq\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l : List \u03b1\n\u22a2 reverse (l ++ reverse l) = l ++ reverse l\n[PROOFSTEP]\nrw [reverse_append, reverse_reverse]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\np : Palindrome l\n\u22a2 reverse (map f l) = map f l\n[PROOFSTEP]\nrw [\u2190 map_reverse, p.reverse_eq]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Palindrome", "llama_tokens": 1642, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998508568416, "lm_q2_score": 0.7461389930307512, "lm_q1q2_score": 0.5799737280012769}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : Countable \u03b2\n\u22a2 Countable (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nrcases exists_injective_nat \u03b1 with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 \u2115\nhf : Injective f\n\u22a2 Countable (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nrcases exists_injective_nat \u03b2 with \u27e8g, hg\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 \u2115\nhf : Injective f\ng : \u03b2 \u2192 \u2115\nhg : Injective g\n\u22a2 Countable (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nexact (Equiv.natSumNatEquivNat.injective.comp <| hf.sum_map hg).countable\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : Countable \u03b2\n\u22a2 Countable (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nrcases exists_injective_nat \u03b1 with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 \u2115\nhf : Injective f\n\u22a2 Countable (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nrcases exists_injective_nat \u03b2 with \u27e8g, hg\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : Countable \u03b2\nf : \u03b1 \u2192 \u2115\nhf : Injective f\ng : \u03b2 \u2192 \u2115\nhg : Injective g\n\u22a2 Countable (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nexact (Nat.pairEquiv.injective.comp <| hf.Prod_map hg).countable\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\n\u22a2 Countable (Sigma \u03c0)\n[PROOFSTEP]\nrcases exists_injective_nat \u03b1 with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nf : \u03b1 \u2192 \u2115\nhf : Injective f\n\u22a2 Countable (Sigma \u03c0)\n[PROOFSTEP]\nchoose g hg using fun a => exists_injective_nat (\u03c0 a)\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03c0 : \u03b1 \u2192 Type w\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nf : \u03b1 \u2192 \u2115\nhf : Injective f\ng : (a : \u03b1) \u2192 \u03c0 a \u2192 \u2115\nhg : \u2200 (a : \u03b1), Injective (g a)\n\u22a2 Countable (Sigma \u03c0)\n[PROOFSTEP]\nexact ((Equiv.sigmaEquivProd \u2115 \u2115).injective.comp <| hf.sigma_map hg).countable\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\n\u22a2 Countable ((a : \u03b1) \u2192 \u03c0 a)\n[PROOFSTEP]\nhave : \u2200 n, Countable (Fin n \u2192 \u2115) := by\n  intro n\n  induction' n with n ihn\n  \u00b7 change Countable (Fin 0 \u2192 \u2115); infer_instance\n  \u00b7 haveI := ihn\n    exact Countable.of_equiv (\u2115 \u00d7 (Fin n \u2192 \u2115)) (Equiv.piFinSucc _ _).symm\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\n\u22a2 \u2200 (n : \u2115), Countable (Fin n \u2192 \u2115)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nn : \u2115\n\u22a2 Countable (Fin n \u2192 \u2115)\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\n\u22a2 Countable (Fin Nat.zero \u2192 \u2115)\n[PROOFSTEP]\nchange Countable (Fin 0 \u2192 \u2115)\n[GOAL]\ncase zero\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\n\u22a2 Countable (Fin 0 \u2192 \u2115)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase succ\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nn : \u2115\nihn : Countable (Fin n \u2192 \u2115)\n\u22a2 Countable (Fin (Nat.succ n) \u2192 \u2115)\n[PROOFSTEP]\nhaveI := ihn\n[GOAL]\ncase succ\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nn : \u2115\nihn this : Countable (Fin n \u2192 \u2115)\n\u22a2 Countable (Fin (Nat.succ n) \u2192 \u2115)\n[PROOFSTEP]\nexact Countable.of_equiv (\u2115 \u00d7 (Fin n \u2192 \u2115)) (Equiv.piFinSucc _ _).symm\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nthis : \u2200 (n : \u2115), Countable (Fin n \u2192 \u2115)\n\u22a2 Countable ((a : \u03b1) \u2192 \u03c0 a)\n[PROOFSTEP]\nrcases Finite.exists_equiv_fin \u03b1 with \u27e8n, \u27e8e\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nthis : \u2200 (n : \u2115), Countable (Fin n \u2192 \u2115)\nn : \u2115\ne : \u03b1 \u2243 Fin n\n\u22a2 Countable ((a : \u03b1) \u2192 \u03c0 a)\n[PROOFSTEP]\nhave f := fun a => (nonempty_embedding_nat (\u03c0 a)).some\n[GOAL]\ncase intro.intro\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03c0 : \u03b1 \u2192 Sort w\ninst\u271d\u00b9 : Finite \u03b1\ninst\u271d : \u2200 (a : \u03b1), Countable (\u03c0 a)\nthis : \u2200 (n : \u2115), Countable (Fin n \u2192 \u2115)\nn : \u2115\ne : \u03b1 \u2243 Fin n\nf : (a : \u03b1) \u2192 \u03c0 a \u21aa \u2115\n\u22a2 Countable ((a : \u03b1) \u2192 \u03c0 a)\n[PROOFSTEP]\nexact ((Embedding.piCongrRight f).trans (Equiv.piCongrLeft' _ e).toEmbedding).countable\n", "meta": {"mathlib_filename": "Mathlib.Data.Countable.Basic", "llama_tokens": 2057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680977182186, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.5796493107202867}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Pow M \u2115\nP : Prop\ninst\u271d : Decidable P\na : M\nb c : \u2115\n\u22a2 (a ^ if P then b else c) = if P then a ^ b else a ^ c\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Pow M \u2115\nP : Prop\ninst\u271d : Decidable P\na : M\nb c : \u2115\nh\u271d : P\n\u22a2 a ^ b = a ^ b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Pow M \u2115\nP : Prop\ninst\u271d : Decidable P\na : M\nb c : \u2115\nh\u271d : \u00acP\n\u22a2 a ^ c = a ^ c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Pow M \u2115\nP : Prop\ninst\u271d : Decidable P\na b : M\nc : \u2115\n\u22a2 (if P then a else b) ^ c = if P then a ^ c else b ^ c\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Pow M \u2115\nP : Prop\ninst\u271d : Decidable P\na b : M\nc : \u2115\nh\u271d : P\n\u22a2 a ^ c = a ^ c\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Pow M \u2115\nP : Prop\ninst\u271d : Decidable P\na b : M\nc : \u2115\nh\u271d : \u00acP\n\u22a2 b ^ c = b ^ c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\nn : \u2115\n\u22a2 n \u2022 0 = 0\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\n\u22a2 Nat.zero \u2022 0 = 0\n[PROOFSTEP]\nexact zero_nsmul _\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\nn : \u2115\nih : n \u2022 0 = 0\n\u22a2 Nat.succ n \u2022 0 = 0\n[PROOFSTEP]\nrw [succ_nsmul, ih, add_zero]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\n\u22a2 1 \u2022 a = a\n[PROOFSTEP]\nrw [succ_nsmul, zero_nsmul, add_zero]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\nm n : \u2115\n\u22a2 (m + n) \u2022 a = m \u2022 a + n \u2022 a\n[PROOFSTEP]\ninduction m with\n| zero => rw [Nat.zero_add, zero_nsmul, zero_add]\n| succ m ih => rw [Nat.succ_add, Nat.succ_eq_add_one, succ_nsmul, ih, succ_nsmul, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\nm n : \u2115\n\u22a2 (m + n) \u2022 a = m \u2022 a + n \u2022 a\n[PROOFSTEP]\ninduction m with\n| zero => rw [Nat.zero_add, zero_nsmul, zero_add]\n| succ m ih => rw [Nat.succ_add, Nat.succ_eq_add_one, succ_nsmul, ih, succ_nsmul, add_assoc]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\nn : \u2115\n\u22a2 (Nat.zero + n) \u2022 a = Nat.zero \u2022 a + n \u2022 a\n[PROOFSTEP]\n\n| zero => rw [Nat.zero_add, zero_nsmul, zero_add]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\nn : \u2115\n\u22a2 (Nat.zero + n) \u2022 a = Nat.zero \u2022 a + n \u2022 a\n[PROOFSTEP]\nrw [Nat.zero_add, zero_nsmul, zero_add]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\nn m : \u2115\nih : (m + n) \u2022 a = m \u2022 a + n \u2022 a\n\u22a2 (Nat.succ m + n) \u2022 a = Nat.succ m \u2022 a + n \u2022 a\n[PROOFSTEP]\n\n| succ m ih => rw [Nat.succ_add, Nat.succ_eq_add_one, succ_nsmul, ih, succ_nsmul, add_assoc]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : A\nn m : \u2115\nih : (m + n) \u2022 a = m \u2022 a + n \u2022 a\n\u22a2 (Nat.succ m + n) \u2022 a = Nat.succ m \u2022 a + n \u2022 a\n[PROOFSTEP]\nrw [Nat.succ_add, Nat.succ_eq_add_one, succ_nsmul, ih, succ_nsmul, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\nn : \u2115\n\u22a2 1 ^ n = 1\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\n\u22a2 1 ^ Nat.zero = 1\n[PROOFSTEP]\nexact pow_zero _\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\nn : \u2115\nih : 1 ^ n = 1\n\u22a2 1 ^ Nat.succ n = 1\n[PROOFSTEP]\nrw [pow_succ, ih, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\n\u22a2 a ^ 1 = a\n[PROOFSTEP]\nrw [pow_succ, pow_zero, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\n\u22a2 a ^ 2 = a * a\n[PROOFSTEP]\nrw [pow_succ, pow_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\n\u22a2 a ^ 3 = a * a * a\n[PROOFSTEP]\nrw [pow_succ', pow_two]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\n\u22a2 a ^ 3 = a * (a * a)\n[PROOFSTEP]\nrw [pow_succ, pow_two]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\n\u22a2 a ^ (m + n) = a ^ m * a ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm : \u2115\n\u22a2 a ^ (m + Nat.zero) = a ^ m * a ^ Nat.zero\n[PROOFSTEP]\nrw [Nat.add_zero, pow_zero, mul_one]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\nih : a ^ (m + n) = a ^ m * a ^ n\n\u22a2 a ^ (m + Nat.succ n) = a ^ m * a ^ Nat.succ n\n[PROOFSTEP]\nrw [pow_succ', \u2190 mul_assoc, \u2190 ih, \u2190 pow_succ', Nat.add_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\n\u22a2 a ^ (m * n) = (a ^ m) ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm : \u2115\n\u22a2 a ^ (m * Nat.zero) = (a ^ m) ^ Nat.zero\n[PROOFSTEP]\nrw [Nat.mul_zero, pow_zero, pow_zero]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\nih : a ^ (m * n) = (a ^ m) ^ n\n\u22a2 a ^ (m * Nat.succ n) = (a ^ m) ^ Nat.succ n\n[PROOFSTEP]\nrw [Nat.mul_succ, pow_add, pow_succ', ih]\n  -- Porting note: we are taking the opportunity to swap the names `mul_nsmul` and `mul_nsmul'`\n  -- using #align, so that in mathlib4 they will match the multiplicative ones.\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\n\u22a2 a ^ (m * n) = (a ^ n) ^ m\n[PROOFSTEP]\nrw [Nat.mul_comm, pow_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : Commute a b\nn : \u2115\n\u22a2 (a * b) ^ n = a ^ n * b ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : Commute a b\n\u22a2 (a * b) ^ Nat.zero = a ^ Nat.zero * b ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero, pow_zero, pow_zero, one_mul]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : Commute a b\nn : \u2115\nih : (a * b) ^ n = a ^ n * b ^ n\n\u22a2 (a * b) ^ Nat.succ n = a ^ Nat.succ n * b ^ Nat.succ n\n[PROOFSTEP]\nsimp only [pow_succ, ih, \u2190 mul_assoc, (h.pow_left n).right_comm]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : AddMonoid A\nP : Prop\ninst\u271d : Decidable P\na : M\n\u22a2 (a ^ if P then 1 else 0) = if P then a else 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\n\u22a2 (a ^ m) ^ n = (a ^ n) ^ m\n[PROOFSTEP]\nrw [\u2190 pow_mul, Nat.mul_comm, pow_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\nh : m \u2264 n\n\u22a2 a ^ m * a ^ (n - m) = a ^ n\n[PROOFSTEP]\nrw [\u2190 pow_add, Nat.add_comm, Nat.sub_add_cancel h]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nm n : \u2115\nh : m \u2264 n\n\u22a2 a ^ (n - m) * a ^ m = a ^ n\n[PROOFSTEP]\nrw [\u2190 pow_add, Nat.sub_add_cancel h]\n[GOAL]\n\u03b1 : Type u_1\nM\u271d : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b2 : Monoid M\u271d\ninst\u271d\u00b9 : AddMonoid A\nM : Type u_2\ninst\u271d : Monoid M\nx : M\nm n : \u2115\nh : x ^ n = 1\n\u22a2 x ^ m = x ^ (m % n)\n[PROOFSTEP]\nhave t : x ^ m = x ^ (n * (m / n) + m % n) :=\n  congr_arg (fun a => x ^ a) ((Nat.add_comm _ _).trans (Nat.mod_add_div _ _)).symm\n[GOAL]\n\u03b1 : Type u_1\nM\u271d : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b2 : Monoid M\u271d\ninst\u271d\u00b9 : AddMonoid A\nM : Type u_2\ninst\u271d : Monoid M\nx : M\nm n : \u2115\nh : x ^ n = 1\nt : x ^ m = x ^ (n * (m / n) + m % n)\n\u22a2 x ^ m = x ^ (m % n)\n[PROOFSTEP]\nrw [t, pow_add, pow_mul, h, one_pow, one_mul]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nn : \u2115\n\u22a2 a ^ bit1 n = a ^ n * a ^ n * a\n[PROOFSTEP]\nrw [bit1, pow_succ', pow_bit0]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nn : \u2115\n\u22a2 a ^ bit0 n = (a * a) ^ n\n[PROOFSTEP]\nrw [pow_bit0, (Commute.refl a).mul_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na : M\nn : \u2115\n\u22a2 a ^ bit1 n = (a * a) ^ n * a\n[PROOFSTEP]\nrw [bit1, pow_succ', pow_bit0']\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nn : \u2115\nh : a * b = 1\n\u22a2 a ^ n * b ^ n = 1\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : a * b = 1\n\u22a2 a ^ Nat.zero * b ^ Nat.zero = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : a * b = 1\nn : \u2115\nhn : a ^ n * b ^ n = 1\n\u22a2 a ^ Nat.succ n * b ^ Nat.succ n = 1\n[PROOFSTEP]\ncalc\n  a ^ n.succ * b ^ n.succ = a ^ n * a * (b * b ^ n) := by rw [pow_succ', pow_succ]\n  _ = a ^ n * (a * b) * b ^ n := by simp only [mul_assoc]\n  _ = 1 := by simp [h, hn]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : a * b = 1\nn : \u2115\nhn : a ^ n * b ^ n = 1\n\u22a2 a ^ Nat.succ n * b ^ Nat.succ n = a ^ n * a * (b * b ^ n)\n[PROOFSTEP]\nrw [pow_succ', pow_succ]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : a * b = 1\nn : \u2115\nhn : a ^ n * b ^ n = 1\n\u22a2 a ^ n * a * (b * b ^ n) = a ^ n * (a * b) * b ^ n\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\na b : M\nh : a * b = 1\nn : \u2115\nhn : a ^ n * b ^ n = 1\n\u22a2 a ^ n * (a * b) * b ^ n = 1\n[PROOFSTEP]\nsimp [h, hn]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\nx y : M\nhxy : x \u2223 y\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 x \u2223 y ^ (n + 1)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : Monoid M\ninst\u271d : AddMonoid A\nx y : M\nhxy : x \u2223 y\nn : \u2115\nx\u271d : n + 1 \u2260 0\n\u22a2 x \u2223 y * y ^ n\n[PROOFSTEP]\nexact hxy.mul_right _\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivInvMonoid G\na : G\n\u22a2 a ^ 1 = a\n[PROOFSTEP]\nconvert pow_one a using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivInvMonoid G\na : G\n\u22a2 a ^ 1 = a ^ 1\n[PROOFSTEP]\nexact zpow_ofNat a 1\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivInvMonoid G\na : G\n\u22a2 a ^ 2 = a * a\n[PROOFSTEP]\nconvert pow_two a using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivInvMonoid G\na : G\n\u22a2 a ^ 2 = a ^ 2\n[PROOFSTEP]\nexact zpow_ofNat a 2\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\n\u22a2 a\u207b\u00b9 ^ 0 = (a ^ 0)\u207b\u00b9\n[PROOFSTEP]\nrw [pow_zero, pow_zero, inv_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2115\n\u22a2 a\u207b\u00b9 ^ (n + 1) = (a ^ (n + 1))\u207b\u00b9\n[PROOFSTEP]\nrw [pow_succ', pow_succ, inv_pow _ n, mul_inv_rev]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 1 ^ \u2191n = 1\n[PROOFSTEP]\nrw [zpow_ofNat, one_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 1 ^ Int.negSucc n = 1\n[PROOFSTEP]\nrw [zpow_negSucc, one_pow, inv_one]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\n\u22a2 a ^ (-0) = (a ^ 0)\u207b\u00b9\n[PROOFSTEP]\nchange a ^ (0 : \u2124) = (a ^ (0 : \u2124))\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\n\u22a2 a ^ 0 = (a ^ 0)\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2115\n\u22a2 a ^ (-Int.negSucc n) = (a ^ Int.negSucc n)\u207b\u00b9\n[PROOFSTEP]\nrw [zpow_negSucc, inv_inv, \u2190 zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2115\n\u22a2 a ^ (-Int.negSucc n) = a ^ \u2191(n + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b\u271d a b : \u03b1\n\u22a2 (a * b) ^ (-1) = b ^ (-1) * a ^ (-1)\n[PROOFSTEP]\nsimp only [zpow_neg, zpow_one, mul_inv_rev]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2115\n\u22a2 a\u207b\u00b9 ^ \u2191n = (a ^ \u2191n)\u207b\u00b9\n[PROOFSTEP]\nrw [zpow_ofNat, zpow_ofNat, inv_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2115\n\u22a2 a\u207b\u00b9 ^ Int.negSucc n = (a ^ Int.negSucc n)\u207b\u00b9\n[PROOFSTEP]\nrw [zpow_negSucc, zpow_negSucc, inv_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2124\n\u22a2 a\u207b\u00b9 ^ n = a ^ (-n)\n[PROOFSTEP]\nrw [inv_zpow, zpow_neg]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2115\n\u22a2 (1 / a) ^ n = 1 / a ^ n\n[PROOFSTEP]\nsimp only [one_div, inv_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na\u271d b a : \u03b1\nn : \u2124\n\u22a2 (1 / a) ^ n = 1 / a ^ n\n[PROOFSTEP]\nsimp only [one_div, inv_zpow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na b : \u03b1\nh : Commute a b\nn : \u2115\n\u22a2 (a * b) ^ \u2191n = a ^ \u2191n * b ^ \u2191n\n[PROOFSTEP]\nsimp [zpow_ofNat, h.mul_pow n]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na b : \u03b1\nh : Commute a b\nn : \u2115\n\u22a2 (a * b) ^ Int.negSucc n = a ^ Int.negSucc n * b ^ Int.negSucc n\n[PROOFSTEP]\nsimp [h.mul_pow, (h.pow_pow _ _).eq, mul_inv_rev]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionCommMonoid \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 (a / b) ^ n = a ^ n / b ^ n\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_pow, inv_pow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionCommMonoid \u03b1\na b : \u03b1\nn : \u2124\n\u22a2 (a / b) ^ n = a ^ n / b ^ n\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, mul_zpow, inv_zpow]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : AddGroup B\na : G\nm n : \u2115\nh : n \u2264 m\n\u22a2 a ^ (m - n) * a ^ n = a ^ m\n[PROOFSTEP]\nrw [\u2190 pow_add, Nat.sub_add_cancel h]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\ninst\u271d\u00b9 : AddGroup A\ninst\u271d : AddGroup B\na : G\nm n : \u2115\nh : n \u2264 m\n\u22a2 a\u207b\u00b9 ^ (m - n) = (a ^ m)\u207b\u00b9 * a ^ n\n[PROOFSTEP]\nrw [pow_sub a\u207b\u00b9 h, inv_pow, inv_pow, inv_inv]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Monoid R\na : R\nm n : \u2115\nh : m \u2264 n\n\u22a2 a ^ n = a ^ m * a ^ (n - m)\n[PROOFSTEP]\nrw [\u2190 pow_add, Nat.add_comm, Nat.sub_add_cancel h]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na x y : G\nh : SemiconjBy a x y\nn : \u2115\n\u22a2 SemiconjBy a (x ^ \u2191n) (y ^ \u2191n)\n[PROOFSTEP]\nsimp [zpow_ofNat, h.pow_right n]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na x y : G\nh : SemiconjBy a x y\nn : \u2115\n\u22a2 SemiconjBy a (x ^ Int.negSucc n) (y ^ Int.negSucc n)\n[PROOFSTEP]\nsimp only [zpow_negSucc, inv_right_iff]\n[GOAL]\n\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Group G\na x y : G\nh : SemiconjBy a x y\nn : \u2115\n\u22a2 SemiconjBy a (x ^ (n + 1)) (y ^ (n + 1))\n[PROOFSTEP]\napply pow_right h\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupPower.Basic", "llama_tokens": 10248, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321796478255, "lm_q2_score": 0.7185943985973773, "lm_q1q2_score": 0.5789227716247234}}
{"text": "[GOAL]\nm n\u271d a\u271d b\u271d c d : \u2124\na b n : \u2115\n\u22a2 \u2191a \u2261 \u2191b [ZMOD \u2191n] \u2194 a \u2261 b [MOD n]\n[PROOFSTEP]\nunfold ModEq Nat.ModEq\n[GOAL]\nm n\u271d a\u271d b\u271d c d : \u2124\na b n : \u2115\n\u22a2 \u2191a % \u2191n = \u2191b % \u2191n \u2194 a % n = b % n\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_inj]\n[GOAL]\nm n\u271d a\u271d b\u271d c d : \u2124\na b n : \u2115\n\u22a2 \u2191a % \u2191n = \u2191b % \u2191n \u2194 \u2191(a % n) = \u2191(b % n)\n[PROOFSTEP]\nsimp [coe_nat_mod]\n[GOAL]\nm n a b c d : \u2124\n\u22a2 a \u2261 0 [ZMOD n] \u2194 n \u2223 a\n[PROOFSTEP]\nrw [ModEq, zero_emod, dvd_iff_emod_eq_zero]\n[GOAL]\nm n a b c d : \u2124\n\u22a2 a \u2261 b [ZMOD n] \u2194 n \u2223 b - a\n[PROOFSTEP]\nrw [ModEq, eq_comm]\n[GOAL]\nm n a b c d : \u2124\n\u22a2 b % n = a % n \u2194 n \u2223 b - a\n[PROOFSTEP]\nsimp [emod_eq_emod_iff_emod_sub_eq_zero, dvd_iff_emod_eq_zero]\n[GOAL]\nm n\u271d a\u271d b\u271d c d a b n : \u2124\n\u22a2 a \u2261 b [ZMOD n] \u2194 \u2203 t, b = a + n * t\n[PROOFSTEP]\nrw [modEq_iff_dvd]\n[GOAL]\nm n\u271d a\u271d b\u271d c d a b n : \u2124\n\u22a2 n \u2223 b - a \u2194 \u2203 t, b = a + n * t\n[PROOFSTEP]\nexact exists_congr fun t => sub_eq_iff_eq_add'\n[GOAL]\nm n a b c d : \u2124\n\u22a2 -a \u2261 -b [ZMOD n] \u2194 a \u2261 b [ZMOD n]\n[PROOFSTEP]\nsimp [-sub_neg_eq_add, neg_sub_neg, modEq_iff_dvd, dvd_sub_comm]\n[GOAL]\nm n a b c d : \u2124\n\u22a2 a \u2261 b [ZMOD -n] \u2194 a \u2261 b [ZMOD n]\n[PROOFSTEP]\nsimp [modEq_iff_dvd]\n[GOAL]\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 c * a \u2261 c * b [ZMOD c * n]\n[PROOFSTEP]\nobtain hc | rfl | hc := lt_trichotomy c 0\n[GOAL]\ncase inl\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : c < 0\n\u22a2 c * a \u2261 c * b [ZMOD c * n]\n[PROOFSTEP]\nrw [\u2190 neg_modEq_neg, \u2190 modEq_neg, \u2190 neg_mul, \u2190 neg_mul, \u2190 neg_mul]\n[GOAL]\ncase inl\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : c < 0\n\u22a2 -c * a \u2261 -c * b [ZMOD -c * n]\n[PROOFSTEP]\nsimp only [ModEq, mul_emod_mul_of_pos _ _ (neg_pos.2 hc), h.eq]\n[GOAL]\ncase inr.inl\nm n a b d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 0 * a \u2261 0 * b [ZMOD 0 * n]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\nhc : 0 < c\n\u22a2 c * a \u2261 c * b [ZMOD c * n]\n[PROOFSTEP]\nsimp only [ModEq, mul_emod_mul_of_pos _ _ hc, h.eq]\n[GOAL]\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 a * c \u2261 b * c [ZMOD n * c]\n[PROOFSTEP]\nrw [mul_comm a, mul_comm b, mul_comm n]\n[GOAL]\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 c * a \u2261 c * b [ZMOD c * n]\n[PROOFSTEP]\nexact h.mul_left'\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : c \u2261 d [ZMOD n]\n\u22a2 n \u2223 b + d - (a + c)\n[PROOFSTEP]\nconvert dvd_add h\u2081.dvd h\u2082.dvd using 1\n[GOAL]\ncase h.e'_4\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : c \u2261 d [ZMOD n]\n\u22a2 b + d - (a + c) = b - a + (d - c)\n[PROOFSTEP]\nring\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : a + c \u2261 b + d [ZMOD n]\n\u22a2 d - c = b + d - (a + c) - (b - a)\n[PROOFSTEP]\nring\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : a + c \u2261 b + d [ZMOD n]\nthis : d - c = b + d - (a + c) - (b - a)\n\u22a2 n \u2223 d - c\n[PROOFSTEP]\nrw [this]\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : a + c \u2261 b + d [ZMOD n]\nthis : d - c = b + d - (a + c) - (b - a)\n\u22a2 n \u2223 b + d - (a + c) - (b - a)\n[PROOFSTEP]\nexact dvd_sub h\u2082.dvd h\u2081.dvd\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : c \u2261 d [ZMOD n]\nh\u2082 : a + c \u2261 b + d [ZMOD n]\n\u22a2 a \u2261 b [ZMOD n]\n[PROOFSTEP]\nrw [add_comm a, add_comm b] at h\u2082 \n[GOAL]\nm n a b c d : \u2124\nh\u2081 : c \u2261 d [ZMOD n]\nh\u2082 : c + a \u2261 d + b [ZMOD n]\n\u22a2 a \u2261 b [ZMOD n]\n[PROOFSTEP]\nexact h\u2081.add_left_cancel h\u2082\n[GOAL]\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 a + -a \u2261 b + -b [ZMOD n]\n[PROOFSTEP]\nsimp_rw [\u2190 sub_eq_add_neg, sub_self]\n[GOAL]\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 0 \u2261 0 [ZMOD n]\n[PROOFSTEP]\nrfl\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : c \u2261 d [ZMOD n]\n\u22a2 a - c \u2261 b - d [ZMOD n]\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg]\n[GOAL]\nm n a b c d : \u2124\nh\u2081 : a \u2261 b [ZMOD n]\nh\u2082 : c \u2261 d [ZMOD n]\n\u22a2 a + -c \u2261 b + -d [ZMOD n]\n[PROOFSTEP]\nexact h\u2081.add h\u2082.neg\n[GOAL]\nm\u271d n a b c d : \u2124\nm : \u2115\nh : a \u2261 b [ZMOD n]\n\u22a2 a ^ m \u2261 b ^ m [ZMOD n]\n[PROOFSTEP]\ninduction' m with d hd\n[GOAL]\ncase zero\nm n a b c d : \u2124\nh : a \u2261 b [ZMOD n]\n\u22a2 a ^ Nat.zero \u2261 b ^ Nat.zero [ZMOD n]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nm n a b c d\u271d : \u2124\nh : a \u2261 b [ZMOD n]\nd : \u2115\nhd : a ^ d \u2261 b ^ d [ZMOD n]\n\u22a2 a ^ Nat.succ d \u2261 b ^ Nat.succ d [ZMOD n]\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\nm n a b c d\u271d : \u2124\nh : a \u2261 b [ZMOD n]\nd : \u2115\nhd : a ^ d \u2261 b ^ d [ZMOD n]\n\u22a2 a * a ^ d \u2261 b * b ^ d [ZMOD n]\n[PROOFSTEP]\nexact h.mul hd\n[GOAL]\nm\u271d n a b c d m : \u2124\nh : a \u2261 b [ZMOD m * n]\n\u22a2 a \u2261 b [ZMOD n]\n[PROOFSTEP]\nrw [modEq_iff_dvd] at *\n[GOAL]\nm\u271d n a b c d m : \u2124\nh : m * n \u2223 b - a\n\u22a2 n \u2223 b - a\n[PROOFSTEP]\nexact (dvd_mul_left n m).trans h\n[GOAL]\nm n a b c d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]\n[PROOFSTEP]\nletI d := gcd m c\n[GOAL]\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]\n[PROOFSTEP]\nhave hmd := gcd_dvd_left m c\n[GOAL]\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]\n[PROOFSTEP]\nhave hcd := gcd_dvd_right m c\n[GOAL]\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : a * c \u2261 b * c [ZMOD m]\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)]\n[PROOFSTEP]\nrw [modEq_iff_dvd] at h \u22a2\n  -- porting note: removed `show` due to leanprover-community/mathlib4#3305\n[GOAL]\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191(gcd m c) \u2223 b - a\n[PROOFSTEP]\nrefine Int.dvd_of_dvd_mul_right_of_gcd_one (?_ : m / d \u2223 c / d * (b - a)) ?_\n[GOAL]\ncase refine_1\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191d \u2223 c / \u2191d * (b - a)\n[PROOFSTEP]\nrw [mul_comm, \u2190 Int.mul_ediv_assoc (b - a) hcd, sub_mul]\n[GOAL]\ncase refine_1\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 m / \u2191d \u2223 (b * c - a * c) / \u2191(gcd m c)\n[PROOFSTEP]\nexact Int.ediv_dvd_ediv hmd h\n[GOAL]\ncase refine_2\nm n a b c d\u271d : \u2124\nhm : 0 < m\nh : m \u2223 b * c - a * c\nd : \u2115 := gcd m c\nhmd : \u2191(gcd m c) \u2223 m\nhcd : \u2191(gcd m c) \u2223 c\n\u22a2 gcd (m / \u2191(gcd m c)) (c / \u2191d) = 1\n[PROOFSTEP]\nrw [gcd_div hmd hcd, natAbs_ofNat, Nat.div_self (gcd_pos_of_ne_zero_left c hm.ne')]\n[GOAL]\nm n a b c d : \u2124\nhm : 0 < m\nh : c * a \u2261 c * b [ZMOD m]\n\u22a2 a * c \u2261 b * c [ZMOD m]\n[PROOFSTEP]\nsimpa [mul_comm] using h\n[GOAL]\nm n a b c d : \u2124\nh : a / c \u2261 b / c [ZMOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 a \u2261 b [ZMOD m]\n[PROOFSTEP]\nconvert h.mul_left'\n[GOAL]\ncase h.e'_1\nm n a b c d : \u2124\nh : a / c \u2261 b / c [ZMOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 m = ?m.21777 * (m / c)\n[PROOFSTEP]\nrwa [Int.mul_ediv_cancel']\n[GOAL]\ncase h.e'_2\nm n a b c d : \u2124\nh : a / c \u2261 b / c [ZMOD 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 [Int.mul_ediv_cancel']\n[GOAL]\ncase h.e'_3\nm n a b c d : \u2124\nh : a / c \u2261 b / c [ZMOD 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 [Int.mul_ediv_cancel']\n[GOAL]\nm n a b c d : \u2124\n\u22a2 a \u2261 b [ZMOD 0] \u2194 a = b\n[PROOFSTEP]\nrw [ModEq, emod_zero, emod_zero]\n[GOAL]\nm n a b c d : \u2124\n\u22a2 n \u2223 n + a - a\n[PROOFSTEP]\nsimp\n[GOAL]\nm n a b c d : \u2124\n\u22a2 n \u2223 a + n - a\n[PROOFSTEP]\nsimp\n[GOAL]\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : a \u2261 b [ZMOD m] \u2227 a \u2261 b [ZMOD n]\n\u22a2 a \u2261 b [ZMOD m * n]\n[PROOFSTEP]\nrw [modEq_iff_dvd, modEq_iff_dvd] at h \n[GOAL]\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m \u2223 b - a \u2227 n \u2223 b - a\n\u22a2 a \u2261 b [ZMOD m * n]\n[PROOFSTEP]\nrw [modEq_iff_dvd, \u2190 natAbs_dvd, \u2190 dvd_natAbs, coe_nat_dvd, natAbs_mul]\n[GOAL]\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m \u2223 b - a \u2227 n \u2223 b - a\n\u22a2 natAbs m * natAbs n \u2223 natAbs (b - a)\n[PROOFSTEP]\nrefine' hmn.mul_dvd_of_dvd_of_dvd _ _\n[GOAL]\ncase refine'_1\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m \u2223 b - a \u2227 n \u2223 b - a\n\u22a2 natAbs m \u2223 natAbs (b - a)\n[PROOFSTEP]\nrw [\u2190 coe_nat_dvd, natAbs_dvd, dvd_natAbs]\n[GOAL]\ncase refine'_2\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m \u2223 b - a \u2227 n \u2223 b - a\n\u22a2 natAbs n \u2223 natAbs (b - a)\n[PROOFSTEP]\nrw [\u2190 coe_nat_dvd, natAbs_dvd, dvd_natAbs]\n[GOAL]\ncase refine'_1\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m \u2223 b - a \u2227 n \u2223 b - a\n\u22a2 m \u2223 b - a\n[PROOFSTEP]\ntauto\n[GOAL]\ncase refine'_2\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2124\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m \u2223 b - a \u2227 n \u2223 b - a\n\u22a2 n \u2223 b - a\n[PROOFSTEP]\ntauto\n[GOAL]\nm n a\u271d b\u271d c d : \u2124\na b : \u2115\n\u22a2 \u2191a * Nat.gcdA a b \u2261 \u2191(Nat.gcd a b) [ZMOD \u2191b]\n[PROOFSTEP]\nrw [\u2190 add_zero ((a : \u2124) * _), Nat.gcd_eq_gcd_ab]\n[GOAL]\nm n a\u271d b\u271d c d : \u2124\na b : \u2115\n\u22a2 \u2191a * Nat.gcdA a b + 0 \u2261 \u2191a * Nat.gcdA a b + \u2191b * Nat.gcdB a b [ZMOD \u2191b]\n[PROOFSTEP]\nexact (dvd_mul_right _ _).zero_modEq_int.add_left _\n[GOAL]\nm n\u271d a\u271d b\u271d c\u271d d a b n c : \u2124\nha : a \u2261 b [ZMOD n]\n\u22a2 b + 0 \u2261 b [ZMOD n]\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\nm n\u271d a\u271d b\u271d c\u271d d a b n c : \u2124\nha : a \u2261 b [ZMOD n]\n\u22a2 a - n * c \u2261 b [ZMOD n]\n[PROOFSTEP]\nconvert Int.modEq_add_fac (-c) ha using 1\n[GOAL]\ncase h.e'_2\nm n\u271d a\u271d b\u271d c\u271d d a b n c : \u2124\nha : a \u2261 b [ZMOD n]\n\u22a2 a - n * c = a + n * -c\n[PROOFSTEP]\nring\n[GOAL]\nm n a\u271d b\u271d c d : \u2124\na b : \u2115\nhab : Nat.coprime a b\nhgcd : Nat.gcd a b = 1\n\u22a2 \u2191a * Nat.gcdA a b + \u2191b * Nat.gcdB a b \u2261 1 [ZMOD \u2191b]\n[PROOFSTEP]\nrw [\u2190 Nat.gcd_eq_gcd_ab, hgcd]\n[GOAL]\nm n a\u271d b\u271d c d : \u2124\na b : \u2115\nhab : Nat.coprime a b\nhgcd : Nat.gcd a b = 1\n\u22a2 \u21911 \u2261 1 [ZMOD \u2191b]\n[PROOFSTEP]\nrfl\n[GOAL]\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\n\u22a2 a % b < b\n[PROOFSTEP]\nhave : a % b < |b| := emod_lt _ (ne_of_gt hb)\n[GOAL]\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\nthis : a % b < |b|\n\u22a2 a % b < b\n[PROOFSTEP]\nrwa [abs_of_pos hb] at this \n[GOAL]\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\n\u22a2 a % b \u2261 a [ZMOD b]\n[PROOFSTEP]\nsimp [ModEq]\n[GOAL]\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\nz : \u2124\nhz1 : 0 \u2264 z\nhz2 : z < b\nhz3 : z \u2261 a [ZMOD b]\n\u22a2 \u2191(natAbs z) < b \u2227 \u2191(natAbs z) \u2261 a [ZMOD b]\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\nz : \u2124\nhz1 : 0 \u2264 z\nhz2 : z < b\nhz3 : z \u2261 a [ZMOD b]\n\u22a2 \u2191(natAbs z) < b\n[PROOFSTEP]\nrw [ofNat_natAbs_eq_of_nonneg z hz1]\n[GOAL]\ncase right\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\nz : \u2124\nhz1 : 0 \u2264 z\nhz2 : z < b\nhz3 : z \u2261 a [ZMOD b]\n\u22a2 \u2191(natAbs z) \u2261 a [ZMOD b]\n[PROOFSTEP]\nrw [ofNat_natAbs_eq_of_nonneg z hz1]\n[GOAL]\ncase left\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\nz : \u2124\nhz1 : 0 \u2264 z\nhz2 : z < b\nhz3 : z \u2261 a [ZMOD b]\n\u22a2 z < b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase right\nm n a\u271d b\u271d c d a b : \u2124\nhb : 0 < b\nz : \u2124\nhz1 : 0 \u2264 z\nhz2 : z < b\nhz3 : z \u2261 a [ZMOD b]\n\u22a2 z \u2261 a [ZMOD b]\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.ModEq", "llama_tokens": 6366, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.831143060406073, "lm_q2_score": 0.6959583187272711, "lm_q1q2_score": 0.5784409269420493}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u22a2 #(WType \u03b2) = sum fun a => #(WType \u03b2) ^ #(\u03b2 a)\n[PROOFSTEP]\nsimp only [Cardinal.power_def, \u2190 Cardinal.mk_sigma]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u22a2 #(WType \u03b2) = #((i : \u03b1) \u00d7 (\u03b2 i \u2192 WType \u03b2))\n[PROOFSTEP]\nexact mk_congr (equivSigma \u03b2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u22a2 #(WType \u03b2) \u2264 \u03ba\n[PROOFSTEP]\ninduction' \u03ba using Cardinal.inductionOn with \u03b3\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : (sum fun a => #\u03b3 ^ #(\u03b2 a)) \u2264 #\u03b3\n\u22a2 #(WType \u03b2) \u2264 #\u03b3\n[PROOFSTEP]\nsimp only [Cardinal.power_def, \u2190 Cardinal.mk_sigma, Cardinal.le_def] at h\u03ba \n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : Nonempty ((i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3)\n\u22a2 #(WType \u03b2) \u2264 #\u03b3\n[PROOFSTEP]\ncases' h\u03ba with h\u03ba\n[GOAL]\ncase h.intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\n\u03ba : Cardinal.{u}\nh\u03ba\u271d : (sum fun a => \u03ba ^ #(\u03b2 a)) \u2264 \u03ba\n\u03b3 : Type u\nh\u03ba : (i : \u03b1) \u00d7 (\u03b2 i \u2192 \u03b3) \u21aa \u03b3\n\u22a2 #(WType \u03b2) \u2264 #\u03b3\n[PROOFSTEP]\nexact Cardinal.mk_le_of_injective (elim_injective _ h\u03ba.1 h\u03ba.2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\n\u22a2 IsEmpty \u03b1 \u2192 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nh : IsEmpty \u03b1\n\u22a2 #(WType \u03b2) \u2264 max #\u03b1 \u2135\u2080\n[PROOFSTEP]\nrw [Cardinal.mk_eq_zero (WType \u03b2)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nh : IsEmpty \u03b1\n\u22a2 0 \u2264 max #\u03b1 \u2135\u2080\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nhn : Nonempty \u03b1\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\n\u22a2 Order.succ 0 \u2264 \u2a06 (a : \u03b1), m ^ #(\u03b2 a)\n[PROOFSTEP]\nrw [succ_zero]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nhn : Nonempty \u03b1\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\n\u22a2 1 \u2264 \u2a06 (a : \u03b1), m ^ #(\u03b2 a)\n[PROOFSTEP]\nobtain \u27e8a\u27e9 : Nonempty \u03b1 := hn\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 1 \u2264 \u2a06 (a : \u03b1), m ^ #(\u03b2 a)\n[PROOFSTEP]\nrefine' le_trans _ (le_ciSup (bddAbove_range.{u, u} _) a)\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 1 \u2264 m ^ #(\u03b2 a)\n[PROOFSTEP]\nrw [\u2190 power_zero]\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type u\ninst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a)\nm : Cardinal.{u} := max #\u03b1 \u2135\u2080\na : \u03b1\n\u22a2 ?m.2506 ^ 0 \u2264 m ^ #(\u03b2 a)\n\u03b1 : Type u \u03b2 : \u03b1 \u2192 Type u inst\u271d : \u2200 (a : \u03b1), Finite (\u03b2 a) m : Cardinal.{u} := max #\u03b1 \u2135\u2080 a : \u03b1 \u22a2 Cardinal.{u}\n[PROOFSTEP]\nexact power_le_power_left (pos_iff_ne_zero.1 (aleph0_pos.trans_le (le_max_right _ _))) (zero_le _)\n", "meta": {"mathlib_filename": "Mathlib.Data.W.Cardinal", "llama_tokens": 1400, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8577681158979306, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.5767046948634705}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 (Fin 3 \u2192 R) \u2192\u2097[R] (Fin 3 \u2192 R) \u2192\u2097[R] Fin 3 \u2192 R\n[PROOFSTEP]\napply LinearMap.mk\u2082 R fun a b : Fin 3 \u2192 R => ![a 1 * b 2 - a 2 * b 1, a 2 * b 0 - a 0 * b 2, a 0 * b 1 - a 1 * b 0]\n[GOAL]\ncase H1\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (m\u2081 m\u2082 n : Fin 3 \u2192 R),\n    ![(m\u2081 + m\u2082) 1 * n 2 - (m\u2081 + m\u2082) 2 * n 1, (m\u2081 + m\u2082) 2 * n 0 - (m\u2081 + m\u2082) 0 * n 2,\n        (m\u2081 + m\u2082) 0 * n 1 - (m\u2081 + m\u2082) 1 * n 0] =\n      ![m\u2081 1 * n 2 - m\u2081 2 * n 1, m\u2081 2 * n 0 - m\u2081 0 * n 2, m\u2081 0 * n 1 - m\u2081 1 * n 0] +\n        ![m\u2082 1 * n 2 - m\u2082 2 * n 1, m\u2082 2 * n 0 - m\u2082 0 * n 2, m\u2082 0 * n 1 - m\u2082 1 * n 0]\n[PROOFSTEP]\nintros\n[GOAL]\ncase H1\nR : Type u_1\ninst\u271d : CommRing R\nm\u2081\u271d m\u2082\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 ![(m\u2081\u271d + m\u2082\u271d) 1 * n\u271d 2 - (m\u2081\u271d + m\u2082\u271d) 2 * n\u271d 1, (m\u2081\u271d + m\u2082\u271d) 2 * n\u271d 0 - (m\u2081\u271d + m\u2082\u271d) 0 * n\u271d 2,\n      (m\u2081\u271d + m\u2082\u271d) 0 * n\u271d 1 - (m\u2081\u271d + m\u2082\u271d) 1 * n\u271d 0] =\n    ![m\u2081\u271d 1 * n\u271d 2 - m\u2081\u271d 2 * n\u271d 1, m\u2081\u271d 2 * n\u271d 0 - m\u2081\u271d 0 * n\u271d 2, m\u2081\u271d 0 * n\u271d 1 - m\u2081\u271d 1 * n\u271d 0] +\n      ![m\u2082\u271d 1 * n\u271d 2 - m\u2082\u271d 2 * n\u271d 1, m\u2082\u271d 2 * n\u271d 0 - m\u2082\u271d 0 * n\u271d 2, m\u2082\u271d 0 * n\u271d 1 - m\u2082\u271d 1 * n\u271d 0]\n[PROOFSTEP]\nsimp_rw [vec3_add, Pi.add_apply]\n[GOAL]\ncase H1\nR : Type u_1\ninst\u271d : CommRing R\nm\u2081\u271d m\u2082\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 ![(m\u2081\u271d 1 + m\u2082\u271d 1) * n\u271d 2 - (m\u2081\u271d 2 + m\u2082\u271d 2) * n\u271d 1, (m\u2081\u271d 2 + m\u2082\u271d 2) * n\u271d 0 - (m\u2081\u271d 0 + m\u2082\u271d 0) * n\u271d 2,\n      (m\u2081\u271d 0 + m\u2082\u271d 0) * n\u271d 1 - (m\u2081\u271d 1 + m\u2082\u271d 1) * n\u271d 0] =\n    ![m\u2081\u271d 1 * n\u271d 2 - m\u2081\u271d 2 * n\u271d 1 + (m\u2082\u271d 1 * n\u271d 2 - m\u2082\u271d 2 * n\u271d 1),\n      m\u2081\u271d 2 * n\u271d 0 - m\u2081\u271d 0 * n\u271d 2 + (m\u2082\u271d 2 * n\u271d 0 - m\u2082\u271d 0 * n\u271d 2),\n      m\u2081\u271d 0 * n\u271d 1 - m\u2081\u271d 1 * n\u271d 0 + (m\u2082\u271d 0 * n\u271d 1 - m\u2082\u271d 1 * n\u271d 0)]\n[PROOFSTEP]\napply vec3_eq\n[GOAL]\ncase H1.h\u2080\nR : Type u_1\ninst\u271d : CommRing R\nm\u2081\u271d m\u2082\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 (m\u2081\u271d 1 + m\u2082\u271d 1) * n\u271d 2 - (m\u2081\u271d 2 + m\u2082\u271d 2) * n\u271d 1 = m\u2081\u271d 1 * n\u271d 2 - m\u2081\u271d 2 * n\u271d 1 + (m\u2082\u271d 1 * n\u271d 2 - m\u2082\u271d 2 * n\u271d 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase H1.h\u2081\nR : Type u_1\ninst\u271d : CommRing R\nm\u2081\u271d m\u2082\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 (m\u2081\u271d 2 + m\u2082\u271d 2) * n\u271d 0 - (m\u2081\u271d 0 + m\u2082\u271d 0) * n\u271d 2 = m\u2081\u271d 2 * n\u271d 0 - m\u2081\u271d 0 * n\u271d 2 + (m\u2082\u271d 2 * n\u271d 0 - m\u2082\u271d 0 * n\u271d 2)\n[PROOFSTEP]\nring\n[GOAL]\ncase H1.h\u2082\nR : Type u_1\ninst\u271d : CommRing R\nm\u2081\u271d m\u2082\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 (m\u2081\u271d 0 + m\u2082\u271d 0) * n\u271d 1 - (m\u2081\u271d 1 + m\u2082\u271d 1) * n\u271d 0 = m\u2081\u271d 0 * n\u271d 1 - m\u2081\u271d 1 * n\u271d 0 + (m\u2082\u271d 0 * n\u271d 1 - m\u2082\u271d 1 * n\u271d 0)\n[PROOFSTEP]\nring\n[GOAL]\ncase H2\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (c : R) (m n : Fin 3 \u2192 R),\n    ![(c \u2022 m) 1 * n 2 - (c \u2022 m) 2 * n 1, (c \u2022 m) 2 * n 0 - (c \u2022 m) 0 * n 2, (c \u2022 m) 0 * n 1 - (c \u2022 m) 1 * n 0] =\n      c \u2022 ![m 1 * n 2 - m 2 * n 1, m 2 * n 0 - m 0 * n 2, m 0 * n 1 - m 1 * n 0]\n[PROOFSTEP]\nintros\n[GOAL]\ncase H2\nR : Type u_1\ninst\u271d : CommRing R\nc\u271d : R\nm\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 ![(c\u271d \u2022 m\u271d) 1 * n\u271d 2 - (c\u271d \u2022 m\u271d) 2 * n\u271d 1, (c\u271d \u2022 m\u271d) 2 * n\u271d 0 - (c\u271d \u2022 m\u271d) 0 * n\u271d 2,\n      (c\u271d \u2022 m\u271d) 0 * n\u271d 1 - (c\u271d \u2022 m\u271d) 1 * n\u271d 0] =\n    c\u271d \u2022 ![m\u271d 1 * n\u271d 2 - m\u271d 2 * n\u271d 1, m\u271d 2 * n\u271d 0 - m\u271d 0 * n\u271d 2, m\u271d 0 * n\u271d 1 - m\u271d 1 * n\u271d 0]\n[PROOFSTEP]\nsimp_rw [smul_vec3, Pi.smul_apply, smul_sub, smul_mul_assoc]\n[GOAL]\ncase H3\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (m n\u2081 n\u2082 : Fin 3 \u2192 R),\n    ![m 1 * (n\u2081 + n\u2082) 2 - m 2 * (n\u2081 + n\u2082) 1, m 2 * (n\u2081 + n\u2082) 0 - m 0 * (n\u2081 + n\u2082) 2,\n        m 0 * (n\u2081 + n\u2082) 1 - m 1 * (n\u2081 + n\u2082) 0] =\n      ![m 1 * n\u2081 2 - m 2 * n\u2081 1, m 2 * n\u2081 0 - m 0 * n\u2081 2, m 0 * n\u2081 1 - m 1 * n\u2081 0] +\n        ![m 1 * n\u2082 2 - m 2 * n\u2082 1, m 2 * n\u2082 0 - m 0 * n\u2082 2, m 0 * n\u2082 1 - m 1 * n\u2082 0]\n[PROOFSTEP]\nintros\n[GOAL]\ncase H3\nR : Type u_1\ninst\u271d : CommRing R\nm\u271d n\u2081\u271d n\u2082\u271d : Fin 3 \u2192 R\n\u22a2 ![m\u271d 1 * (n\u2081\u271d + n\u2082\u271d) 2 - m\u271d 2 * (n\u2081\u271d + n\u2082\u271d) 1, m\u271d 2 * (n\u2081\u271d + n\u2082\u271d) 0 - m\u271d 0 * (n\u2081\u271d + n\u2082\u271d) 2,\n      m\u271d 0 * (n\u2081\u271d + n\u2082\u271d) 1 - m\u271d 1 * (n\u2081\u271d + n\u2082\u271d) 0] =\n    ![m\u271d 1 * n\u2081\u271d 2 - m\u271d 2 * n\u2081\u271d 1, m\u271d 2 * n\u2081\u271d 0 - m\u271d 0 * n\u2081\u271d 2, m\u271d 0 * n\u2081\u271d 1 - m\u271d 1 * n\u2081\u271d 0] +\n      ![m\u271d 1 * n\u2082\u271d 2 - m\u271d 2 * n\u2082\u271d 1, m\u271d 2 * n\u2082\u271d 0 - m\u271d 0 * n\u2082\u271d 2, m\u271d 0 * n\u2082\u271d 1 - m\u271d 1 * n\u2082\u271d 0]\n[PROOFSTEP]\nsimp_rw [vec3_add, Pi.add_apply]\n[GOAL]\ncase H3\nR : Type u_1\ninst\u271d : CommRing R\nm\u271d n\u2081\u271d n\u2082\u271d : Fin 3 \u2192 R\n\u22a2 ![m\u271d 1 * (n\u2081\u271d 2 + n\u2082\u271d 2) - m\u271d 2 * (n\u2081\u271d 1 + n\u2082\u271d 1), m\u271d 2 * (n\u2081\u271d 0 + n\u2082\u271d 0) - m\u271d 0 * (n\u2081\u271d 2 + n\u2082\u271d 2),\n      m\u271d 0 * (n\u2081\u271d 1 + n\u2082\u271d 1) - m\u271d 1 * (n\u2081\u271d 0 + n\u2082\u271d 0)] =\n    ![m\u271d 1 * n\u2081\u271d 2 - m\u271d 2 * n\u2081\u271d 1 + (m\u271d 1 * n\u2082\u271d 2 - m\u271d 2 * n\u2082\u271d 1),\n      m\u271d 2 * n\u2081\u271d 0 - m\u271d 0 * n\u2081\u271d 2 + (m\u271d 2 * n\u2082\u271d 0 - m\u271d 0 * n\u2082\u271d 2),\n      m\u271d 0 * n\u2081\u271d 1 - m\u271d 1 * n\u2081\u271d 0 + (m\u271d 0 * n\u2082\u271d 1 - m\u271d 1 * n\u2082\u271d 0)]\n[PROOFSTEP]\napply vec3_eq\n[GOAL]\ncase H3.h\u2080\nR : Type u_1\ninst\u271d : CommRing R\nm\u271d n\u2081\u271d n\u2082\u271d : Fin 3 \u2192 R\n\u22a2 m\u271d 1 * (n\u2081\u271d 2 + n\u2082\u271d 2) - m\u271d 2 * (n\u2081\u271d 1 + n\u2082\u271d 1) = m\u271d 1 * n\u2081\u271d 2 - m\u271d 2 * n\u2081\u271d 1 + (m\u271d 1 * n\u2082\u271d 2 - m\u271d 2 * n\u2082\u271d 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase H3.h\u2081\nR : Type u_1\ninst\u271d : CommRing R\nm\u271d n\u2081\u271d n\u2082\u271d : Fin 3 \u2192 R\n\u22a2 m\u271d 2 * (n\u2081\u271d 0 + n\u2082\u271d 0) - m\u271d 0 * (n\u2081\u271d 2 + n\u2082\u271d 2) = m\u271d 2 * n\u2081\u271d 0 - m\u271d 0 * n\u2081\u271d 2 + (m\u271d 2 * n\u2082\u271d 0 - m\u271d 0 * n\u2082\u271d 2)\n[PROOFSTEP]\nring\n[GOAL]\ncase H3.h\u2082\nR : Type u_1\ninst\u271d : CommRing R\nm\u271d n\u2081\u271d n\u2082\u271d : Fin 3 \u2192 R\n\u22a2 m\u271d 0 * (n\u2081\u271d 1 + n\u2082\u271d 1) - m\u271d 1 * (n\u2081\u271d 0 + n\u2082\u271d 0) = m\u271d 0 * n\u2081\u271d 1 - m\u271d 1 * n\u2081\u271d 0 + (m\u271d 0 * n\u2082\u271d 1 - m\u271d 1 * n\u2082\u271d 0)\n[PROOFSTEP]\nring\n[GOAL]\ncase H4\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2200 (c : R) (m n : Fin 3 \u2192 R),\n    ![m 1 * (c \u2022 n) 2 - m 2 * (c \u2022 n) 1, m 2 * (c \u2022 n) 0 - m 0 * (c \u2022 n) 2, m 0 * (c \u2022 n) 1 - m 1 * (c \u2022 n) 0] =\n      c \u2022 ![m 1 * n 2 - m 2 * n 1, m 2 * n 0 - m 0 * n 2, m 0 * n 1 - m 1 * n 0]\n[PROOFSTEP]\nintros\n[GOAL]\ncase H4\nR : Type u_1\ninst\u271d : CommRing R\nc\u271d : R\nm\u271d n\u271d : Fin 3 \u2192 R\n\u22a2 ![m\u271d 1 * (c\u271d \u2022 n\u271d) 2 - m\u271d 2 * (c\u271d \u2022 n\u271d) 1, m\u271d 2 * (c\u271d \u2022 n\u271d) 0 - m\u271d 0 * (c\u271d \u2022 n\u271d) 2,\n      m\u271d 0 * (c\u271d \u2022 n\u271d) 1 - m\u271d 1 * (c\u271d \u2022 n\u271d) 0] =\n    c\u271d \u2022 ![m\u271d 1 * n\u271d 2 - m\u271d 2 * n\u271d 1, m\u271d 2 * n\u271d 0 - m\u271d 0 * n\u271d 2, m\u271d 0 * n\u271d 1 - m\u271d 1 * n\u271d 0]\n[PROOFSTEP]\nsimp_rw [smul_vec3, Pi.smul_apply, smul_sub, mul_smul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv w : Fin 3 \u2192 R\n\u22a2 -\u2191(\u2191crossProduct v) w = \u2191(\u2191crossProduct w) v\n[PROOFSTEP]\nsimp [cross_apply, mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv w : Fin 3 \u2192 R\n\u22a2 \u2191(\u2191crossProduct v) w + \u2191(\u2191crossProduct w) v = 0\n[PROOFSTEP]\nrw [add_eq_zero_iff_eq_neg, cross_anticomm]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv : Fin 3 \u2192 R\n\u22a2 \u2191(\u2191crossProduct v) v = 0\n[PROOFSTEP]\nsimp_rw [cross_apply, mul_comm, cons_eq_zero_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv : Fin 3 \u2192 R\n\u22a2 v 1 * v 2 - v 1 * v 2 = 0 \u2227 v 0 * v 2 - v 0 * v 2 = 0 \u2227 v 0 * v 1 - v 0 * v 1 = 0 \u2227 ![] = 0\n[PROOFSTEP]\nexact \u27e8sub_self _, sub_self _, sub_self _, zero_empty.symm\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv w : Fin 3 \u2192 R\n\u22a2 v \u2b1d\u1d65 \u2191(\u2191crossProduct v) w = 0\n[PROOFSTEP]\nrw [cross_apply, vec3_dotProduct]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv w : Fin 3 \u2192 R\n\u22a2 v 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n        v 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n      v 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n    0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv w : Fin 3 \u2192 R\n\u22a2 v 0 * (v 1 * w 2 - v 2 * w 1) + v 1 * (v 2 * w 0 - v 0 * w 2) + v 2 * (v 0 * w 1 - v 1 * w 0) = 0\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nv w : Fin 3 \u2192 R\n\u22a2 w \u2b1d\u1d65 \u2191(\u2191crossProduct v) w = 0\n[PROOFSTEP]\nrw [\u2190 cross_anticomm, Matrix.dotProduct_neg, dot_self_cross, neg_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u \u2b1d\u1d65 \u2191(\u2191crossProduct v) w = v \u2b1d\u1d65 \u2191(\u2191crossProduct w) u\n[PROOFSTEP]\nsimp_rw [cross_apply, vec3_dotProduct]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n        u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n      u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n    v 0 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 0 +\n        v 1 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 1 +\n      v 2 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * (v 1 * w 2 - v 2 * w 1) + u 1 * (v 2 * w 0 - v 0 * w 2) + u 2 * (v 0 * w 1 - v 1 * w 0) =\n    v 0 * (w 1 * u 2 - w 2 * u 1) + v 1 * (w 2 * u 0 - w 0 * u 2) + v 2 * (w 0 * u 1 - w 1 * u 0)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u \u2b1d\u1d65 \u2191(\u2191crossProduct v) w = det ![u, v, w]\n[PROOFSTEP]\nrw [vec3_dotProduct, cross_apply, det_fin_three]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n        u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n      u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n    vecCons u ![v, w] 0 0 * vecCons u ![v, w] 1 1 * vecCons u ![v, w] 2 2 -\n              vecCons u ![v, w] 0 0 * vecCons u ![v, w] 1 2 * vecCons u ![v, w] 2 1 -\n            vecCons u ![v, w] 0 1 * vecCons u ![v, w] 1 0 * vecCons u ![v, w] 2 2 +\n          vecCons u ![v, w] 0 1 * vecCons u ![v, w] 1 2 * vecCons u ![v, w] 2 0 +\n        vecCons u ![v, w] 0 2 * vecCons u ![v, w] 1 0 * vecCons u ![v, w] 2 1 -\n      vecCons u ![v, w] 0 2 * vecCons u ![v, w] 1 1 * vecCons u ![v, w] 2 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * (v 1 * w 2 - v 2 * w 1) + u 1 * (v 2 * w 0 - v 0 * w 2) + u 2 * (v 0 * w 1 - v 1 * w 0) =\n    u 0 * v 1 * w 2 - u 0 * v 2 * w 1 - u 1 * v 0 * w 2 + u 1 * v 2 * w 0 + u 2 * v 0 * w 1 - u 2 * v 1 * w 0\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w x : Fin 3 \u2192 R\n\u22a2 \u2191(\u2191crossProduct u) v \u2b1d\u1d65 \u2191(\u2191crossProduct w) x = u \u2b1d\u1d65 w * v \u2b1d\u1d65 x - u \u2b1d\u1d65 x * v \u2b1d\u1d65 w\n[PROOFSTEP]\nsimp_rw [cross_apply, vec3_dotProduct]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w x : Fin 3 \u2192 R\n\u22a2 vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 *\n          vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 0 +\n        vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 *\n          vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 1 +\n      vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 *\n        vecCons (w 1 * x 2 - w 2 * x 1) ![w 2 * x 0 - w 0 * x 2, w 0 * x 1 - w 1 * x 0] 2 =\n    (u 0 * w 0 + u 1 * w 1 + u 2 * w 2) * (v 0 * x 0 + v 1 * x 1 + v 2 * x 2) -\n      (u 0 * x 0 + u 1 * x 1 + u 2 * x 2) * (v 0 * w 0 + v 1 * w 1 + v 2 * w 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w x : Fin 3 \u2192 R\n\u22a2 (u 1 * v 2 - u 2 * v 1) * (w 1 * x 2 - w 2 * x 1) + (u 2 * v 0 - u 0 * v 2) * (w 2 * x 0 - w 0 * x 2) +\n      (u 0 * v 1 - u 1 * v 0) * (w 0 * x 1 - w 1 * x 0) =\n    (u 0 * w 0 + u 1 * w 1 + u 2 * w 2) * (v 0 * x 0 + v 1 * x 1 + v 2 * x 2) -\n      (u 0 * x 0 + u 1 * x 1 + u 2 * x 2) * (v 0 * w 0 + v 1 * w 1 + v 2 * w 2)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 \u2191(\u2191crossProduct u) (\u2191(\u2191crossProduct v) w) =\n    \u2191(\u2191crossProduct (\u2191(\u2191crossProduct u) v)) w + \u2191(\u2191crossProduct v) (\u2191(\u2191crossProduct u) w)\n[PROOFSTEP]\nsimp_rw [cross_apply, vec3_add]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 ![u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -\n        u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1,\n      u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -\n        u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2,\n      u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -\n        u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0] =\n    ![vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -\n          vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1 +\n        (v 1 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 2 -\n          v 2 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 1),\n      vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -\n          vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2 +\n        (v 2 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 0 -\n          v 0 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 2),\n      vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -\n          vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0 +\n        (v 0 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 1 -\n          v 1 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 0)]\n[PROOFSTEP]\napply vec3_eq\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 -\n      u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 =\n    vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 2 -\n        vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 1 +\n      (v 1 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 2 -\n        v 2 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 -\n      u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n    vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 2 * w 0 -\n        vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 2 +\n      (v 2 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 0 -\n        v 0 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\u2082\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 -\n      u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 =\n    vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 0 * w 1 -\n        vecCons (u 1 * v 2 - u 2 * v 1) ![u 2 * v 0 - u 0 * v 2, u 0 * v 1 - u 1 * v 0] 1 * w 0 +\n      (v 0 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 1 -\n        v 1 * vecCons (u 1 * w 2 - u 2 * w 1) ![u 2 * w 0 - u 0 * w 2, u 0 * w 1 - u 1 * w 0] 0)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 1 * (v 0 * w 1 - v 1 * w 0) - u 2 * (v 2 * w 0 - v 0 * w 2) =\n    (u 2 * v 0 - u 0 * v 2) * w 2 - (u 0 * v 1 - u 1 * v 0) * w 1 +\n      (v 1 * (u 0 * w 1 - u 1 * w 0) - v 2 * (u 2 * w 0 - u 0 * w 2))\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 2 * (v 1 * w 2 - v 2 * w 1) - u 0 * (v 0 * w 1 - v 1 * w 0) =\n    (u 0 * v 1 - u 1 * v 0) * w 0 - (u 1 * v 2 - u 2 * v 1) * w 2 +\n      (v 2 * (u 1 * w 2 - u 2 * w 1) - v 0 * (u 0 * w 1 - u 1 * w 0))\n[PROOFSTEP]\nring\n[GOAL]\ncase h\u2082\nR : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * (v 2 * w 0 - v 0 * w 2) - u 1 * (v 1 * w 2 - v 2 * w 1) =\n    (u 1 * v 2 - u 2 * v 1) * w 1 - (u 2 * v 0 - u 0 * v 2) * w 0 +\n      (v 0 * (u 2 * w 0 - u 0 * w 2) - v 1 * (u 1 * w 2 - u 2 * w 1))\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.CrossProduct", "llama_tokens": 10557, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8577680977182187, "lm_q2_score": 0.6723316860482763, "lm_q1q2_score": 0.5767046713773126}}
{"text": "[GOAL]\nX : Type u_1\nt : TopologicalSpace X\n\u22a2 t = \u2a05 (u : Opens X), induced (fun x => x \u2208 u) sierpinskiSpace\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nX : Type u_1\nt : TopologicalSpace X\n\u22a2 t \u2264 \u2a05 (u : Opens X), induced (fun x => x \u2208 u) sierpinskiSpace\n[PROOFSTEP]\nrw [le_iInf_iff]\n[GOAL]\ncase a\nX : Type u_1\nt : TopologicalSpace X\n\u22a2 \u2200 (i : Opens X), t \u2264 induced (fun x => x \u2208 i) sierpinskiSpace\n[PROOFSTEP]\nexact fun u => Continuous.le_induced (isOpen_iff_continuous_mem.mp u.2)\n[GOAL]\ncase a\nX : Type u_1\nt : TopologicalSpace X\n\u22a2 \u2a05 (u : Opens X), induced (fun x => x \u2208 u) sierpinskiSpace \u2264 t\n[PROOFSTEP]\nintro u h\n[GOAL]\ncase a\nX : Type u_1\nt : TopologicalSpace X\nu : Set X\nh : IsOpen u\n\u22a2 IsOpen u\n[PROOFSTEP]\nrw [\u2190 generateFrom_iUnion_isOpen]\n[GOAL]\ncase a\nX : Type u_1\nt : TopologicalSpace X\nu : Set X\nh : IsOpen u\n\u22a2 IsOpen u\n[PROOFSTEP]\napply isOpen_generateFrom_of_mem\n[GOAL]\ncase a.hs\nX : Type u_1\nt : TopologicalSpace X\nu : Set X\nh : IsOpen u\n\u22a2 u \u2208 \u22c3 (i : Opens X), {s | IsOpen s}\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Set.mem_setOf_eq, isOpen_induced_iff]\n[GOAL]\ncase a.hs\nX : Type u_1\nt : TopologicalSpace X\nu : Set X\nh : IsOpen u\n\u22a2 \u2203 i t_1, IsOpen t_1 \u2227 (fun x => x \u2208 i) \u207b\u00b9' t_1 = u\n[PROOFSTEP]\nexact \u27e8\u27e8u, h\u27e9, { True }, isOpen_singleton_true, by simp [Set.preimage]\u27e9\n[GOAL]\nX : Type u_1\nt : TopologicalSpace X\nu : Set X\nh : IsOpen u\n\u22a2 (fun x => x \u2208 { carrier := u, is_open' := h }) \u207b\u00b9' {True} = u\n[PROOFSTEP]\nsimp [Set.preimage]\n[GOAL]\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u22a2 Inducing \u2191(productOfMemOpens X)\n[PROOFSTEP]\nconvert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x \u2208 u\n[GOAL]\ncase h.e'_3\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u22a2 inst\u271d = \u2a05 (i : Opens X), induced (fun x => x \u2208 i) inferInstance\n[PROOFSTEP]\napply eq_induced_by_maps_to_sierpinski\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : T0Space X\n\u22a2 Function.Injective \u2191(productOfMemOpens X)\n[PROOFSTEP]\nintro x1 x2 h\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : T0Space X\nx1 x2 : X\nh : \u2191(productOfMemOpens X) x1 = \u2191(productOfMemOpens X) x2\n\u22a2 x1 = x2\n[PROOFSTEP]\napply Inseparable.eq\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : T0Space X\nx1 x2 : X\nh : \u2191(productOfMemOpens X) x1 = \u2191(productOfMemOpens X) x2\n\u22a2 Inseparable x1 x2\n[PROOFSTEP]\nrw [\u2190 Inducing.inseparable_iff (productOfMemOpens_inducing X), h]\n", "meta": {"mathlib_filename": "Mathlib.Topology.ContinuousFunction.T0Sierpinski", "llama_tokens": 1094, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619177503205, "lm_q2_score": 0.6992544085240401, "lm_q1q2_score": 0.5765086306470961}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : T1Space B\nx : B\n\u22a2 IsClosed (range (TotalSpace.mk x))\n[PROOFSTEP]\nrw [TotalSpace.range_mk]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b9 : FiberBundle F E\ninst\u271d : T1Space B\nx : B\n\u22a2 IsClosed (TotalSpace.proj \u207b\u00b9' {x})\n[PROOFSTEP]\nexact isClosed_singleton.preimage <| continuous_proj F E\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8ea, hea\u27e9 : \u2203 ea : Trivialization F (\u03c0 F E), a \u2208 ea.baseSet :=\n  \u27e8trivializationAt F E a, mem_baseSet_trivializationAt F E a\u27e9\n    -- If `a < b`, then `[a, b] = \u2205`, and the statement is trivial\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\ncases' lt_or_le b a with hab hab\n[GOAL]\ncase intro.inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : b < a\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nexact\n  \u27e8ea, by simp [*]\u27e9\n    /- Let `s` be the set of points `x \u2208 [a, b]` such that `E` is trivializable over `[a, x]`.\n        We need to show that `b \u2208 s`. Let `c = Sup s`. We will show that `c \u2208 s` and `c = b`. -/\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : b < a\n\u22a2 Icc a b \u2286 ea.baseSet\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nset s : Set B := {x \u2208 Icc a b | \u2203 e : Trivialization F (\u03c0 F E), Icc a x \u2286 e.baseSet}\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nhave ha : a \u2208 s := \u27e8left_mem_Icc.2 hab, ea, by simp [hea]\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\n\u22a2 Icc a a \u2286 ea.baseSet\n[PROOFSTEP]\nsimp [hea]\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nhave sne : s.Nonempty := \u27e8a, ha\u27e9\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nhave hsb : b \u2208 upperBounds s := fun x hx => hx.1.2\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nhave sbd : BddAbove s := \u27e8b, hsb\u27e9\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nset c := sSup s\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nhave hsc : IsLUB s c := isLUB_csSup sne sbd\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nhave hc : c \u2208 Icc a b := \u27e8hsc.1 ha, hsc.2 hsb\u27e9\n[GOAL]\ncase intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8-, ec : Trivialization F (\u03c0 F E), hec : Icc a c \u2286 ec.baseSet\u27e9 : c \u2208 s :=\n  by\n  cases' hc.1.eq_or_lt with heq hlt\n  \u00b7 rwa [\u2190 heq]\n  refine\n    \u27e8hc, ?_\u27e9\n      /- In order to show that `c \u2208 s`, consider a trivialization `ec` of `proj` over a neighborhood\n            of `c`. Its base set includes `(c', c]` for some `c' \u2208 [a, c)`. -/\n  obtain \u27e8ec, hc\u27e9 : \u2203 ec : Trivialization F (\u03c0 F E), c \u2208 ec.baseSet :=\n    \u27e8trivializationAt F E c, mem_baseSet_trivializationAt F E c\u27e9\n  obtain \u27e8c', hc', hc'e\u27e9 : \u2203 c' \u2208 Ico a c, Ioc c' c \u2286 ec.baseSet :=\n    (mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset hlt).1\n      (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet hc)\n        /- Since `c' < c = Sup s`, there exists `d \u2208 s \u2229 (c', c]`. Let `ead` be a trivialization of\n              `proj` over `[a, d]`. Then we can glue `ead` and `ec` into a trivialization over `[a, c]`. -/\n  obtain \u27e8d, \u27e8hdab, ead, had\u27e9, hd\u27e9 : \u2203 d \u2208 s, d \u2208 Ioc c' c := hsc.exists_between hc'.2\n  refine' \u27e8ead.piecewiseLe ec d (had \u27e8hdab.1, le_rfl\u27e9) (hc'e hd), subset_ite.2 _\u27e9\n  refine'\n    \u27e8fun x hx => had \u27e8hx.1.1, hx.2\u27e9, fun x hx => hc'e \u27e8hd.1.trans (not_le.1 hx.2), hx.1.2\u27e9\u27e9\n      /- So, `c \u2208 s`. Let `ec` be a trivialization of `proj` over `[a, c]`.  If `c = b`, then we are\n          done. Otherwise we show that `proj` can be trivialized over a larger interval `[a, d]`,\n          `d \u2208 (c, b]`, hence `c` is not an upper bound of `s`. -/\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\n\u22a2 c \u2208 s\n[PROOFSTEP]\ncases' hc.1.eq_or_lt with heq hlt\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nheq : a = c\n\u22a2 c \u2208 s\n[PROOFSTEP]\nrwa [\u2190 heq]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhlt : a < c\n\u22a2 c \u2208 s\n[PROOFSTEP]\nrefine\n  \u27e8hc, ?_\u27e9\n    /- In order to show that `c \u2208 s`, consider a trivialization `ec` of `proj` over a neighborhood\n          of `c`. Its base set includes `(c', c]` for some `c' \u2208 [a, c)`. -/\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nhlt : a < c\n\u22a2 \u2203 e, Icc a c \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8ec, hc\u27e9 : \u2203 ec : Trivialization F (\u03c0 F E), c \u2208 ec.baseSet :=\n  \u27e8trivializationAt F E c, mem_baseSet_trivializationAt F E c\u27e9\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc\u271d : c \u2208 Icc a b\nhlt : a < c\nec : Trivialization F TotalSpace.proj\nhc : c \u2208 ec.baseSet\n\u22a2 \u2203 e, Icc a c \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8c', hc', hc'e\u27e9 : \u2203 c' \u2208 Ico a c, Ioc c' c \u2286 ec.baseSet :=\n  (mem_nhdsWithin_Iic_iff_exists_mem_Ico_Ioc_subset hlt).1\n    (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet hc)\n      /- Since `c' < c = Sup s`, there exists `d \u2208 s \u2229 (c', c]`. Let `ead` be a trivialization of\n            `proj` over `[a, d]`. Then we can glue `ead` and `ec` into a trivialization over `[a, c]`. -/\n[GOAL]\ncase inr.intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc\u271d : c \u2208 Icc a b\nhlt : a < c\nec : Trivialization F TotalSpace.proj\nhc : c \u2208 ec.baseSet\nc' : B\nhc' : c' \u2208 Ico a c\nhc'e : Ioc c' c \u2286 ec.baseSet\n\u22a2 \u2203 e, Icc a c \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8d, \u27e8hdab, ead, had\u27e9, hd\u27e9 : \u2203 d \u2208 s, d \u2208 Ioc c' c := hsc.exists_between hc'.2\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc\u271d : c \u2208 Icc a b\nhlt : a < c\nec : Trivialization F TotalSpace.proj\nhc : c \u2208 ec.baseSet\nc' : B\nhc' : c' \u2208 Ico a c\nhc'e : Ioc c' c \u2286 ec.baseSet\nd : B\nhd : d \u2208 Ioc c' c\nhdab : d \u2208 Icc a b\nead : Trivialization F TotalSpace.proj\nhad : Icc a d \u2286 ead.baseSet\n\u22a2 \u2203 e, Icc a c \u2286 e.baseSet\n[PROOFSTEP]\nrefine' \u27e8ead.piecewiseLe ec d (had \u27e8hdab.1, le_rfl\u27e9) (hc'e hd), subset_ite.2 _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc\u271d : c \u2208 Icc a b\nhlt : a < c\nec : Trivialization F TotalSpace.proj\nhc : c \u2208 ec.baseSet\nc' : B\nhc' : c' \u2208 Ico a c\nhc'e : Ioc c' c \u2286 ec.baseSet\nd : B\nhd : d \u2208 Ioc c' c\nhdab : d \u2208 Icc a b\nead : Trivialization F TotalSpace.proj\nhad : Icc a d \u2286 ead.baseSet\n\u22a2 Icc a c \u2229 Iic d \u2286 ead.baseSet \u2227\n    Icc a c \\ Iic d \u2286\n      (Trivialization.transFiberHomeomorph ec\n          (Trivialization.coordChangeHomeomorph ec ead (_ : d \u2208 ec.baseSet) (_ : d \u2208 ead.baseSet))).baseSet\n[PROOFSTEP]\nrefine'\n  \u27e8fun x hx => had \u27e8hx.1.1, hx.2\u27e9, fun x hx => hc'e \u27e8hd.1.trans (not_le.1 hx.2), hx.1.2\u27e9\u27e9\n    /- So, `c \u2208 s`. Let `ec` be a trivialization of `proj` over `[a, c]`.  If `c = b`, then we are\n        done. Otherwise we show that `proj` can be trivialized over a larger interval `[a, d]`,\n        `d \u2208 (c, b]`, hence `c` is not an upper bound of `s`. -/\n[GOAL]\ncase intro.inr.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\ncases' hc.2.eq_or_lt with heq hlt\n[GOAL]\ncase intro.inr.intro.intro.inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nheq : c = b\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nexact \u27e8ec, heq \u25b8 hec\u27e9\n[GOAL]\ncase intro.inr.intro.intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nsuffices : \u2203 d \u2208 Ioc c b, \u2203 e : Trivialization F (\u03c0 F E), Icc a d \u2286 e.baseSet\n[GOAL]\ncase intro.inr.intro.intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nthis : \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nrcases this with\n  \u27e8d, hdcb, hd\u27e9\n    -- porting note: todo: use `rsuffices`\n[GOAL]\ncase intro.inr.intro.intro.inr.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : \u2203 e, Icc a d \u2286 e.baseSet\n\u22a2 \u2203 e, Icc a b \u2286 e.baseSet\n[PROOFSTEP]\nexact ((hsc.1 \u27e8\u27e8hc.1.trans hdcb.1.le, hdcb.2\u27e9, hd\u27e9).not_lt hdcb.1).elim\n[GOAL]\ncase this\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8d, hdcb, hd\u27e9 : \u2203 d \u2208 Ioc c b, Ico c d \u2286 ec.baseSet :=\n  (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1\n    (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet (hec \u27e8hc.1, le_rfl\u27e9))\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nhave had : Ico a d \u2286 ec.baseSet := Ico_subset_Icc_union_Ico.trans (union_subset hec hd)\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nby_cases he : Disjoint (Iio d) (Ioi c)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : Disjoint (Iio d) (Ioi c)\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nobtain \u27e8ed, hed\u27e9 : \u2203 ed : Trivialization F (\u03c0 F E), d \u2208 ed.baseSet :=\n  \u27e8trivializationAt F E d, mem_baseSet_trivializationAt F E d\u27e9\n[GOAL]\ncase pos.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : Disjoint (Iio d) (Ioi c)\ned : Trivialization F TotalSpace.proj\nhed : d \u2208 ed.baseSet\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nrefine'\n  \u27e8d, hdcb,\n    (ec.restrOpen (Iio d) isOpen_Iio).disjointUnion (ed.restrOpen (Ioi c) isOpen_Ioi)\n      (he.mono (inter_subset_right _ _) (inter_subset_right _ _)),\n    fun x hx => _\u27e9\n[GOAL]\ncase pos.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : Disjoint (Iio d) (Ioi c)\ned : Trivialization F TotalSpace.proj\nhed : d \u2208 ed.baseSet\nx : B\nhx : x \u2208 Icc a d\n\u22a2 x \u2208\n    (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d)))\n        (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c)))\n        (_ :\n          Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet\n            (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet\n[PROOFSTEP]\nrcases hx.2.eq_or_lt with (rfl | hxd)\n[GOAL]\ncase pos.intro.inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\ned : Trivialization F TotalSpace.proj\nx : B\nhdcb : x \u2208 Ioc c b\nhd : Ico c x \u2286 ec.baseSet\nhad : Ico a x \u2286 ec.baseSet\nhe : Disjoint (Iio x) (Ioi c)\nhed : x \u2208 ed.baseSet\nhx : x \u2208 Icc a x\n\u22a2 x \u2208\n    (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x)))\n        (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c)))\n        (_ :\n          Disjoint (Trivialization.restrOpen ec (Iio x) (_ : IsOpen (Iio x))).baseSet\n            (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet\ncase pos.intro.inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : Disjoint (Iio d) (Ioi c)\ned : Trivialization F TotalSpace.proj\nhed : d \u2208 ed.baseSet\nx : B\nhx : x \u2208 Icc a d\nhxd : x < d\n\u22a2 x \u2208\n    (Trivialization.disjointUnion (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d)))\n        (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c)))\n        (_ :\n          Disjoint (Trivialization.restrOpen ec (Iio d) (_ : IsOpen (Iio d))).baseSet\n            (Trivialization.restrOpen ed (Ioi c) (_ : IsOpen (Ioi c))).baseSet)).baseSet\n[PROOFSTEP]\nexacts [Or.inr \u27e8hed, hdcb.1\u27e9, Or.inl \u27e8had \u27e8hx.1, hxd\u27e9, hxd\u27e9]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : \u00acDisjoint (Iio d) (Ioi c)\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nrw [disjoint_left] at he \n[GOAL]\ncase neg\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : \u00ac\u2200 \u2983a : B\u2984, a \u2208 Iio d \u2192 \u00aca \u2208 Ioi c\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\npush_neg at he \n[GOAL]\ncase neg\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nhe : Exists fun \u2983a\u2984 => a \u2208 Iio d \u2227 a \u2208 Ioi (sSup s)\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nrcases he with \u27e8d', hdd' : d' < d, hd'c\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nE : B \u2192 Type u_5\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (b : B) \u2192 TopologicalSpace (E b)\ninst\u271d\u00b2 : ConditionallyCompleteLinearOrder B\ninst\u271d\u00b9 : OrderTopology B\ninst\u271d : FiberBundle F E\na b : B\nea : Trivialization F TotalSpace.proj\nhea : a \u2208 ea.baseSet\nhab : a \u2264 b\ns : Set B := {x | x \u2208 Icc a b \u2227 \u2203 e, Icc a x \u2286 e.baseSet}\nha : a \u2208 s\nsne : Set.Nonempty s\nhsb : b \u2208 upperBounds s\nsbd : BddAbove s\nc : B := sSup s\nhsc : IsLUB s c\nhc : c \u2208 Icc a b\nec : Trivialization F TotalSpace.proj\nhec : Icc a c \u2286 ec.baseSet\nhlt : c < b\nd : B\nhdcb : d \u2208 Ioc c b\nhd : Ico c d \u2286 ec.baseSet\nhad : Ico a d \u2286 ec.baseSet\nd' : B\nhdd' : d' < d\nhd'c : d' \u2208 Ioi (sSup s)\n\u22a2 \u2203 d, d \u2208 Ioc c b \u2227 \u2203 e, Icc a d \u2286 e.baseSet\n[PROOFSTEP]\nexact \u27e8d', \u27e8hd'c, hdd'.le.trans hdcb.2\u27e9, ec, (Icc_subset_Ico_right hdd').trans had\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\nhp : p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ\n\u22a2 (fun p => (p.fst, coordChange Z i j p.fst p.snd)) p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa using hp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\nhp : p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ\n\u22a2 (fun p => (p.fst, coordChange Z j i p.fst p.snd)) p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa using hp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\n\u22a2 \u2200 \u2983x : B \u00d7 F\u2984,\n    x \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n      (fun p => (p.fst, coordChange Z j i p.fst p.snd)) ((fun p => (p.fst, coordChange Z i j p.fst p.snd)) x) = x\n[PROOFSTEP]\nrintro \u27e8x, v\u27e9 hx\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : (x, v) \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ\n\u22a2 (fun p => (p.fst, coordChange Z j i p.fst p.snd)) ((fun p => (p.fst, coordChange Z i j p.fst p.snd)) (x, v)) = (x, v)\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true, mem_univ] at hx \n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 (fun p => (p.fst, coordChange Z j i p.fst p.snd)) ((fun p => (p.fst, coordChange Z i j p.fst p.snd)) (x, v)) = (x, v)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 (x, coordChange Z j i x (coordChange Z i j x v)) = (x, v)\n[PROOFSTEP]\nrw [coordChange_comp, Z.coordChange_self]\n[GOAL]\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 x \u2208 baseSet Z i\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 x \u2208 baseSet Z i \u2229 baseSet Z j \u2229 baseSet Z i\n[PROOFSTEP]\nexacts [hx.1, \u27e8\u27e8hx.1, hx.2\u27e9, hx.1\u27e9]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\n\u22a2 \u2200 \u2983x : B \u00d7 F\u2984,\n    x \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n      (fun p => (p.fst, coordChange Z i j p.fst p.snd)) ((fun p => (p.fst, coordChange Z j i p.fst p.snd)) x) = x\n[PROOFSTEP]\nrintro \u27e8x, v\u27e9 hx\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : (x, v) \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ\n\u22a2 (fun p => (p.fst, coordChange Z i j p.fst p.snd)) ((fun p => (p.fst, coordChange Z j i p.fst p.snd)) (x, v)) = (x, v)\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ] at hx \n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 (fun p => (p.fst, coordChange Z i j p.fst p.snd)) ((fun p => (p.fst, coordChange Z j i p.fst p.snd)) (x, v)) = (x, v)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 (x, coordChange Z i j x (coordChange Z j i x v)) = (x, v)\n[PROOFSTEP]\nrw [Z.coordChange_comp, Z.coordChange_self]\n[GOAL]\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 x \u2208 baseSet Z j\n[PROOFSTEP]\nexact hx.2\n[GOAL]\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 x \u2208 baseSet Z j \u2229 baseSet Z i \u2229 baseSet Z j\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\n\u22a2 ContinuousOn\n    { toFun := fun p => (p.fst, coordChange Z i j p.fst p.snd),\n        invFun := fun p => (p.fst, coordChange Z j i p.fst p.snd), source := (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ,\n        target := (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ,\n        map_source' :=\n          (_ :\n            \u2200 (p : B \u00d7 F),\n              p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z i j p.fst p.snd)) p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ),\n        map_target' :=\n          (_ :\n            \u2200 (p : B \u00d7 F),\n              p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z j i p.fst p.snd)) p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : B \u00d7 F\u2984,\n              x \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z j i p.fst p.snd))\n                    ((fun p => (p.fst, coordChange Z i j p.fst p.snd)) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : B \u00d7 F\u2984,\n              x \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z i j p.fst p.snd))\n                    ((fun p => (p.fst, coordChange Z j i p.fst p.snd)) x) =\n                  x) }.invFun\n    { toFun := fun p => (p.fst, coordChange Z i j p.fst p.snd),\n        invFun := fun p => (p.fst, coordChange Z j i p.fst p.snd), source := (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ,\n        target := (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ,\n        map_source' :=\n          (_ :\n            \u2200 (p : B \u00d7 F),\n              p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z i j p.fst p.snd)) p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ),\n        map_target' :=\n          (_ :\n            \u2200 (p : B \u00d7 F),\n              p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z j i p.fst p.snd)) p \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ),\n        left_inv' :=\n          (_ :\n            \u2200 \u2983x : B \u00d7 F\u2984,\n              x \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z j i p.fst p.snd))\n                    ((fun p => (p.fst, coordChange Z i j p.fst p.snd)) x) =\n                  x),\n        right_inv' :=\n          (_ :\n            \u2200 \u2983x : B \u00d7 F\u2984,\n              x \u2208 (baseSet Z i \u2229 baseSet Z j) \u00d7\u02e2 univ \u2192\n                (fun p => (p.fst, coordChange Z i j p.fst p.snd))\n                    ((fun p => (p.fst, coordChange Z j i p.fst p.snd)) x) =\n                  x) }.target\n[PROOFSTEP]\nsimpa [inter_comm] using continuous_fst.continuousOn.prod (Z.continuousOn_coordChange j i)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\n\u22a2 p \u2208 (trivChange Z i j).toLocalEquiv.source \u2194 p.fst \u2208 baseSet Z i \u2229 baseSet Z j\n[PROOFSTEP]\nerw [mem_prod]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\n\u22a2 p.fst \u2208 baseSet Z i \u2229 baseSet Z j \u2227 p.snd \u2208 univ \u2194 p.fst \u2208 baseSet Z i \u2229 baseSet Z j\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\np : TotalSpace Z\nhp : p \u2208 proj Z \u207b\u00b9' baseSet Z i\n\u22a2 (fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd)) p \u2208 baseSet Z i \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [Set.mem_preimage, and_true_iff, Set.mem_univ, Set.prod_mk_mem_set_prod_eq] using hp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\np : B \u00d7 F\nhp : p \u2208 baseSet Z i \u00d7\u02e2 univ\n\u22a2 (fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd }) p \u2208 proj Z \u207b\u00b9' baseSet Z i\n[PROOFSTEP]\nsimpa only [Set.mem_preimage, and_true_iff, Set.mem_univ, Set.mem_prod] using hp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\n\u22a2 \u2200 \u2983x : TotalSpace Z\u2984,\n    x \u2208 proj Z \u207b\u00b9' baseSet Z i \u2192\n      (fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd })\n          ((fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd)) x) =\n        x\n[PROOFSTEP]\nrintro \u27e8x, v\u27e9 hx\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : Fiber Z x\nhx : { proj := x, snd := v } \u2208 proj Z \u207b\u00b9' baseSet Z i\n\u22a2 (fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd })\n      ((fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd)) { proj := x, snd := v }) =\n    { proj := x, snd := v }\n[PROOFSTEP]\nreplace hx : x \u2208 Z.baseSet i := hx\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : Fiber Z x\nhx : x \u2208 baseSet Z i\n\u22a2 (fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd })\n      ((fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd)) { proj := x, snd := v }) =\n    { proj := x, snd := v }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : Fiber Z x\nhx : x \u2208 baseSet Z i\n\u22a2 { proj := x, snd := coordChange Z i (indexAt Z x) x (coordChange Z (indexAt Z x) i x v) } = { proj := x, snd := v }\n[PROOFSTEP]\nrw [Z.coordChange_comp, Z.coordChange_self]\n[GOAL]\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : Fiber Z x\nhx : x \u2208 baseSet Z i\n\u22a2 x \u2208 baseSet Z (indexAt Z x)\n[PROOFSTEP]\napply_rules [mem_baseSet_at, mem_inter]\n[GOAL]\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : Fiber Z x\nhx : x \u2208 baseSet Z i\n\u22a2 x \u2208 baseSet Z (indexAt Z x) \u2229 baseSet Z i \u2229 baseSet Z (indexAt Z x)\n[PROOFSTEP]\napply_rules [mem_baseSet_at, mem_inter]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\n\u22a2 \u2200 \u2983x : B \u00d7 F\u2984,\n    x \u2208 baseSet Z i \u00d7\u02e2 univ \u2192\n      (fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd))\n          ((fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd }) x) =\n        x\n[PROOFSTEP]\nrintro \u27e8x, v\u27e9 hx\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : F\nhx : (x, v) \u2208 baseSet Z i \u00d7\u02e2 univ\n\u22a2 (fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd))\n      ((fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd }) (x, v)) =\n    (x, v)\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, and_true_iff, mem_univ] at hx \n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i\n\u22a2 (fun p => (p.proj, coordChange Z (indexAt Z p.proj) i p.proj p.snd))\n      ((fun p => { proj := p.fst, snd := coordChange Z i (indexAt Z p.fst) p.fst p.snd }) (x, v)) =\n    (x, v)\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i\n\u22a2 (x, coordChange Z (indexAt Z x) i x (coordChange Z i (indexAt Z x) x v)) = (x, v)\n[PROOFSTEP]\nrw [Z.coordChange_comp, Z.coordChange_self]\n[GOAL]\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i\n\u22a2 x \u2208 baseSet Z i\ncase mk.a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i\n\u22a2 x \u2208 baseSet Z i \u2229 baseSet Z (indexAt Z x) \u2229 baseSet Z i\n[PROOFSTEP]\nexacts [hx, \u27e8\u27e8hx, Z.mem_baseSet_at _\u27e9, hx\u27e9]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\np : B \u00d7 F\n\u22a2 p \u2208 (localTrivAsLocalEquiv Z i).target \u2194 p.fst \u2208 baseSet Z i\n[PROOFSTEP]\nerw [mem_prod]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\np : B \u00d7 F\n\u22a2 p.fst \u2208 baseSet Z i \u2227 p.snd \u2208 univ \u2194 p.fst \u2208 baseSet Z i\n[PROOFSTEP]\nsimp only [and_true_iff, mem_univ]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\n\u22a2 LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j) \u2248\n    (trivChange Z i j).toLocalEquiv\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\n\u22a2 (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source =\n    (trivChange Z i j).toLocalEquiv.source\n[PROOFSTEP]\next x\n[GOAL]\ncase left.h\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\nx : B \u00d7 F\n\u22a2 x \u2208 (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source \u2194\n    x \u2208 (trivChange Z i j).toLocalEquiv.source\n[PROOFSTEP]\nsimp only [mem_localTrivAsLocalEquiv_target, mfld_simps]\n[GOAL]\ncase left.h\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\nx : B \u00d7 F\n\u22a2 x.fst \u2208 baseSet Z i \u2227 \u2191(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) x \u2208 (localTrivAsLocalEquiv Z j).source \u2194\n    x.fst \u2208 baseSet Z i \u2227 x.fst \u2208 baseSet Z j\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\n\u22a2 EqOn (\u2191(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)))\n    (\u2191(trivChange Z i j).toLocalEquiv)\n    (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source\n[PROOFSTEP]\nrintro \u27e8x, v\u27e9 hx\n[GOAL]\ncase right.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\nx : B\nv : F\nhx : (x, v) \u2208 (LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)).source\n\u22a2 \u2191(LocalEquiv.trans (LocalEquiv.symm (localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z j)) (x, v) =\n    \u2191(trivChange Z i j).toLocalEquiv (x, v)\n[PROOFSTEP]\nsimp only [trivChange, localTrivAsLocalEquiv, LocalEquiv.symm, true_and_iff, Prod.mk.inj_iff, prod_mk_mem_set_prod_eq,\n  LocalEquiv.trans_source, mem_inter_iff, and_true_iff, mem_preimage, proj, mem_univ, eq_self_iff_true, (\u00b7 \u2218 \u00b7),\n  LocalEquiv.coe_trans, TotalSpace.proj] at hx \u22a2\n[GOAL]\ncase right.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d i j : \u03b9\nx : B\nv : F\nhx : x \u2208 baseSet Z i \u2227 x \u2208 baseSet Z j\n\u22a2 coordChange Z (indexAt Z x) j x (coordChange Z i (indexAt Z x) x v) = coordChange Z i j x v\n[PROOFSTEP]\nsimp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 IsOpen (localTrivAsLocalEquiv Z i).source\n[PROOFSTEP]\napply TopologicalSpace.GenerateOpen.basic\n[GOAL]\ncase a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 (localTrivAsLocalEquiv Z i).source \u2208\n    \u22c3 (i : \u03b9) (s : Set (B \u00d7 F)) (_ : IsOpen s),\n      {(localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' s}\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, mem_singleton_iff]\n[GOAL]\ncase a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 \u2203 i_1 i_2,\n    IsOpen i_2 \u2227\n      (localTrivAsLocalEquiv Z i).source = (localTrivAsLocalEquiv Z i_1).source \u2229 \u2191(localTrivAsLocalEquiv Z i_1) \u207b\u00b9' i_2\n[PROOFSTEP]\nrefine \u27e8i, Z.baseSet i \u00d7\u02e2 univ, (Z.isOpen_baseSet i).prod isOpen_univ, ?_\u27e9\n[GOAL]\ncase a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 (localTrivAsLocalEquiv Z i).source =\n    (localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' baseSet Z i \u00d7\u02e2 univ\n[PROOFSTEP]\next p\n[GOAL]\ncase a.h\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\np : TotalSpace Z\n\u22a2 p \u2208 (localTrivAsLocalEquiv Z i).source \u2194\n    p \u2208 (localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' baseSet Z i \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [localTrivAsLocalEquiv_apply, prod_mk_mem_set_prod_eq, mem_inter_iff, and_self_iff,\n  mem_localTrivAsLocalEquiv_source, and_true, mem_univ, mem_preimage]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 ContinuousOn (\u2191(localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z i).source\n[PROOFSTEP]\nrw [continuousOn_open_iff (Z.open_source' i)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 \u2200 (t : Set (B \u00d7 F)), IsOpen t \u2192 IsOpen ((localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' t)\n[PROOFSTEP]\nintro s s_open\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\n\u22a2 IsOpen ((localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' s)\n[PROOFSTEP]\napply TopologicalSpace.GenerateOpen.basic\n[GOAL]\ncase a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\n\u22a2 (localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' s \u2208\n    \u22c3 (i : \u03b9) (s : Set (B \u00d7 F)) (_ : IsOpen s),\n      {(localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' s}\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, mem_singleton_iff]\n[GOAL]\ncase a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\n\u22a2 \u2203 i_1 i_2,\n    IsOpen i_2 \u2227\n      (localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' s =\n        (localTrivAsLocalEquiv Z i_1).source \u2229 \u2191(localTrivAsLocalEquiv Z i_1) \u207b\u00b9' i_2\n[PROOFSTEP]\nexact \u27e8i, s, s_open, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\n\u22a2 ContinuousOn (localTrivAsLocalEquiv Z i).invFun (localTrivAsLocalEquiv Z i).target\n[PROOFSTEP]\nrefine continuousOn_open_of_generateFrom fun t ht \u21a6 ?_\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nht :\n  t \u2208\n    \u22c3 (i : \u03b9) (s : Set (B \u00d7 F)) (_ : IsOpen s),\n      {(localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' s}\n\u22a2 IsOpen ((localTrivAsLocalEquiv Z i).target \u2229 (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' t)\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, mem_singleton_iff] at ht \n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nht : \u2203 i i_1, IsOpen i_1 \u2227 t = (localTrivAsLocalEquiv Z i).source \u2229 \u2191(localTrivAsLocalEquiv Z i) \u207b\u00b9' i_1\n\u22a2 IsOpen ((localTrivAsLocalEquiv Z i).target \u2229 (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' t)\n[PROOFSTEP]\nobtain \u27e8j, s, s_open, ts\u27e9 :\n  \u2203 j s, IsOpen s \u2227 t = (localTrivAsLocalEquiv Z j).source \u2229 localTrivAsLocalEquiv Z j \u207b\u00b9' s := ht\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\n\u22a2 IsOpen ((localTrivAsLocalEquiv Z i).target \u2229 (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' t)\n[PROOFSTEP]\nrw [ts]\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\n\u22a2 IsOpen\n    ((localTrivAsLocalEquiv Z i).target \u2229\n      (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' ((localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s))\n[PROOFSTEP]\nsimp only [LocalEquiv.right_inv, preimage_inter, LocalEquiv.left_inv]\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\n\u22a2 IsOpen\n    ((localTrivAsLocalEquiv Z i).target \u2229\n      ((localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)))\n[PROOFSTEP]\nlet e := Z.localTrivAsLocalEquiv i\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\n\u22a2 IsOpen\n    ((localTrivAsLocalEquiv Z i).target \u2229\n      ((localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)))\n[PROOFSTEP]\nlet e' := Z.localTrivAsLocalEquiv j\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\n\u22a2 IsOpen\n    ((localTrivAsLocalEquiv Z i).target \u2229\n      ((localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)))\n[PROOFSTEP]\nlet f := e.symm.trans e'\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\nf : LocalEquiv (B \u00d7 F) (B \u00d7 F) := LocalEquiv.trans (LocalEquiv.symm e) e'\n\u22a2 IsOpen\n    ((localTrivAsLocalEquiv Z i).target \u2229\n      ((localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)))\n[PROOFSTEP]\nhave : IsOpen (f.source \u2229 f \u207b\u00b9' s) :=\n  by\n  rw [LocalEquiv.EqOnSource.source_inter_preimage_eq (Z.localTrivAsLocalEquiv_trans i j)]\n  exact (continuousOn_open_iff (Z.trivChange i j).open_source).1 (Z.trivChange i j).continuousOn _ s_open\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\nf : LocalEquiv (B \u00d7 F) (B \u00d7 F) := LocalEquiv.trans (LocalEquiv.symm e) e'\n\u22a2 IsOpen (f.source \u2229 \u2191f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [LocalEquiv.EqOnSource.source_inter_preimage_eq (Z.localTrivAsLocalEquiv_trans i j)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\nf : LocalEquiv (B \u00d7 F) (B \u00d7 F) := LocalEquiv.trans (LocalEquiv.symm e) e'\n\u22a2 IsOpen ((trivChange Z i j).toLocalEquiv.source \u2229 \u2191(trivChange Z i j).toLocalEquiv \u207b\u00b9' s)\n[PROOFSTEP]\nexact (continuousOn_open_iff (Z.trivChange i j).open_source).1 (Z.trivChange i j).continuousOn _ s_open\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\nf : LocalEquiv (B \u00d7 F) (B \u00d7 F) := LocalEquiv.trans (LocalEquiv.symm e) e'\nthis : IsOpen (f.source \u2229 \u2191f \u207b\u00b9' s)\n\u22a2 IsOpen\n    ((localTrivAsLocalEquiv Z i).target \u2229\n      ((localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)))\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\nf : LocalEquiv (B \u00d7 F) (B \u00d7 F) := LocalEquiv.trans (LocalEquiv.symm e) e'\nthis : IsOpen (f.source \u2229 \u2191f \u207b\u00b9' s)\n\u22a2 (localTrivAsLocalEquiv Z i).target \u2229\n      ((localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        (localTrivAsLocalEquiv Z i).invFun \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)) =\n    f.source \u2229 \u2191f \u207b\u00b9' s\n[PROOFSTEP]\ndsimp [LocalEquiv.trans_source]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\nt : Set (TotalSpace Z)\nj : \u03b9\ns : Set (B \u00d7 F)\ns_open : IsOpen s\nts : t = (localTrivAsLocalEquiv Z j).source \u2229 \u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s\ne : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z i\ne' : LocalEquiv (TotalSpace Z) (B \u00d7 F) := localTrivAsLocalEquiv Z j\nf : LocalEquiv (B \u00d7 F) (B \u00d7 F) := LocalEquiv.trans (LocalEquiv.symm e) e'\nthis : IsOpen (f.source \u2229 \u2191f \u207b\u00b9' s)\n\u22a2 (localTrivAsLocalEquiv Z i).target \u2229\n      (\u2191(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n        \u2191(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) \u207b\u00b9' (\u2191(localTrivAsLocalEquiv Z j) \u207b\u00b9' s)) =\n    (localTrivAsLocalEquiv Z i).target \u2229\n        \u2191(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) \u207b\u00b9' (localTrivAsLocalEquiv Z j).source \u2229\n      \u2191(localTrivAsLocalEquiv Z j) \u2218 \u2191(LocalEquiv.symm (localTrivAsLocalEquiv Z i)) \u207b\u00b9' s\n[PROOFSTEP]\nrw [\u2190 preimage_comp, inter_assoc]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\np : TotalSpace Z\nx\u271d :\n  p \u2208\n    { toLocalEquiv := localTrivAsLocalEquiv Z i, open_source := (_ : IsOpen (localTrivAsLocalEquiv Z i).source),\n          open_target := (_ : IsOpen (baseSet Z i \u00d7\u02e2 univ)),\n          continuous_toFun := (_ : ContinuousOn (\u2191(localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z i).source),\n          continuous_invFun :=\n            (_ :\n              ContinuousOn (localTrivAsLocalEquiv Z i).invFun (localTrivAsLocalEquiv Z i).target) }.toLocalEquiv.source\n\u22a2 (\u2191{ toLocalEquiv := localTrivAsLocalEquiv Z i, open_source := (_ : IsOpen (localTrivAsLocalEquiv Z i).source),\n            open_target := (_ : IsOpen (baseSet Z i \u00d7\u02e2 univ)),\n            continuous_toFun := (_ : ContinuousOn (\u2191(localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z i).source),\n            continuous_invFun :=\n              (_ : ContinuousOn (localTrivAsLocalEquiv Z i).invFun (localTrivAsLocalEquiv Z i).target) }\n        p).fst =\n    proj Z p\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni\u271d : \u03b9\nb : B\na : F\ni : \u03b9\np : TotalSpace Z\nx\u271d :\n  p \u2208\n    { toLocalEquiv := localTrivAsLocalEquiv Z i, open_source := (_ : IsOpen (localTrivAsLocalEquiv Z i).source),\n          open_target := (_ : IsOpen (baseSet Z i \u00d7\u02e2 univ)),\n          continuous_toFun := (_ : ContinuousOn (\u2191(localTrivAsLocalEquiv Z i)) (localTrivAsLocalEquiv Z i).source),\n          continuous_invFun :=\n            (_ :\n              ContinuousOn (localTrivAsLocalEquiv Z i).invFun (localTrivAsLocalEquiv Z i).target) }.toLocalEquiv.source\n\u22a2 (\u2191(localTrivAsLocalEquiv Z i) p).fst = p.proj\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\n\u22a2 Continuous\n    (let_fun this := fun x => { proj := x, snd := v };\n    this)\n[PROOFSTEP]\nrefine continuous_iff_continuousAt.2 fun x => ?_\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\n\u22a2 ContinuousAt\n    (let_fun this := fun x => { proj := x, snd := v };\n    this)\n    x\n[PROOFSTEP]\nhave A : Z.baseSet (Z.indexAt x) \u2208 \ud835\udcdd x := IsOpen.mem_nhds (Z.isOpen_baseSet (Z.indexAt x)) (Z.mem_baseSet_at x)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\n\u22a2 ContinuousAt\n    (let_fun this := fun x => { proj := x, snd := v };\n    this)\n    x\n[PROOFSTEP]\nrefine ((Z.localTrivAt x).toLocalHomeomorph.continuousAt_iff_continuousAt_comp_left ?_).2 ?_\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\n\u22a2 (let_fun this := fun x => { proj := x, snd := v };\n      this) \u207b\u00b9'\n      (localTrivAt Z x).toLocalHomeomorph.toLocalEquiv.source \u2208\n    \ud835\udcdd x\n[PROOFSTEP]\nexact A\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\n\u22a2 ContinuousAt\n    (\u2191(localTrivAt Z x).toLocalHomeomorph \u2218\n      let_fun this := fun x => { proj := x, snd := v };\n      this)\n    x\n[PROOFSTEP]\napply continuousAt_id.prod\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\n\u22a2 ContinuousAt\n    (fun x_1 =>\n      coordChange Z\n        (indexAt Z\n          ((let_fun this := fun x => { proj := x, snd := v };\n              this)\n              x_1).proj)\n        (indexAt Z x)\n        ((let_fun this := fun x => { proj := x, snd := v };\n            this)\n            x_1).proj\n        ((let_fun this := fun x => { proj := x, snd := v };\n            this)\n            x_1).snd)\n    x\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), mfld_simps, localTrivAt_snd]\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\n\u22a2 ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x\n[PROOFSTEP]\nhave : ContinuousOn (fun _ : B => v) (Z.baseSet (Z.indexAt x)) := continuousOn_const\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\nthis : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x))\n\u22a2 ContinuousAt (fun x_1 => coordChange Z (indexAt Z x_1) (indexAt Z x) x_1 v) x\n[PROOFSTEP]\nrefine (this.congr fun y hy \u21a6 ?_).continuousAt A\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na v : F\nh : \u2200 (i j : \u03b9) (x : B), x \u2208 baseSet Z i \u2229 baseSet Z j \u2192 coordChange Z i j x v = v\nx : B\nA : baseSet Z (indexAt Z x) \u2208 \ud835\udcdd x\nthis : ContinuousOn (fun x => v) (baseSet Z (indexAt Z x))\ny : B\nhy : y \u2208 baseSet Z (indexAt Z x)\n\u22a2 coordChange Z (indexAt Z y) (indexAt Z x) y v = v\n[PROOFSTEP]\nexact h _ _ _ \u27e8mem_baseSet_at _ _, hy\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na : F\np : TotalSpace Z\n\u22a2 \u2191(localTrivAt Z p.proj) p = (p.proj, p.snd)\n[PROOFSTEP]\nrw [localTrivAt, localTriv_apply, coordChange_self]\n[GOAL]\ncase a\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na : F\np : TotalSpace Z\n\u22a2 p.proj \u2208 baseSet Z (indexAt Z p.proj)\n[PROOFSTEP]\nexact Z.mem_baseSet_at p.1\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\n\u22a2 b \u2208 (localTrivAt Z b).baseSet\n[PROOFSTEP]\nrw [localTrivAt, \u2190 baseSet_at]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\n\u22a2 b \u2208 baseSet Z (indexAt Z b)\n[PROOFSTEP]\nexact Z.mem_baseSet_at b\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb : B\na : F\n\u22a2 { proj := b, snd := a } \u2208 (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source\n[PROOFSTEP]\nsimp only [mfld_simps]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\nx : Fiber Z b\n\u22a2 \ud835\udcdd x = comap (TotalSpace.mk b) (\ud835\udcdd { proj := b, snd := x })\n[PROOFSTEP]\nrw [(Z.localTrivAt b).nhds_eq_comap_inf_principal (mk_mem_localTrivAt_source _ _ _), comap_inf, comap_principal,\n  comap_comap]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\nx : Fiber Z b\n\u22a2 \ud835\udcdd x =\n    comap (\u2191(localTrivAt Z b).toLocalHomeomorph \u2218 TotalSpace.mk b)\n        (\ud835\udcdd (\u2191(localTrivAt Z b).toLocalHomeomorph { proj := b, snd := x })) \u2293\n      \ud835\udcdf (TotalSpace.mk b \u207b\u00b9' (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source)\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), localTrivAt_apply_mk, Trivialization.coe_coe, \u2190 (embedding_prod_mk b).nhds_eq_comap]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\nx : Fiber Z b\n\u22a2 \ud835\udcdd x = \ud835\udcdd x \u2293 \ud835\udcdf (TotalSpace.mk b \u207b\u00b9' (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source)\n[PROOFSTEP]\nconvert_to \ud835\udcdd x = \ud835\udcdd x \u2293 \ud835\udcdf univ\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\nx : Fiber Z b\n\u22a2 \ud835\udcdd x \u2293 \ud835\udcdf (TotalSpace.mk b \u207b\u00b9' (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source) = \ud835\udcdd x \u2293 \ud835\udcdf univ\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_3.e_a.e_s\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\nx : Fiber Z b\n\u22a2 TotalSpace.mk b \u207b\u00b9' (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source = univ\n[PROOFSTEP]\nexact eq_univ_of_forall (mk_mem_localTrivAt_source Z _)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni : \u03b9\nb\u271d : B\na : F\nb : B\nx : Fiber Z b\n\u22a2 \ud835\udcdd x = \ud835\udcdd x \u2293 \ud835\udcdf univ\n[PROOFSTEP]\nrw [principal_univ, inf_top_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\n\u22a2 ContinuousOn (\u2191(LocalEquiv.symm e.toLocalEquiv)) e.target\n[PROOFSTEP]\nrefine' fun z H U h => preimage_nhdsWithin_coinduced' H (le_def.1 (nhds_mono _) U h)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nz : B \u00d7 F\nH : z \u2208 e.target\nU : Set (TotalSpace F E)\nh : U \u2208 \ud835\udcdd (\u2191(LocalEquiv.symm e.toLocalEquiv) z)\n\u22a2 coinduced (fun x => \u2191(LocalEquiv.symm e.toLocalEquiv) \u2191x) inferInstance \u2264 totalSpaceTopology a\n[PROOFSTEP]\nexact le_iSup\u2082 (\u03b1 := TopologicalSpace (TotalSpace F E)) e he\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e : Pretrivialization F TotalSpace.proj\n\u22a2 IsOpen e.source\n[PROOFSTEP]\nrefine isOpen_iSup_iff.mpr fun e' => isOpen_iSup_iff.mpr fun _ => ?_\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nx\u271d : e' \u2208 a.pretrivializationAtlas\n\u22a2 IsOpen e.source\n[PROOFSTEP]\nrefine' isOpen_coinduced.mpr (isOpen_induced_iff.mpr \u27e8e.target, e.open_target, _\u27e9)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nx\u271d : e' \u2208 a.pretrivializationAtlas\n\u22a2 Subtype.val \u207b\u00b9' e.target = Pretrivialization.setSymm e' \u207b\u00b9' e.source\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nx\u271d : e' \u2208 a.pretrivializationAtlas\nx : B \u00d7 F\nhx : x \u2208 e'.target\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' e.target \u2194\n    { val := x, property := hx } \u2208 Pretrivialization.setSymm e' \u207b\u00b9' e.source\n[PROOFSTEP]\nsimp only [mem_preimage, Pretrivialization.setSymm, restrict, e.mem_target, e.mem_source, e'.proj_symm_apply hx]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\n\u22a2 IsOpen (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n[PROOFSTEP]\nletI := a.totalSpaceTopology\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 IsOpen (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n[PROOFSTEP]\nobtain \u27e8u, hu1, hu2\u27e9 :=\n  continuousOn_iff'.mp (a.continuous_symm_of_mem_pretrivializationAtlas he') e.source (a.isOpen_source e)\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 : \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source \u2229 e'.target = u \u2229 e'.target\n\u22a2 IsOpen (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n[PROOFSTEP]\nrw [inter_comm, hu2]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d e e' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 : \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source \u2229 e'.target = u \u2229 e'.target\n\u22a2 IsOpen (u \u2229 e'.target)\n[PROOFSTEP]\nexact hu1.inter e'.open_target\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 ContinuousOn (\u2191e.toLocalEquiv) e.source\n[PROOFSTEP]\nrefine\n  continuousOn_iff'.mpr fun s hs =>\n    \u27e8e \u207b\u00b9' s \u2229 e.source, isOpen_iSup_iff.mpr fun e' => ?_, by rw [inter_assoc, inter_self]; rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\n\u22a2 \u2191e.toLocalEquiv \u207b\u00b9' s \u2229 e.source = \u2191e \u207b\u00b9' s \u2229 e.source \u2229 e.source\n[PROOFSTEP]\nrw [inter_assoc, inter_self]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\n\u22a2 \u2191e.toLocalEquiv \u207b\u00b9' s \u2229 e.source = \u2191e \u207b\u00b9' s \u2229 e.source\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\n\u22a2 IsOpen (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nrefine isOpen_iSup_iff.mpr fun he' => ?_\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\n\u22a2 IsOpen (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nrw [isOpen_coinduced, isOpen_induced_iff]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\n\u22a2 \u2203 t, IsOpen t \u2227 Subtype.val \u207b\u00b9' t = Pretrivialization.setSymm e' \u207b\u00b9' (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nobtain \u27e8u, hu1, hu2\u27e9 := continuousOn_iff'.mp (a.continuous_trivChange _ he _ he') s hs\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n\u22a2 \u2203 t, IsOpen t \u2227 Subtype.val \u207b\u00b9' t = Pretrivialization.setSymm e' \u207b\u00b9' (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nhave hu3 := congr_arg (fun s => (fun x : e'.target => (x : B \u00d7 F)) \u207b\u00b9' s) hu2\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\nhu3 :\n  (fun s => (fun x => \u2191x) \u207b\u00b9' s)\n      (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)) =\n    (fun s => (fun x => \u2191x) \u207b\u00b9' s) (u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source))\n\u22a2 \u2203 t, IsOpen t \u2227 Subtype.val \u207b\u00b9' t = Pretrivialization.setSymm e' \u207b\u00b9' (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nsimp only [Subtype.coe_preimage_self, preimage_inter, univ_inter] at hu3 \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\nhu3 :\n  (fun x => \u2191x) \u207b\u00b9' (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s) \u2229\n      (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    (fun x => \u2191x) \u207b\u00b9' u \u2229 (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n\u22a2 \u2203 t, IsOpen t \u2227 Subtype.val \u207b\u00b9' t = Pretrivialization.setSymm e' \u207b\u00b9' (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nrefine\n  \u27e8u \u2229 e'.toLocalEquiv.target \u2229 e'.toLocalEquiv.symm \u207b\u00b9' e.source, ?_, by\n    simp only [preimage_inter, inter_univ, Subtype.coe_preimage_self, hu3.symm]; rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\nhu3 :\n  (fun x => \u2191x) \u207b\u00b9' (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s) \u2229\n      (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    (fun x => \u2191x) \u207b\u00b9' u \u2229 (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n\u22a2 Subtype.val \u207b\u00b9' (u \u2229 e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    Pretrivialization.setSymm e' \u207b\u00b9' (\u2191e \u207b\u00b9' s \u2229 e.source)\n[PROOFSTEP]\nsimp only [preimage_inter, inter_univ, Subtype.coe_preimage_self, hu3.symm]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\nhu3 :\n  (fun x => \u2191x) \u207b\u00b9' (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s) \u2229\n      (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    (fun x => \u2191x) \u207b\u00b9' u \u2229 (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n\u22a2 (fun x => \u2191x) \u207b\u00b9' (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s) \u2229\n      (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    Pretrivialization.setSymm e' \u207b\u00b9' (\u2191e \u207b\u00b9' s) \u2229 Pretrivialization.setSymm e' \u207b\u00b9' e.source\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\nhu3 :\n  (fun x => \u2191x) \u207b\u00b9' (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s) \u2229\n      (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    (fun x => \u2191x) \u207b\u00b9' u \u2229 (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n\u22a2 IsOpen (u \u2229 e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n[PROOFSTEP]\nrw [inter_assoc]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nx\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ns : Set (B \u00d7 F)\nhs : IsOpen s\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nu : Set (B \u00d7 F)\nhu1 : IsOpen u\nhu2 :\n  \u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\nhu3 :\n  (fun x => \u2191x) \u207b\u00b9' (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' s) \u2229\n      (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source) =\n    (fun x => \u2191x) \u207b\u00b9' u \u2229 (fun x => \u2191x) \u207b\u00b9' (\u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n\u22a2 IsOpen (u \u2229 (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source))\n[PROOFSTEP]\nexact hu1.inter (a.isOpen_target_of_mem_pretrivializationAtlas_inter e e' he')\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nx : E b\n\u22a2 { proj := b, snd := x } \u2208 (pretrivializationAt a b).source\n[PROOFSTEP]\nsimp only [(a.pretrivializationAt b).source_eq, mem_preimage, TotalSpace.proj]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nx : E b\n\u22a2 b \u2208 (pretrivializationAt a b).baseSet\n[PROOFSTEP]\nexact a.mem_base_pretrivializationAt b\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\n\u22a2 Continuous (TotalSpace.mk b)\n[PROOFSTEP]\nletI := a.totalSpaceTopology\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 Continuous (TotalSpace.mk b)\n[PROOFSTEP]\nlet e := a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nb : B\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a b \u2208 a.pretrivializationAtlas)\n\u22a2 Continuous (TotalSpace.mk b)\n[PROOFSTEP]\nrw [e.toLocalHomeomorph.continuous_iff_continuous_comp_left (a.totalSpaceMk_preimage_source b)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nb : B\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a b \u2208 a.pretrivializationAtlas)\n\u22a2 Continuous (\u2191e.toLocalHomeomorph \u2218 TotalSpace.mk b)\n[PROOFSTEP]\nexact continuous_iff_le_induced.mpr (le_antisymm_iff.mp (a.totalSpaceMk_inducing b).induced).1\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nh : Inducing (\u2191(pretrivializationAt a b) \u2218 TotalSpace.mk b)\n\u22a2 Inducing (TotalSpace.mk b)\n[PROOFSTEP]\nletI := a.totalSpaceTopology\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nh : Inducing (\u2191(pretrivializationAt a b) \u2218 TotalSpace.mk b)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 Inducing (TotalSpace.mk b)\n[PROOFSTEP]\nrw [\u2190 restrict_comp_codRestrict (a.mem_pretrivializationAt_source b)] at h \n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nh :\n  Inducing\n    (restrict (pretrivializationAt a b).source \u2191(pretrivializationAt a b) \u2218\n      codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source\n        (_ : \u2200 (x : E b), { proj := b, snd := x } \u2208 (pretrivializationAt a b).source))\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 Inducing (TotalSpace.mk b)\n[PROOFSTEP]\napply Inducing.of_codRestrict (a.mem_pretrivializationAt_source b)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nh :\n  Inducing\n    (restrict (pretrivializationAt a b).source \u2191(pretrivializationAt a b) \u2218\n      codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source\n        (_ : \u2200 (x : E b), { proj := b, snd := x } \u2208 (pretrivializationAt a b).source))\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 Inducing\n    (codRestrict (fun a => { proj := b, snd := a }) (pretrivializationAt a b).source\n      (_ : \u2200 (x : E b), { proj := b, snd := x } \u2208 (pretrivializationAt a b).source))\n[PROOFSTEP]\nrefine\n  inducing_of_inducing_compose ?_\n    (continuousOn_iff_continuous_restrict.mp\n      (a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas b)).continuous_toFun)\n    h\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nb : B\nh :\n  Inducing\n    (restrict (pretrivializationAt a b).source \u2191(pretrivializationAt a b) \u2218\n      codRestrict (fun x => { proj := b, snd := x }) (pretrivializationAt a b).source\n        (_ : \u2200 (x : E b), { proj := b, snd := x } \u2208 (pretrivializationAt a b).source))\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 Continuous\n    (codRestrict (fun a => { proj := b, snd := a }) (pretrivializationAt a b).source\n      (_ : \u2200 (x : E b), { proj := b, snd := x } \u2208 (pretrivializationAt a b).source))\n[PROOFSTEP]\nexact (a.continuous_totalSpaceMk b).codRestrict (a.mem_pretrivializationAt_source b)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\n\u22a2 Continuous TotalSpace.proj\n[PROOFSTEP]\nletI := a.totalSpaceTopology\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 Continuous TotalSpace.proj\n[PROOFSTEP]\nletI := a.toFiberBundle\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\nE : B \u2192 Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n\u22a2 Continuous TotalSpace.proj\n[PROOFSTEP]\nexact FiberBundle.continuous_proj F E\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\n\u22a2 ContinuousOn f (TotalSpace.proj \u207b\u00b9' s)\n[PROOFSTEP]\nletI := a.totalSpaceTopology\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\n\u22a2 ContinuousOn f (TotalSpace.proj \u207b\u00b9' s)\n[PROOFSTEP]\nintro z hz\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\n\u22a2 ContinuousWithinAt f (TotalSpace.proj \u207b\u00b9' s) z\n[PROOFSTEP]\nlet e : Trivialization F (\u03c0 F E) := a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas z.proj)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 ContinuousWithinAt f (TotalSpace.proj \u207b\u00b9' s) z\n[PROOFSTEP]\nrefine' (e.continuousAt_of_comp_right _ ((hf z.proj hz).continuousAt (IsOpen.mem_nhds _ _))).continuousWithinAt\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 z.proj \u2208 e.baseSet\n[PROOFSTEP]\nexact a.mem_base_pretrivializationAt z.proj\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 IsOpen ((s \u2229 (pretrivializationAt a z.proj).baseSet) \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact (hs.inter (a.pretrivializationAt z.proj).open_baseSet).prod isOpen_univ\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 \u2191e z \u2208 (s \u2229 (pretrivializationAt a z.proj).baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nrefine' \u27e8_, mem_univ _\u27e9\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 (\u2191e z).fst \u2208 s \u2229 (pretrivializationAt a z.proj).baseSet\n[PROOFSTEP]\nrw [e.coe_fst]\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 z.proj \u2208 s \u2229 (pretrivializationAt a z.proj).baseSet\n[PROOFSTEP]\nexact \u27e8hz, a.mem_base_pretrivializationAt z.proj\u27e9\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 z \u2208 e.source\n[PROOFSTEP]\nrw [e.mem_source]\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX\u271d : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u271d\nE : B \u2192 Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\na : FiberPrebundle F E\ne\u271d : Pretrivialization F TotalSpace.proj\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : TotalSpace F E \u2192 X\ns : Set B\nhs : IsOpen s\nhf :\n  \u2200 (b : B),\n    b \u2208 s \u2192\n      ContinuousOn (f \u2218 \u2191(LocalEquiv.symm (pretrivializationAt a b).toLocalEquiv))\n        ((s \u2229 (pretrivializationAt a b).baseSet) \u00d7\u02e2 univ)\nthis : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nz : TotalSpace F E\nhz : z \u2208 TotalSpace.proj \u207b\u00b9' s\ne : Trivialization F TotalSpace.proj :=\n  trivializationOfMemPretrivializationAtlas a (_ : pretrivializationAt a z.proj \u2208 a.pretrivializationAtlas)\n\u22a2 z.proj \u2208 e.baseSet\n[PROOFSTEP]\nexact a.mem_base_pretrivializationAt z.proj\n", "meta": {"mathlib_filename": "Mathlib.Topology.FiberBundle.Basic", "llama_tokens": 49332, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835411997897, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.5759034976808096}}
{"text": "[GOAL]\nK : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field F\ninst\u271d : Algebra F K\n\u22a2 Splits (algebraMap F K) (X ^ Fintype.card K - X)\n[PROOFSTEP]\nhave h : (X ^ Fintype.card K - X : K[X]).natDegree = Fintype.card K :=\n  FiniteField.X_pow_card_sub_X_natDegree_eq K Fintype.one_lt_card\n[GOAL]\nK : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field F\ninst\u271d : Algebra F K\nh : natDegree (X ^ Fintype.card K - X) = Fintype.card K\n\u22a2 Splits (algebraMap F K) (X ^ Fintype.card K - X)\n[PROOFSTEP]\nrw [\u2190 splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow, map_X, h,\n  FiniteField.roots_X_pow_card_sub_X K, \u2190 Finset.card_def, Finset.card_univ]\n[GOAL]\nK : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field F\ninst\u271d : Algebra F K\n\u22a2 Algebra.adjoin F (rootSet (X ^ Fintype.card K - X) K) = \u22a4\n[PROOFSTEP]\nclassical\ntrans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)\n\u00b7 simp only [rootSet, Polynomial.map_pow, map_X, Polynomial.map_sub]\n\u00b7 rw [FiniteField.roots_X_pow_card_sub_X, val_toFinset, coe_univ, Algebra.adjoin_univ]\n[GOAL]\nK : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field F\ninst\u271d : Algebra F K\n\u22a2 Algebra.adjoin F (rootSet (X ^ Fintype.card K - X) K) = \u22a4\n[PROOFSTEP]\ntrans Algebra.adjoin F ((roots (X ^ Fintype.card K - X : K[X])).toFinset : Set K)\n[GOAL]\nK : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field F\ninst\u271d : Algebra F K\n\u22a2 Algebra.adjoin F (rootSet (X ^ Fintype.card K - X) K) =\n    Algebra.adjoin F \u2191(Multiset.toFinset (roots (X ^ Fintype.card K - X)))\n[PROOFSTEP]\nsimp only [rootSet, Polynomial.map_pow, map_X, Polynomial.map_sub]\n[GOAL]\nK : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field F\ninst\u271d : Algebra F K\n\u22a2 Algebra.adjoin F \u2191(Multiset.toFinset (roots (X ^ Fintype.card K - X))) = \u22a4\n[PROOFSTEP]\nrw [FiniteField.roots_X_pow_card_sub_X, val_toFinset, coe_univ, Algebra.adjoin_univ]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\np q : \u2115\ninst\u271d : CharP K p\nh : p \u2223 q\n\u22a2 Separable (X ^ q - X)\n[PROOFSTEP]\nuse 1, X ^ q - X - 1\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\np q : \u2115\ninst\u271d : CharP K p\nh : p \u2223 q\n\u22a2 1 * (X ^ q - X) + (X ^ q - X - 1) * \u2191derivative (X ^ q - X) = 1\n[PROOFSTEP]\nrw [\u2190 CharP.cast_eq_zero_iff K[X] p] at h \n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\np q : \u2115\ninst\u271d : CharP K p\nh : \u2191q = 0\n\u22a2 1 * (X ^ q - X) + (X ^ q - X - 1) * \u2191derivative (X ^ q - X) = 1\n[PROOFSTEP]\nrw [derivative_sub, derivative_X_pow, derivative_X, C_eq_nat_cast, h]\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\np q : \u2115\ninst\u271d : CharP K p\nh : \u2191q = 0\n\u22a2 1 * (X ^ q - X) + (X ^ q - X - 1) * (0 * X ^ (q - 1) - 1) = 1\n[PROOFSTEP]\nring\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 CharP (ZMod p) p\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 FiniteDimensional (ZMod p) (GaloisField p n)\n[PROOFSTEP]\ndsimp only [GaloisField]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 FiniteDimensional (ZMod p) (SplittingField (X ^ p ^ n - X))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 Fintype (GaloisField p n)\n[PROOFSTEP]\ndsimp only [GaloisField]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 Fintype (SplittingField (X ^ p ^ n - X))\n[PROOFSTEP]\nexact FiniteDimensional.fintypeOfFintype (ZMod p) (GaloisField p n)\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nset g_poly := (X ^ p ^ n - X : (ZMod p)[X])\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nhave hp : 1 < p := h_prime.out.one_lt\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nhave aux : g_poly \u2260 0 := FiniteField.X_pow_card_pow_sub_X_ne_zero _ h hp\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nhave key : Fintype.card (g_poly.rootSet (GaloisField p n)) = g_poly.natDegree :=\n  card_rootSet_eq_natDegree (galois_poly_separable p _ (dvd_pow (dvd_refl p) h))\n    (SplittingField.splits (X ^ p ^ n - X : (ZMod p)[X]))\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = natDegree g_poly\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nhave nat_degree_eq : g_poly.natDegree = p ^ n := FiniteField.X_pow_card_pow_sub_X_natDegree_eq _ h hp\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = natDegree g_poly\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nrw [nat_degree_eq] at key \n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nsuffices g_poly.rootSet (GaloisField p n) = Set.univ\n  by\n  simp_rw [this, \u2190 Fintype.ofEquiv_card (Equiv.Set.univ _)] at key \n  rw [@card_eq_pow_finrank (ZMod p) _ _ _ _ _ (_), ZMod.card] at key \n  exact Nat.pow_right_injective (Nat.Prime.one_lt' p).out key\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\nthis : rootSet g_poly (GaloisField p n) = Set.univ\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nsimp_rw [this, \u2190 Fintype.ofEquiv_card (Equiv.Set.univ _)] at key \n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nnat_degree_eq : natDegree g_poly = p ^ n\nthis : rootSet g_poly (GaloisField p n) = Set.univ\nkey : Fintype.card (GaloisField p n) = p ^ n\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nrw [@card_eq_pow_finrank (ZMod p) _ _ _ _ _ (_), ZMod.card] at key \n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nnat_degree_eq : natDegree g_poly = p ^ n\nthis : rootSet g_poly (GaloisField p n) = Set.univ\nkey\u271d : Fintype.card (GaloisField p n) = p ^ n\nkey : p ^ FiniteDimensional.finrank (ZMod p) (GaloisField p n) = p ^ n\n\u22a2 FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n\n[PROOFSTEP]\nexact Nat.pow_right_injective (Nat.Prime.one_lt' p).out key\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 rootSet g_poly (GaloisField p n) = Set.univ\n[PROOFSTEP]\nrw [Set.eq_univ_iff_forall]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 \u2200 (x : GaloisField p n), x \u2208 rootSet g_poly (GaloisField p n)\n[PROOFSTEP]\nsuffices\n  \u2200 (x) (hx : x \u2208 (\u22a4 : Subalgebra (ZMod p) (GaloisField p n))),\n    x \u2208 (X ^ p ^ n - X : (ZMod p)[X]).rootSet (GaloisField p n)\n  by simpa\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\nthis : \u2200 (x : GaloisField p n), x \u2208 \u22a4 \u2192 x \u2208 rootSet (X ^ p ^ n - X) (GaloisField p n)\n\u22a2 \u2200 (x : GaloisField p n), x \u2208 rootSet g_poly (GaloisField p n)\n[PROOFSTEP]\nsimpa\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 \u2200 (x : GaloisField p n), x \u2208 \u22a4 \u2192 x \u2208 rootSet (X ^ p ^ n - X) (GaloisField p n)\n[PROOFSTEP]\nrw [\u2190 SplittingField.adjoin_rootSet]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 \u2200 (x : GaloisField p n),\n    x \u2208 Algebra.adjoin (ZMod p) (rootSet (X ^ p ^ n - X) (SplittingField (X ^ p ^ n - X))) \u2192\n      x \u2208 rootSet (X ^ p ^ n - X) (GaloisField p n)\n[PROOFSTEP]\nsimp_rw [Algebra.mem_adjoin_iff]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\n\u22a2 \u2200 (x : GaloisField p n),\n    x \u2208\n        Subring.closure\n          (Set.range \u2191(algebraMap (ZMod p) (GaloisField p n)) \u222a\n            rootSet (X ^ p ^ n - X) (SplittingField (X ^ p ^ n - X))) \u2192\n      x \u2208 rootSet (X ^ p ^ n - X) (GaloisField p n)\n[PROOFSTEP]\nintro x hx\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d\u00b9 p : \u2115\nh_prime : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh : n \u2260 0\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField p n)) = p ^ n\nnat_degree_eq : natDegree g_poly = p ^ n\nx : GaloisField p n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod p) (GaloisField p n)) \u222a rootSet (X ^ p ^ n - X) (SplittingField (X ^ p ^ n - X)))\n\u22a2 x \u2208 rootSet (X ^ p ^ n - X) (GaloisField p n)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase zero\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b9 n\u271d n : \u2115\nh : n \u2260 0\nh_prime : Fact (Nat.Prime Nat.zero)\ng_poly : (ZMod Nat.zero)[X] := X ^ Nat.zero ^ n - X\nhp : 1 < Nat.zero\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField Nat.zero n)) = Nat.zero ^ n\nnat_degree_eq : natDegree g_poly = Nat.zero ^ n\nx : GaloisField Nat.zero n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod Nat.zero) (GaloisField Nat.zero n)) \u222a\n        rootSet (X ^ Nat.zero ^ n - X) (SplittingField (X ^ Nat.zero ^ n - X)))\n\u22a2 x \u2208 rootSet (X ^ Nat.zero ^ n - X) (GaloisField Nat.zero n)\ncase succ\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\ncases hp\n[GOAL]\ncase succ\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nrefine Subring.closure_induction hx ?_ ?_ ?_ ?_ ?_ ?_\n[GOAL]\ncase succ.refine_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x : GaloisField (Nat.succ n\u271d) n),\n    x \u2208\n        Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n          rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)) \u2192\n      x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nsimp_rw [mem_rootSet_of_ne aux]\n[GOAL]\ncase succ.refine_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 0 \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nsimp_rw [mem_rootSet_of_ne aux]\n[GOAL]\ncase succ.refine_3\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 1 \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nsimp_rw [mem_rootSet_of_ne aux]\n[GOAL]\ncase succ.refine_4\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x y : GaloisField (Nat.succ n\u271d) n),\n    x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n) \u2192\n      y \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n) \u2192\n        x + y \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nsimp_rw [mem_rootSet_of_ne aux]\n[GOAL]\ncase succ.refine_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x : GaloisField (Nat.succ n\u271d) n),\n    x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n) \u2192\n      -x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nsimp_rw [mem_rootSet_of_ne aux]\n[GOAL]\ncase succ.refine_6\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x y : GaloisField (Nat.succ n\u271d) n),\n    x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n) \u2192\n      y \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n) \u2192\n        x * y \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)\n[PROOFSTEP]\nsimp_rw [mem_rootSet_of_ne aux]\n[GOAL]\ncase succ.refine_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x : GaloisField (Nat.succ n\u271d) n),\n    x \u2208\n        Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n          rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)) \u2192\n      \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nrintro x (\u27e8r, rfl\u27e9 | hx)\n[GOAL]\ncase succ.refine_1.inl.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nr : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(aeval (\u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) r)) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nsimp only [map_sub, map_pow, aeval_X]\n[GOAL]\ncase succ.refine_1.inl.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nr : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) r ^ Nat.succ n\u271d ^ n -\n      \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) r =\n    0\n[PROOFSTEP]\nrw [\u2190 map_pow, ZMod.pow_card_pow, sub_self]\n[GOAL]\ncase succ.refine_1.inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx : GaloisField (Nat.succ n\u271d) n\nhx : x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X))\n\u22a2 \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\ndsimp only [GaloisField] at hx \n[GOAL]\ncase succ.refine_1.inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx : GaloisField (Nat.succ n\u271d) n\nhx : x \u2208 rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X))\n\u22a2 \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nrwa [mem_rootSet_of_ne aux] at hx \n[GOAL]\ncase succ.refine_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2191(aeval 0) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nrw [\u2190 coeff_zero_eq_aeval_zero']\n[GOAL]\ncase succ.refine_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) (coeff (X ^ Nat.succ n\u271d ^ n - X) 0) = 0\n[PROOFSTEP]\nsimp only [coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff, one_ne_zero, coeff_sub]\n[GOAL]\ncase succ.refine_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 0 = Nat.succ n\u271d ^ n \u2192 False\n[PROOFSTEP]\nintro hn\n[GOAL]\ncase succ.refine_2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nhn : 0 = Nat.succ n\u271d ^ n\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_lt_zero 1 (pow_eq_zero hn.symm \u25b8 hp)\n[GOAL]\ncase succ.refine_3\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2191(aeval 1) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.refine_4\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x y : GaloisField (Nat.succ n\u271d) n),\n    \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0 \u2192\n      \u2191(aeval y) (X ^ Nat.succ n\u271d ^ n - X) = 0 \u2192 \u2191(aeval (x + y)) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nsimp only [aeval_X_pow, aeval_X, AlgHom.map_sub, add_pow_char_pow, sub_eq_zero]\n[GOAL]\ncase succ.refine_4\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x y : GaloisField (Nat.succ n\u271d) n),\n    x ^ Nat.succ n\u271d ^ n = x \u2192 y ^ Nat.succ n\u271d ^ n = y \u2192 x ^ Nat.succ n\u271d ^ n + y ^ Nat.succ n\u271d ^ n = x + y\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase succ.refine_4\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx y : GaloisField (Nat.succ n\u271d) n\nhx : x ^ Nat.succ n\u271d ^ n = x\nhy : y ^ Nat.succ n\u271d ^ n = y\n\u22a2 x ^ Nat.succ n\u271d ^ n + y ^ Nat.succ n\u271d ^ n = x + y\n[PROOFSTEP]\nrw [hx, hy]\n[GOAL]\ncase succ.refine_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x : GaloisField (Nat.succ n\u271d) n),\n    \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0 \u2192 \u2191(aeval (-x)) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase succ.refine_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx : GaloisField (Nat.succ n\u271d) n\nhx : \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0\n\u22a2 \u2191(aeval (-x)) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nsimp only [sub_eq_zero, aeval_X_pow, aeval_X, AlgHom.map_sub, sub_neg_eq_add] at *\n[GOAL]\ncase succ.refine_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : X ^ Nat.succ n\u271d ^ n - X \u2260 0\nkey : Fintype.card \u2191(rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree (X ^ Nat.succ n\u271d ^ n - X) = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx : GaloisField (Nat.succ n\u271d) n\nhx : x ^ Nat.succ n\u271d ^ n = x\n\u22a2 (-x) ^ Nat.succ n\u271d ^ n + x = 0\n[PROOFSTEP]\nrw [neg_pow, hx, CharP.neg_one_pow_char_pow]\n[GOAL]\ncase succ.refine_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : X ^ Nat.succ n\u271d ^ n - X \u2260 0\nkey : Fintype.card \u2191(rootSet (X ^ Nat.succ n\u271d ^ n - X) (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree (X ^ Nat.succ n\u271d ^ n - X) = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx : GaloisField (Nat.succ n\u271d) n\nhx : x ^ Nat.succ n\u271d ^ n = x\n\u22a2 -1 * x + x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.refine_6\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x y : GaloisField (Nat.succ n\u271d) n),\n    \u2191(aeval x) (X ^ Nat.succ n\u271d ^ n - X) = 0 \u2192\n      \u2191(aeval y) (X ^ Nat.succ n\u271d ^ n - X) = 0 \u2192 \u2191(aeval (x * y)) (X ^ Nat.succ n\u271d ^ n - X) = 0\n[PROOFSTEP]\nsimp only [aeval_X_pow, aeval_X, AlgHom.map_sub, mul_pow, sub_eq_zero]\n[GOAL]\ncase succ.refine_6\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx : GaloisField (Nat.succ n\u271d) n\nhx :\n  x \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\n\u22a2 \u2200 (x y : GaloisField (Nat.succ n\u271d) n),\n    x ^ Nat.succ n\u271d ^ n = x \u2192 y ^ Nat.succ n\u271d ^ n = y \u2192 x ^ Nat.succ n\u271d ^ n * y ^ Nat.succ n\u271d ^ n = x * y\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase succ.refine_6\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b2 n\u271d\u00b9 n : \u2115\nh : n \u2260 0\nn\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\ng_poly : (ZMod (Nat.succ n\u271d))[X] := X ^ Nat.succ n\u271d ^ n - X\nhp : 1 < Nat.succ n\u271d\naux : g_poly \u2260 0\nkey : Fintype.card \u2191(rootSet g_poly (GaloisField (Nat.succ n\u271d) n)) = Nat.succ n\u271d ^ n\nnat_degree_eq : natDegree g_poly = Nat.succ n\u271d ^ n\nx\u271d : GaloisField (Nat.succ n\u271d) n\nhx\u271d :\n  x\u271d \u2208\n    Subring.closure\n      (Set.range \u2191(algebraMap (ZMod (Nat.succ n\u271d)) (GaloisField (Nat.succ n\u271d) n)) \u222a\n        rootSet (X ^ Nat.succ n\u271d ^ n - X) (SplittingField (X ^ Nat.succ n\u271d ^ n - X)))\nx y : GaloisField (Nat.succ n\u271d) n\nhx : x ^ Nat.succ n\u271d ^ n = x\nhy : y ^ Nat.succ n\u271d ^ n = y\n\u22a2 x ^ Nat.succ n\u271d ^ n * y ^ Nat.succ n\u271d ^ n = x * y\n[PROOFSTEP]\nrw [hx, hy]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nh : n \u2260 0\n\u22a2 Fintype.card (GaloisField p n) = p ^ n\n[PROOFSTEP]\nlet b := IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nh : n \u2260 0\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) (GaloisField p n) } (ZMod p) (GaloisField p n) :=\n  IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)\n\u22a2 Fintype.card (GaloisField p n) = p ^ n\n[PROOFSTEP]\nrw [Module.card_fintype b, \u2190 FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 Splits (RingHom.id (ZMod p)) (X ^ p - X)\n[PROOFSTEP]\nhave hp : 1 < p := h_prime.out.one_lt\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nhp : 1 < p\n\u22a2 Splits (RingHom.id (ZMod p)) (X ^ p - X)\n[PROOFSTEP]\nhave h1 : roots (X ^ p - X : (ZMod p)[X]) = Finset.univ.val :=\n  by\n  convert FiniteField.roots_X_pow_card_sub_X (ZMod p)\n  exact (ZMod.card p).symm\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nhp : 1 < p\n\u22a2 roots (X ^ p - X) = univ.val\n[PROOFSTEP]\nconvert FiniteField.roots_X_pow_card_sub_X (ZMod p)\n[GOAL]\ncase h.e'_2.h.e'_4.h.h.e'_5.h.e'_6\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nhp : 1 < p\ne_2\u271d : ZMod.commRing p = EuclideanDomain.toCommRing\n\u22a2 p = Fintype.card (ZMod p)\n[PROOFSTEP]\nexact (ZMod.card p).symm\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nhp : 1 < p\nh1 : roots (X ^ p - X) = univ.val\n\u22a2 Splits (RingHom.id (ZMod p)) (X ^ p - X)\n[PROOFSTEP]\nhave h2 := FiniteField.X_pow_card_sub_X_natDegree_eq (ZMod p) hp\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nhp : 1 < p\nh1 : roots (X ^ p - X) = univ.val\nh2 : natDegree (X ^ p - X) = p\n\u22a2 Splits (RingHom.id (ZMod p)) (X ^ p - X)\n[PROOFSTEP]\ncases p\n[GOAL]\ncase zero\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d n : \u2115\nh_prime : Fact (Nat.Prime Nat.zero)\nhp : 1 < Nat.zero\nh1 : roots (X ^ Nat.zero - X) = univ.val\nh2 : natDegree (X ^ Nat.zero - X) = Nat.zero\n\u22a2 Splits (RingHom.id (ZMod Nat.zero)) (X ^ Nat.zero - X)\ncase succ\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b9 n n\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\nhp : 1 < Nat.succ n\u271d\nh1 : roots (X ^ Nat.succ n\u271d - X) = univ.val\nh2 : natDegree (X ^ Nat.succ n\u271d - X) = Nat.succ n\u271d\n\u22a2 Splits (RingHom.id (ZMod (Nat.succ n\u271d))) (X ^ Nat.succ n\u271d - X)\n[PROOFSTEP]\ncases hp\n[GOAL]\ncase succ\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nn\u271d\u00b9 n n\u271d : \u2115\nh_prime : Fact (Nat.Prime (Nat.succ n\u271d))\nhp : 1 < Nat.succ n\u271d\nh1 : roots (X ^ Nat.succ n\u271d - X) = univ.val\nh2 : natDegree (X ^ Nat.succ n\u271d - X) = Nat.succ n\u271d\n\u22a2 Splits (RingHom.id (ZMod (Nat.succ n\u271d))) (X ^ Nat.succ n\u271d - X)\n[PROOFSTEP]\nrw [splits_iff_card_roots, h1, \u2190 Finset.card_def, Finset.card_univ, h2, ZMod.card]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 X ^ p ^ 1 = X ^ Fintype.card (ZMod p)\n[PROOFSTEP]\nrw [pow_one, ZMod.card p]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nh : X ^ p ^ 1 = X ^ Fintype.card (ZMod p) :=\n  Eq.mpr (id (pow_one p \u25b8 Eq.refl (X ^ p ^ 1 = X ^ Fintype.card (ZMod p))))\n    (Eq.mpr (id (ZMod.card p \u25b8 Eq.refl (X ^ p = X ^ Fintype.card (ZMod p)))) (Eq.refl (X ^ p)))\n\u22a2 IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X)\n[PROOFSTEP]\nrw [h]\n[GOAL]\np\u271d : \u2115\ninst\u271d : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nh : X ^ p ^ 1 = X ^ Fintype.card (ZMod p) :=\n  Eq.mpr (id (pow_one p \u25b8 Eq.refl (X ^ p ^ 1 = X ^ Fintype.card (ZMod p))))\n    (Eq.mpr (id (ZMod.card p \u25b8 Eq.refl (X ^ p = X ^ Fintype.card (ZMod p)))) (Eq.refl (X ^ p)))\n\u22a2 IsSplittingField (ZMod p) (ZMod p) (X ^ Fintype.card (ZMod p) - X)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2077 : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\ninst\u271d\u2075 : Fintype K\u271d\ninst\u271d\u2074 : Algebra (ZMod p) K\u271d\nK : Type u_2\nK' : Type u_3\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field K'\ninst\u271d\u00b9 : Finite K'\ninst\u271d : Algebra K K'\n\u22a2 IsGalois K K'\n[PROOFSTEP]\ncases nonempty_fintype K'\n[GOAL]\ncase intro\np\u271d : \u2115\ninst\u271d\u2077 : Fact (Nat.Prime p\u271d)\nn\u271d p : \u2115\nh_prime : Fact (Nat.Prime p)\nn : \u2115\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\ninst\u271d\u2075 : Fintype K\u271d\ninst\u271d\u2074 : Algebra (ZMod p) K\u271d\nK : Type u_2\nK' : Type u_3\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field K'\ninst\u271d\u00b9 : Finite K'\ninst\u271d : Algebra K K'\nval\u271d : Fintype K'\n\u22a2 IsGalois K K'\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := CharP.exists K\n[GOAL]\ncase intro.intro\np\u271d\u00b9 : \u2115\ninst\u271d\u2077 : Fact (Nat.Prime p\u271d\u00b9)\nn\u271d p\u271d : \u2115\nh_prime : Fact (Nat.Prime p\u271d)\nn : \u2115\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\ninst\u271d\u2075 : Fintype K\u271d\ninst\u271d\u2074 : Algebra (ZMod p\u271d) K\u271d\nK : Type u_2\nK' : Type u_3\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field K'\ninst\u271d\u00b9 : Finite K'\ninst\u271d : Algebra K K'\nval\u271d : Fintype K'\np : \u2115\nhp : CharP K p\n\u22a2 IsGalois K K'\n[PROOFSTEP]\nhaveI : CharP K p := hp\n[GOAL]\ncase intro.intro\np\u271d\u00b9 : \u2115\ninst\u271d\u2077 : Fact (Nat.Prime p\u271d\u00b9)\nn\u271d p\u271d : \u2115\nh_prime : Fact (Nat.Prime p\u271d)\nn : \u2115\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\ninst\u271d\u2075 : Fintype K\u271d\ninst\u271d\u2074 : Algebra (ZMod p\u271d) K\u271d\nK : Type u_2\nK' : Type u_3\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field K'\ninst\u271d\u00b9 : Finite K'\ninst\u271d : Algebra K K'\nval\u271d : Fintype K'\np : \u2115\nhp : CharP K p\nthis : CharP K p\n\u22a2 IsGalois K K'\n[PROOFSTEP]\nhaveI : CharP K' p := charP_of_injective_algebraMap' K K' p\n[GOAL]\ncase intro.intro\np\u271d\u00b9 : \u2115\ninst\u271d\u2077 : Fact (Nat.Prime p\u271d\u00b9)\nn\u271d p\u271d : \u2115\nh_prime : Fact (Nat.Prime p\u271d)\nn : \u2115\nK\u271d : Type u_1\ninst\u271d\u2076 : Field K\u271d\ninst\u271d\u2075 : Fintype K\u271d\ninst\u271d\u2074 : Algebra (ZMod p\u271d) K\u271d\nK : Type u_2\nK' : Type u_3\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field K'\ninst\u271d\u00b9 : Finite K'\ninst\u271d : Algebra K K'\nval\u271d : Fintype K'\np : \u2115\nhp : CharP K p\nthis\u271d : CharP K p\nthis : CharP K' p\n\u22a2 IsGalois K K'\n[PROOFSTEP]\nexact\n  IsGalois.of_separable_splitting_field\n    (galois_poly_separable p (Fintype.card K')\n      (let \u27e8n, _, hn\u27e9 := FiniteField.card K' p\n      hn.symm \u25b8 dvd_pow_self p n.ne_zero))\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nhave : CharP K p := by rw [\u2190 Algebra.charP_iff (ZMod p) K p]; exact ZMod.charP p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\n\u22a2 CharP K p\n[PROOFSTEP]\nrw [\u2190 Algebra.charP_iff (ZMod p) K p]\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\n\u22a2 CharP (ZMod p) p\n[PROOFSTEP]\nexact ZMod.charP p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\nthis : CharP K p\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nhave : CharP K' p := by rw [\u2190 Algebra.charP_iff (ZMod p) K' p]; exact ZMod.charP p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\nthis : CharP K p\n\u22a2 CharP K' p\n[PROOFSTEP]\nrw [\u2190 Algebra.charP_iff (ZMod p) K' p]\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\nthis : CharP K p\n\u22a2 CharP (ZMod p) p\n[PROOFSTEP]\nexact ZMod.charP p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\nthis\u271d : CharP K p\nthis : CharP K' p\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nchoose n a hK using FiniteField.card K p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\nthis\u271d : CharP K p\nthis : CharP K' p\nn : \u2115+\na : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nchoose n' a' hK' using FiniteField.card K' p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nhKK' : Fintype.card K = Fintype.card K'\nthis\u271d : CharP K p\nthis : CharP K' p\nn : \u2115+\na : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\na' : Nat.Prime p\nhK' : Fintype.card K' = p ^ \u2191n'\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nrw [hK, hK'] at hKK' \n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nthis\u271d : CharP K p\nthis : CharP K' p\nn : \u2115+\na : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhKK' : p ^ \u2191n = p ^ \u2191n'\na' : Nat.Prime p\nhK' : Fintype.card K' = p ^ \u2191n'\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nhave hGalK := GaloisField.algEquivGaloisField p n hK\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nthis\u271d : CharP K p\nthis : CharP K' p\nn : \u2115+\na : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhKK' : p ^ \u2191n = p ^ \u2191n'\na' : Nat.Prime p\nhK' : Fintype.card K' = p ^ \u2191n'\nhGalK : K \u2243\u2090[ZMod p] GaloisField p \u2191n\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nhave hK'Gal := (GaloisField.algEquivGaloisField p n' hK').symm\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nthis\u271d : CharP K p\nthis : CharP K' p\nn : \u2115+\na : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhKK' : p ^ \u2191n = p ^ \u2191n'\na' : Nat.Prime p\nhK' : Fintype.card K' = p ^ \u2191n'\nhGalK : K \u2243\u2090[ZMod p] GaloisField p \u2191n\nhK'Gal : GaloisField p \u2191n' \u2243\u2090[ZMod p] K'\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nrw [Nat.pow_right_injective h_prime.out.one_lt hKK'] at *\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2076 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Fintype K\nK' : Type u_2\ninst\u271d\u00b3 : Field K'\ninst\u271d\u00b2 : Fintype K'\np : \u2115\nh_prime : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Algebra (ZMod p) K\ninst\u271d : Algebra (ZMod p) K'\nthis\u271d : CharP K p\nthis : CharP K' p\nn : \u2115+\na : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhKK' : p ^ \u2191n' = p ^ \u2191n'\na' : Nat.Prime p\nhK' : Fintype.card K' = p ^ \u2191n'\nhGalK : K \u2243\u2090[ZMod p] GaloisField p \u2191n'\nhK'Gal : GaloisField p \u2191n' \u2243\u2090[ZMod p] K'\n\u22a2 K \u2243\u2090[ZMod p] K'\n[PROOFSTEP]\nexact AlgEquiv.trans hGalK hK'Gal\n[GOAL]\np : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p)\nn : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nchoose p _char_p_K using CharP.exists K\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nchoose p' _char_p'_K' using CharP.exists K'\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nchoose n hp hK using FiniteField.card K p\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nchoose n' hp' hK' using FiniteField.card K' p'\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nhave hpp' : p = p' := by\n  by_contra hne\n  have h2 := Nat.coprime_pow_primes n n' hp hp' hne\n  rw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2 \n  exact Nat.Prime.ne_one hp' h2\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\n\u22a2 p = p'\n[PROOFSTEP]\nby_contra hne\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhne : \u00acp = p'\n\u22a2 False\n[PROOFSTEP]\nhave h2 := Nat.coprime_pow_primes n n' hp hp' hne\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhne : \u00acp = p'\nh2 : Nat.coprime (p ^ \u2191n) (p' ^ \u2191n')\n\u22a2 False\n[PROOFSTEP]\nrw [(Eq.congr hK hK').mp hKK', Nat.coprime_self, pow_eq_one_iff (PNat.ne_zero n')] at h2 \n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhne : \u00acp = p'\nh2 : p' = 1\n\u22a2 False\n[PROOFSTEP]\nexact Nat.Prime.ne_one hp' h2\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p'\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhpp' : p = p'\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nrw [\u2190 hpp'] at _char_p'_K' \n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhpp' : p = p'\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nhaveI := fact_iff.2 hp\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhpp' : p = p'\nthis : Fact (Nat.Prime p)\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nletI : Algebra (ZMod p) K := ZMod.algebra _ _\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhpp' : p = p'\nthis\u271d : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) K := ZMod.algebra K p\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nletI : Algebra (ZMod p) K' := ZMod.algebra _ _\n[GOAL]\np\u271d : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p\u271d)\nn\u271d : \u2115\nK : Type u_1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Fintype K\nK' : Type u_2\ninst\u271d\u00b9 : Field K'\ninst\u271d : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : \u2115\n_char_p_K : CharP K p\np' : \u2115\n_char_p'_K' : CharP K' p\nn : \u2115+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ \u2191n\nn' : \u2115+\nhp' : Nat.Prime p'\nhK' : Fintype.card K' = p' ^ \u2191n'\nhpp' : p = p'\nthis\u271d\u00b9 : Fact (Nat.Prime p)\nthis\u271d : Algebra (ZMod p) K := ZMod.algebra K p\nthis : Algebra (ZMod p) K' := ZMod.algebra K' p\n\u22a2 K \u2243+* K'\n[PROOFSTEP]\nexact \u2191(algEquivOfCardEq p hKK')\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Finite.GaloisField", "llama_tokens": 28707, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430562234878, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.5756845747313021}}
{"text": "[GOAL]\n\u22a2 NoncompactSpace \u210d\n[PROOFSTEP]\nrefine' \u27e8fun h => _\u27e9\n[GOAL]\nh : IsCompact univ\n\u22a2 False\n[PROOFSTEP]\nhave : IsCompact (Complex.im \u207b\u00b9' Ioi 0) := isCompact_iff_isCompact_univ.2 h\n[GOAL]\nh : IsCompact univ\nthis : IsCompact (Complex.im \u207b\u00b9' Ioi 0)\n\u22a2 False\n[PROOFSTEP]\nreplace := this.isClosed.closure_eq\n[GOAL]\nh : IsCompact univ\nthis : closure (Complex.im \u207b\u00b9' Ioi 0) = Complex.im \u207b\u00b9' Ioi 0\n\u22a2 False\n[PROOFSTEP]\nrw [closure_preimage_im, closure_Ioi, Set.ext_iff] at this \n[GOAL]\nh : IsCompact univ\nthis : \u2200 (x : \u2102), x \u2208 Complex.im \u207b\u00b9' Ici 0 \u2194 x \u2208 Complex.im \u207b\u00b9' Ioi 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd ((this 0).1 (@left_mem_Ici \u211d _ 0)) (@lt_irrefl \u211d _ 0)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.UpperHalfPlane.Topology", "llama_tokens": 324, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511543206819, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.5754158608289776}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxz := hxy.trans hyz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxz : x < z\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nrw [\u2190 sub_pos] at hxy hxz hyz \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nsuffices f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n  by\n  ring_nf at this \u22a2\n  linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nthis : f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nring_nf at this \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nthis : f y * (y - x)\u207b\u00b9 + f y * (-y + z)\u207b\u00b9 \u2264 (y - x)\u207b\u00b9 * f x + (-y + z)\u207b\u00b9 * f z\n\u22a2 f y * (y - x)\u207b\u00b9 - (y - x)\u207b\u00b9 * f x \u2264 -(f y * (-y + z)\u207b\u00b9) + (-y + z)\u207b\u00b9 * f z\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nset a := (z - y) / (z - x)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nset b := (y - x) / (z - x)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nhave hy : a \u2022 x + b \u2022 z = y := by\n  field_simp\n  rw [div_eq_iff] <;> [ring; linarith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 a \u2022 x + b \u2022 z = y\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 ((z - y) * x + (y - x) * z) / (z - x) = y\n[PROOFSTEP]\nrw [div_eq_iff] <;> [ring; linarith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 ((z - y) * x + (y - x) * z) / (z - x) = y\n[PROOFSTEP]\nrw [div_eq_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 (z - y) * x + (y - x) * z = y * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 z - x \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nhave key :=\n  hf.2 hx hz (show 0 \u2264 a by apply div_nonneg <;> linarith) (show 0 \u2264 b by apply div_nonneg <;> linarith)\n    (show a + b = 1 by\n      field_simp\n      rw [div_eq_iff] <;> [ring; linarith])\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 0 \u2264 a\n[PROOFSTEP]\napply div_nonneg\n[GOAL]\ncase ha\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 0 \u2264 z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hb\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 0 \u2264 z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 0 \u2264 b\n[PROOFSTEP]\napply div_nonneg\n[GOAL]\ncase ha\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 0 \u2264 y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hb\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 0 \u2264 z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 a + b = 1\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 (z - x) / (z - x) = 1\n[PROOFSTEP]\nrw [div_eq_iff] <;> [ring; linarith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 (z - x) / (z - x) = 1\n[PROOFSTEP]\nrw [div_eq_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 z - x = 1 * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 z - x \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nrw [hy] at key \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y \u2264 a \u2022 f x + b \u2022 f z\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nreplace key := mul_le_mul_of_nonneg_left key hxz.le\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : (z - x) * f y \u2264 (z - x) * (a \u2022 f x + b \u2022 f z)\n\u22a2 f y / (y - x) + f y / (z - y) \u2264 f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nfield_simp [hxy.ne', hyz.ne', hxz.ne', mul_comm (z - x) _] at key \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y * (z - x) \u2264 (z - y) * f x + (y - x) * f z\n\u22a2 (f y * (z - y) + f y * (y - x)) / ((y - x) * (z - y)) \u2264 (f x * (z - y) + f z * (y - x)) / ((y - x) * (z - y))\n[PROOFSTEP]\nrw [div_le_div_right]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y * (z - x) \u2264 (z - y) * f x + (y - x) * f z\n\u22a2 f y * (z - y) + f y * (y - x) \u2264 f x * (z - y) + f z * (y - x)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y * (z - x) \u2264 (z - y) * f x + (y - x) * f z\n\u22a2 0 < (y - x) * (z - y)\n[PROOFSTEP]\nnlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConcaveOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\n[PROOFSTEP]\nhave := neg_le_neg (ConvexOn.slope_mono_adjacent hf.neg hx hz hxy hyz)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConcaveOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) \u2264 -(((-f) y - (-f) x) / (y - x))\n\u22a2 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\n[PROOFSTEP]\nsimp only [Pi.neg_apply, \u2190 neg_div, neg_sub', neg_neg] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConcaveOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\n\u22a2 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\n[PROOFSTEP]\nexact this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxz := hxy.trans hyz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxz : x < z\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxz' := hxz.ne\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxz : x < z\nhxz' : x \u2260 z\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nrw [\u2190 sub_pos] at hxy hxz hyz \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nsuffices f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n  by\n  ring_nf at this \u22a2\n  linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\nthis : f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nring_nf at this \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\nthis : f y * (y - x)\u207b\u00b9 + f y * (-y + z)\u207b\u00b9 < (y - x)\u207b\u00b9 * f x + (-y + z)\u207b\u00b9 * f z\n\u22a2 f y * (y - x)\u207b\u00b9 - (y - x)\u207b\u00b9 * f x < -(f y * (-y + z)\u207b\u00b9) + (-y + z)\u207b\u00b9 * f z\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nset a := (z - y) / (z - x)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nset b := (y - x) / (z - x)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nhave hy : a \u2022 x + b \u2022 z = y := by\n  field_simp\n  rw [div_eq_iff] <;> [ring; linarith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 a \u2022 x + b \u2022 z = y\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 ((z - y) * x + (y - x) * z) / (z - x) = y\n[PROOFSTEP]\nrw [div_eq_iff] <;> [ring; linarith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 ((z - y) * x + (y - x) * z) / (z - x) = y\n[PROOFSTEP]\nrw [div_eq_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 (z - y) * x + (y - x) * z = y * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\n\u22a2 z - x \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nhave key :=\n  hf.2 hx hz hxz' (div_pos hyz hxz) (div_pos hxy hxz)\n    (show a + b = 1 by\n      field_simp\n      rw [div_eq_iff] <;> [ring; linarith])\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 a + b = 1\n[PROOFSTEP]\nfield_simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 (z - x) / (z - x) = 1\n[PROOFSTEP]\nrw [div_eq_iff] <;> [ring; linarith]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 (z - x) / (z - x) = 1\n[PROOFSTEP]\nrw [div_eq_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 z - x = 1 * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\n\u22a2 z - x \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f (((z - y) / (z - x)) \u2022 x + ((y - x) / (z - x)) \u2022 z) < ((z - y) / (z - x)) \u2022 f x + ((y - x) / (z - x)) \u2022 f z\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nrw [hy] at key \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y < ((z - y) / (z - x)) \u2022 f x + ((y - x) / (z - x)) \u2022 f z\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nreplace key := mul_lt_mul_of_pos_left key hxz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : (z - x) * f y < (z - x) * (((z - y) / (z - x)) \u2022 f x + ((y - x) / (z - x)) \u2022 f z)\n\u22a2 f y / (y - x) + f y / (z - y) < f x / (y - x) + f z / (z - y)\n[PROOFSTEP]\nfield_simp [hxy.ne', hyz.ne', hxz.ne', mul_comm (z - x) _] at key \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y * (z - x) < (z - y) * f x + (y - x) * f z\n\u22a2 (f y * (z - y) + f y * (y - x)) / ((y - x) * (z - y)) < (f x * (z - y) + f z * (y - x)) / ((y - x) * (z - y))\n[PROOFSTEP]\nrw [div_lt_div_right]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y * (z - x) < (z - y) * f x + (y - x) * f z\n\u22a2 f y * (z - y) + f y * (y - x) < f x * (z - y) + f z * (y - x)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy\u271d : x < y\nhxy : 0 < y - x\nhyz\u271d : y < z\nhyz : 0 < z - y\nhxz\u271d : x < z\nhxz : 0 < z - x\nhxz' : x \u2260 z\na : \ud835\udd5c := (z - y) / (z - x)\nb : \ud835\udd5c := (y - x) / (z - x)\nhy : a \u2022 x + b \u2022 z = y\nkey : f y * (z - x) < (z - y) * f x + (y - x) * f z\n\u22a2 0 < (y - x) * (z - y)\n[PROOFSTEP]\nnlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f z - f y) / (z - y) < (f y - f x) / (y - x)\n[PROOFSTEP]\nhave := neg_lt_neg (StrictConvexOn.slope_strict_mono_adjacent hf.neg hx hz hxy hyz)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) < -(((-f) y - (-f) x) / (y - x))\n\u22a2 (f z - f y) / (z - y) < (f y - f x) / (y - x)\n[PROOFSTEP]\nsimp only [Pi.neg_apply, \u2190 neg_div, neg_sub', neg_neg] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) < (f y - f x) / (y - x)\n\u22a2 (f z - f y) / (z - y) < (f y - f x) / (y - x)\n[PROOFSTEP]\nexact this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nlet y := a * x + b * z\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hxy : x < y := by\n  rw [\u2190 one_mul x, \u2190 hab, add_mul]\n  exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\n\u22a2 x < y\n[PROOFSTEP]\nrw [\u2190 one_mul x, \u2190 hab, add_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\n\u22a2 a * x + b * x < y\n[PROOFSTEP]\nexact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hyz : y < z := by\n  rw [\u2190 one_mul z, \u2190 hab, add_mul]\n  exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\n\u22a2 y < z\n[PROOFSTEP]\nrw [\u2190 one_mul z, \u2190 hab, add_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\n\u22a2 y < a * z + b * z\n[PROOFSTEP]\nexact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x) :=\n  (div_le_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hxz : 0 < z - x := sub_pos.2 (hxy.trans hyz)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave ha : (z - y) / (z - x) = a := by\n  rw [eq_comm, \u2190 sub_eq_iff_eq_add'] at hab \n  simp_rw [div_eq_iff hxz.ne', \u2190 hab]\n  ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 (z - y) / (z - x) = a\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_iff_eq_add'] at hab \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 1 - a = b\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 (z - y) / (z - x) = a\n[PROOFSTEP]\nsimp_rw [div_eq_iff hxz.ne', \u2190 hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 1 - a = b\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 z - (a * x + (1 - a) * z) = a * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hb : (y - x) / (z - x) = b := by\n  rw [eq_comm, \u2190 sub_eq_iff_eq_add] at hab \n  simp_rw [div_eq_iff hxz.ne', \u2190 hab]\n  ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 (y - x) / (z - x) = b\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_iff_eq_add] at hab \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : 1 - b = a\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 (y - x) / (z - x) = b\n[PROOFSTEP]\nsimp_rw [div_eq_iff hxz.ne', \u2190 hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : 1 - b = a\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 (1 - b) * x + b * z - x = b * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb\u271d : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) \u2264 (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\nhb : (y - x) / (z - x) = b\n\u22a2 f (a \u2022 x + b \u2022 z) \u2264 a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nrwa [sub_mul, sub_mul, sub_le_iff_le_add', \u2190 add_sub_assoc, le_sub_iff_add_le, \u2190 mul_add, sub_add_sub_cancel, \u2190\n  le_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x), mul_comm (f z), ha, hb] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\n\u22a2 ConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nrw [\u2190 neg_convexOn_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\n\u22a2 ConvexOn \ud835\udd5c s (-f)\n[PROOFSTEP]\nrefine' convexOn_of_slope_mono_adjacent hs fun hx hz hxy hyz => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\nx\u271d y\u271d z\u271d : \ud835\udd5c\nhx : x\u271d \u2208 s\nhz : z\u271d \u2208 s\nhxy : x\u271d < y\u271d\nhyz : y\u271d < z\u271d\n\u22a2 ((-f) y\u271d - (-f) x\u271d) / (y\u271d - x\u271d) \u2264 ((-f) z\u271d - (-f) y\u271d) / (z\u271d - y\u271d)\n[PROOFSTEP]\nrw [\u2190 neg_le_neg_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\nx\u271d y\u271d z\u271d : \ud835\udd5c\nhx : x\u271d \u2208 s\nhz : z\u271d \u2208 s\nhxy : x\u271d < y\u271d\nhyz : y\u271d < z\u271d\n\u22a2 -(((-f) z\u271d - (-f) y\u271d) / (z\u271d - y\u271d)) \u2264 -(((-f) y\u271d - (-f) x\u271d) / (y\u271d - x\u271d))\n[PROOFSTEP]\nsimp_rw [\u2190 neg_div, neg_sub, Pi.neg_apply, neg_sub_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) \u2264 (f y - f x) / (y - x)\nx\u271d y\u271d z\u271d : \ud835\udd5c\nhx : x\u271d \u2208 s\nhz : z\u271d \u2208 s\nhxy : x\u271d < y\u271d\nhyz : y\u271d < z\u271d\n\u22a2 (f z\u271d - f y\u271d) / (z\u271d - y\u271d) \u2264 (f y\u271d - f x\u271d) / (y\u271d - x\u271d)\n[PROOFSTEP]\nexact hf hx hz hxy hyz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nlet y := a * x + b * z\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hxy : x < y := by\n  rw [\u2190 one_mul x, \u2190 hab, add_mul]\n  exact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\n\u22a2 x < y\n[PROOFSTEP]\nrw [\u2190 one_mul x, \u2190 hab, add_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\n\u22a2 a * x + b * x < y\n[PROOFSTEP]\nexact add_lt_add_left ((mul_lt_mul_left hb).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hyz : y < z := by\n  rw [\u2190 one_mul z, \u2190 hab, add_mul]\n  exact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\n\u22a2 y < z\n[PROOFSTEP]\nrw [\u2190 one_mul z, \u2190 hab, add_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\n\u22a2 y < a * z + b * z\n[PROOFSTEP]\nexact add_lt_add_right ((mul_lt_mul_left ha).2 hxz) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave : (f y - f x) * (z - y) < (f z - f y) * (y - x) :=\n  (div_lt_div_iff (sub_pos.2 hxy) (sub_pos.2 hyz)).1 (hf hx hz hxy hyz)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hxz : 0 < z - x := sub_pos.2 (hxy.trans hyz)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave ha : (z - y) / (z - x) = a := by\n  rw [eq_comm, \u2190 sub_eq_iff_eq_add'] at hab \n  simp_rw [div_eq_iff hxz.ne', \u2190 hab]\n  ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 (z - y) / (z - x) = a\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_iff_eq_add'] at hab \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 1 - a = b\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 (z - y) / (z - x) = a\n[PROOFSTEP]\nsimp_rw [div_eq_iff hxz.ne', \u2190 hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 1 - a = b\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\n\u22a2 z - (a * x + (1 - a) * z) = a * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nhave hb : (y - x) / (z - x) = b := by\n  rw [eq_comm, \u2190 sub_eq_iff_eq_add] at hab \n  simp_rw [div_eq_iff hxz.ne', \u2190 hab]\n  ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 (y - x) / (z - x) = b\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_iff_eq_add] at hab \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : 1 - b = a\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 (y - x) / (z - x) = b\n[PROOFSTEP]\nsimp_rw [div_eq_iff hxz.ne', \u2190 hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb : 0 < b\nhab : 1 - b = a\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\n\u22a2 (1 - b) * x + b * z - x = b * (z - x)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : \ud835\udd5c\nhx : x \u2208 s\nz : \ud835\udd5c\nhz : z \u2208 s\nhxz\u271d : x < z\na b : \ud835\udd5c\nha\u271d : 0 < a\nhb\u271d : 0 < b\nhab : a + b = 1\ny : \ud835\udd5c := a * x + b * z\nhxy : x < y\nhyz : y < z\nthis : (f y - f x) * (z - y) < (f z - f y) * (y - x)\nhxz : 0 < z - x\nha : (z - y) / (z - x) = a\nhb : (y - x) / (z - x) = b\n\u22a2 f (a \u2022 x + b \u2022 z) < a \u2022 f x + b \u2022 f z\n[PROOFSTEP]\nrwa [sub_mul, sub_mul, sub_lt_iff_lt_add', \u2190 add_sub_assoc, lt_sub_iff_add_lt, \u2190 mul_add, sub_add_sub_cancel, \u2190\n  lt_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x), mul_comm (f z), ha, hb] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) < (f y - f x) / (y - x)\n\u22a2 StrictConcaveOn \ud835\udd5c s f\n[PROOFSTEP]\nrw [\u2190 neg_strictConvexOn_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) < (f y - f x) / (y - x)\n\u22a2 StrictConvexOn \ud835\udd5c s (-f)\n[PROOFSTEP]\nrefine' strictConvexOn_of_slope_strict_mono_adjacent hs fun hx hz hxy hyz => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) < (f y - f x) / (y - x)\nx\u271d y\u271d z\u271d : \ud835\udd5c\nhx : x\u271d \u2208 s\nhz : z\u271d \u2208 s\nhxy : x\u271d < y\u271d\nhyz : y\u271d < z\u271d\n\u22a2 ((-f) y\u271d - (-f) x\u271d) / (y\u271d - x\u271d) < ((-f) z\u271d - (-f) y\u271d) / (z\u271d - y\u271d)\n[PROOFSTEP]\nrw [\u2190 neg_lt_neg_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) < (f y - f x) / (y - x)\nx\u271d y\u271d z\u271d : \ud835\udd5c\nhx : x\u271d \u2208 s\nhz : z\u271d \u2208 s\nhxy : x\u271d < y\u271d\nhyz : y\u271d < z\u271d\n\u22a2 -(((-f) z\u271d - (-f) y\u271d) / (z\u271d - y\u271d)) < -(((-f) y\u271d - (-f) x\u271d) / (y\u271d - x\u271d))\n[PROOFSTEP]\nsimp_rw [\u2190 neg_div, neg_sub, Pi.neg_apply, neg_sub_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhs : Convex \ud835\udd5c s\nhf : \u2200 {x y z : \ud835\udd5c}, x \u2208 s \u2192 z \u2208 s \u2192 x < y \u2192 y < z \u2192 (f z - f y) / (z - y) < (f y - f x) / (y - x)\nx\u271d y\u271d z\u271d : \ud835\udd5c\nhx : x\u271d \u2208 s\nhz : z\u271d \u2208 s\nhxy : x\u271d < y\u271d\nhyz : y\u271d < z\u271d\n\u22a2 (f z\u271d - f y\u271d) / (z\u271d - y\u271d) < (f y\u271d - f x\u271d) / (y\u271d - x\u271d)\n[PROOFSTEP]\nexact hf hx hz hxy hyz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (z - x) * f y \u2264 (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nhave hxy' : 0 < y - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 (z - x) * f y \u2264 (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nhave hyz' : 0 < z - y := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n\u22a2 (z - x) * f y \u2264 (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nhave hxz' : 0 < z - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n\u22a2 0 < z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 (z - x) * f y \u2264 (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nrw [\u2190 le_div_iff' hxz']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 f y \u2264 ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\nhave ha : 0 \u2264 (z - y) / (z - x) := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 0 \u2264 (z - y) / (z - x)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\n\u22a2 f y \u2264 ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\nhave hb : 0 \u2264 (y - x) / (z - x) := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\n\u22a2 0 \u2264 (y - x) / (z - x)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 f y \u2264 ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\ncalc\n  f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) := ?_\n  _ \u2264 (z - y) / (z - x) * f x + (y - x) / (z - x) * f z := (hf.2 hx hz ha hb ?_)\n  _ = ((z - y) * f x + (y - x) * f z) / (z - x) := ?_\n[GOAL]\ncase calc_1\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase calc_1.e_a\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 y = (z - y) / (z - x) * x + (y - x) / (z - x) * z\n[PROOFSTEP]\nfield_simp [hxy'.ne', hyz'.ne', hxz'.ne']\n[GOAL]\ncase calc_1.e_a\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 y * (z - x) = (z - y) * x + (y - x) * z\n[PROOFSTEP]\nring\n[GOAL]\ncase calc_2\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 Div.div (z - y) (z - x) + Div.div (y - x) (z - x) = 1\n[PROOFSTEP]\nshow (z - y) / (z - x) + (y - x) / (z - x) = 1\n[GOAL]\ncase calc_2\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 (z - y) / (z - x) + (y - x) / (z - x) = 1\n[PROOFSTEP]\nfield_simp [hxy'.ne', hyz'.ne', hxz'.ne']\n[GOAL]\ncase calc_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 \u2264 (z - y) / (z - x)\nhb : 0 \u2264 (y - x) / (z - x)\n\u22a2 (z - y) / (z - x) * f x + (y - x) / (z - x) * f z = ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\nfield_simp [hxy'.ne', hyz'.ne', hxz'.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f x) / (z - x)\n[PROOFSTEP]\nhave hxy' : 0 < y - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f x) / (z - x)\n[PROOFSTEP]\nhave hxz' : 0 < z - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 0 < z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhxz' : 0 < z - x\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f x) / (z - x)\n[PROOFSTEP]\nrw [div_le_div_iff hxy' hxz']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhxz' : 0 < z - x\n\u22a2 (f y - f x) * (z - x) \u2264 (f z - f x) * (y - x)\n[PROOFSTEP]\nlinarith only [hf.secant_mono_aux1 hx hz hxy hyz]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f z - f x) / (z - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hyz' : 0 < z - y := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n\u22a2 (f z - f x) / (z - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxz' : 0 < z - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n\u22a2 0 < z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 (f z - f x) / (z - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nrw [div_le_div_iff hxz' hyz']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 (f z - f x) * (z - y) \u2264 (f z - f y) * (z - x)\n[PROOFSTEP]\nlinarith only [hf.secant_mono_aux1 hx hz hxy hyz]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x \u2264 y\n\u22a2 (f x - f a) / (x - a) \u2264 (f y - f a) / (y - a)\n[PROOFSTEP]\nrcases eq_or_lt_of_le hxy with (rfl | hxy)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhxa : x \u2260 a\nhy : x \u2208 s\nhya : x \u2260 a\nhxy : x \u2264 x\n\u22a2 (f x - f a) / (x - a) \u2264 (f x - f a) / (x - a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\n\u22a2 (f x - f a) / (x - a) \u2264 (f y - f a) / (y - a)\n[PROOFSTEP]\ncases' lt_or_gt_of_ne hxa with hxa hxa\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\n\u22a2 (f x - f a) / (x - a) \u2264 (f y - f a) / (y - a)\n[PROOFSTEP]\ncases' lt_or_gt_of_ne hya with hya hya\n[GOAL]\ncase inr.inl.inl\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 (f x - f a) / (x - a) \u2264 (f y - f a) / (y - a)\n[PROOFSTEP]\nconvert hf.secant_mono_aux3 hx ha hxy hya using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 (f x - f a) / (x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nrw [\u2190 neg_div_neg_eq]\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 (f y - f a) / (y - a) = (f a - f y) / (a - y)\n[PROOFSTEP]\nrw [\u2190 neg_div_neg_eq]\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 -(f x - f a) / -(x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 -(f y - f a) / -(y - a) = (f a - f y) / (a - y)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase inr.inl.inr\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y > a\n\u22a2 (f x - f a) / (x - a) \u2264 (f y - f a) / (y - a)\n[PROOFSTEP]\nconvert hf.slope_mono_adjacent hx hy hxa hya using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y > a\n\u22a2 (f x - f a) / (x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nrw [\u2190 neg_div_neg_eq]\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x < a\nhya : y > a\n\u22a2 -(f x - f a) / -(x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase inr.inr\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya : y \u2260 a\nhxy\u271d : x \u2264 y\nhxy : x < y\nhxa : x > a\n\u22a2 (f x - f a) / (x - a) \u2264 (f y - f a) / (y - a)\n[PROOFSTEP]\nexact hf.secant_mono_aux2 ha hy hxa hxy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (z - x) * f y < (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nhave hxy' : 0 < y - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 (z - x) * f y < (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nhave hyz' : 0 < z - y := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n\u22a2 (z - x) * f y < (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nhave hxz' : 0 < z - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n\u22a2 0 < z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 (z - x) * f y < (z - y) * f x + (y - x) * f z\n[PROOFSTEP]\nrw [\u2190 lt_div_iff' hxz']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 f y < ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\nhave ha : 0 < (z - y) / (z - x) := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 0 < (z - y) / (z - x)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\n\u22a2 f y < ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\nhave hb : 0 < (y - x) / (z - x) := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\n\u22a2 0 < (y - x) / (z - x)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 f y < ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\ncalc\n  f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z) := ?_\n  _ < (z - y) / (z - x) * f x + (y - x) / (z - x) * f z := (hf.2 hx hz (by linarith) ha hb ?_)\n  _ = ((z - y) * f x + (y - x) * f z) / (z - x) := ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 x \u2260 z\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase calc_1\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 f y = f ((z - y) / (z - x) * x + (y - x) / (z - x) * z)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase calc_1.e_a\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 y = (z - y) / (z - x) * x + (y - x) / (z - x) * z\n[PROOFSTEP]\nfield_simp [hxy'.ne', hyz'.ne', hxz'.ne']\n[GOAL]\ncase calc_1.e_a\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 y * (z - x) = (z - y) * x + (y - x) * z\n[PROOFSTEP]\nring\n[GOAL]\ncase calc_2\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 Div.div (z - y) (z - x) + Div.div (y - x) (z - x) = 1\n[PROOFSTEP]\nshow (z - y) / (z - x) + (y - x) / (z - x) = 1\n[GOAL]\ncase calc_2\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 (z - y) / (z - x) + (y - x) / (z - x) = 1\n[PROOFSTEP]\nfield_simp [hxy'.ne', hyz'.ne', hxz'.ne']\n[GOAL]\ncase calc_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\nhxz' : 0 < z - x\nha : 0 < (z - y) / (z - x)\nhb : 0 < (y - x) / (z - x)\n\u22a2 (z - y) / (z - x) * f x + (y - x) / (z - x) * f z = ((z - y) * f x + (y - x) * f z) / (z - x)\n[PROOFSTEP]\nfield_simp [hxy'.ne', hyz'.ne', hxz'.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f y - f x) / (y - x) < (f z - f x) / (z - x)\n[PROOFSTEP]\nhave hxy' : 0 < y - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 (f y - f x) / (y - x) < (f z - f x) / (z - x)\n[PROOFSTEP]\nhave hxz' : 0 < z - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n\u22a2 0 < z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhxz' : 0 < z - x\n\u22a2 (f y - f x) / (y - x) < (f z - f x) / (z - x)\n[PROOFSTEP]\nrw [div_lt_div_iff hxy' hxz']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhxz' : 0 < z - x\n\u22a2 (f y - f x) * (z - x) < (f z - f x) * (y - x)\n[PROOFSTEP]\nlinarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 (f z - f x) / (z - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hyz' : 0 < z - y := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n\u22a2 (f z - f x) / (z - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxz' : 0 < z - x := by linarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n\u22a2 0 < z - x\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 (f z - f x) / (z - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nrw [div_lt_div_iff hxz' hyz']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\nx y z : \ud835\udd5c\nhx : x \u2208 s\nhz : z \u2208 s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\nhxz' : 0 < z - x\n\u22a2 (f z - f x) * (z - y) < (f z - f y) * (z - x)\n[PROOFSTEP]\nlinarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\n\u22a2 (f x - f a) / (x - a) < (f y - f a) / (y - a)\n[PROOFSTEP]\ncases' lt_or_gt_of_ne hxa with hxa hxa\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nhxa : x < a\n\u22a2 (f x - f a) / (x - a) < (f y - f a) / (y - a)\n[PROOFSTEP]\ncases' lt_or_gt_of_ne hya with hya hya\n[GOAL]\ncase inl.inl\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 (f x - f a) / (x - a) < (f y - f a) / (y - a)\n[PROOFSTEP]\nconvert hf.secant_strict_mono_aux3 hx ha hxy hya using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 (f x - f a) / (x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nrw [\u2190 neg_div_neg_eq]\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 (f y - f a) / (y - a) = (f a - f y) / (a - y)\n[PROOFSTEP]\nrw [\u2190 neg_div_neg_eq]\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 -(f x - f a) / -(x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y < a\n\u22a2 -(f y - f a) / -(y - a) = (f a - f y) / (a - y)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase inl.inr\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y > a\n\u22a2 (f x - f a) / (x - a) < (f y - f a) / (y - a)\n[PROOFSTEP]\nconvert hf.slope_strict_mono_adjacent hx hy hxa hya using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y > a\n\u22a2 (f x - f a) / (x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nrw [\u2190 neg_div_neg_eq]\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya\u271d : y \u2260 a\nhxy : x < y\nhxa : x < a\nhya : y > a\n\u22a2 -(f x - f a) / -(x - a) = (f a - f x) / (a - x)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConvexOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa\u271d : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nhxa : x > a\n\u22a2 (f x - f a) / (x - a) < (f y - f a) / (y - a)\n[PROOFSTEP]\nexact hf.secant_strict_mono_aux2 ha hy hxa hxy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\n\u22a2 (f y - f a) / (y - a) < (f x - f a) / (x - a)\n[PROOFSTEP]\nhave key := hf.neg.secant_strict_mono ha hx hy hxa hya hxy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : ((-f) x - (-f) a) / (x - a) < ((-f) y - (-f) a) / (y - a)\n\u22a2 (f y - f a) / (y - a) < (f x - f a) / (x - a)\n[PROOFSTEP]\nsimp only [Pi.neg_apply] at key \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : (-f x - -f a) / (x - a) < (-f y - -f a) / (y - a)\n\u22a2 (f y - f a) / (y - a) < (f x - f a) / (x - a)\n[PROOFSTEP]\nrw [\u2190 neg_lt_neg_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : (-f x - -f a) / (x - a) < (-f y - -f a) / (y - a)\n\u22a2 -((f x - f a) / (x - a)) < -((f y - f a) / (y - a))\n[PROOFSTEP]\nconvert key using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : (-f x - -f a) / (x - a) < (-f y - -f a) / (y - a)\n\u22a2 -((f x - f a) / (x - a)) = (-f x - -f a) / (x - a)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : (-f x - -f a) / (x - a) < (-f y - -f a) / (y - a)\n\u22a2 -((f y - f a) / (y - a)) = (-f y - -f a) / (y - a)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : (-f x - -f a) / (x - a) < (-f y - -f a) / (y - a)\n\u22a2 (f a - f x) / (x - a) = (-f x + f a) / (x - a)\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : StrictConcaveOn \ud835\udd5c s f\na x y : \ud835\udd5c\nha : a \u2208 s\nhx : x \u2208 s\nhy : y \u2208 s\nhxa : x \u2260 a\nhya : y \u2260 a\nhxy : x < y\nkey : (-f x - -f a) / (x - a) < (-f y - -f a) / (y - a)\n\u22a2 (f a - f y) / (y - a) = (-f y + f a) / (y - a)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\n\u22a2 StrictMonoOn f (s \u2229 Set.Ici y)\n[PROOFSTEP]\nintro u hu v hv huv\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\n\u22a2 f u < f v\n[PROOFSTEP]\nhave step1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z :=\n  by\n  intros z hz\n  refine hf.lt_right_of_left_lt hx hz.1 ?_ hxy'\n  rw [openSegment_eq_Ioo (hxy.trans hz.2)]\n  exact \u27e8hxy, hz.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\n\u22a2 \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\n[PROOFSTEP]\nintros z hz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nz : \ud835\udd5c\nhz : z \u2208 s \u2229 Set.Ioi y\n\u22a2 f y < f z\n[PROOFSTEP]\nrefine hf.lt_right_of_left_lt hx hz.1 ?_ hxy'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nz : \ud835\udd5c\nhz : z \u2208 s \u2229 Set.Ioi y\n\u22a2 y \u2208 openSegment \ud835\udd5c x z\n[PROOFSTEP]\nrw [openSegment_eq_Ioo (hxy.trans hz.2)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nz : \ud835\udd5c\nhz : z \u2208 s \u2229 Set.Ioi y\n\u22a2 y \u2208 Set.Ioo x z\n[PROOFSTEP]\nexact \u27e8hxy, hz.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\n\u22a2 f u < f v\n[PROOFSTEP]\nrcases eq_or_lt_of_le hu.2 with (rfl | hu2)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu : y \u2208 s \u2229 Set.Ici y\nhuv : y < v\n\u22a2 f y < f v\n[PROOFSTEP]\nexact step1 \u27e8hv.1, huv\u27e9\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu2 : y < u\n\u22a2 f u < f v\n[PROOFSTEP]\nrefine' hf.lt_right_of_left_lt _ hv.1 _ (step1 \u27e8hu.1, hu2\u27e9)\n[GOAL]\ncase inr.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu2 : y < u\n\u22a2 y \u2208 s\n[PROOFSTEP]\napply hf.1.segment_subset hx hu.1\n[GOAL]\ncase inr.refine'_1.a\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu2 : y < u\n\u22a2 y \u2208 segment \ud835\udd5c x u\n[PROOFSTEP]\nrw [segment_eq_Icc (hxy.le.trans hu.2)]\n[GOAL]\ncase inr.refine'_1.a\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu2 : y < u\n\u22a2 y \u2208 Set.Icc x u\n[PROOFSTEP]\nexact \u27e8hxy.le, hu.2\u27e9\n[GOAL]\ncase inr.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu2 : y < u\n\u22a2 u \u2208 openSegment \ud835\udd5c y v\n[PROOFSTEP]\nrw [openSegment_eq_Ioo (hu2.trans huv)]\n[GOAL]\ncase inr.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d : LinearOrderedField \ud835\udd5c\ns : Set \ud835\udd5c\nf : \ud835\udd5c \u2192 \ud835\udd5c\nhf : ConvexOn \ud835\udd5c s f\nx y : \ud835\udd5c\nhx : x \u2208 s\nhxy : x < y\nhxy' : f x < f y\nu : \ud835\udd5c\nhu : u \u2208 s \u2229 Set.Ici y\nv : \ud835\udd5c\nhv : v \u2208 s \u2229 Set.Ici y\nhuv : u < v\nstep1 : \u2200 {z : \ud835\udd5c}, z \u2208 s \u2229 Set.Ioi y \u2192 f y < f z\nhu2 : y < u\n\u22a2 u \u2208 Set.Ioo y v\n[PROOFSTEP]\nexact \u27e8hu2, huv\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Slope", "llama_tokens": 44727, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117940706734, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.5753569554868264}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\n\u22a2 rayleighQuotient T (c \u2022 x) = rayleighQuotient T x\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : x = 0\n\u22a2 rayleighQuotient T (c \u2022 x) = rayleighQuotient T x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : \u00acx = 0\n\u22a2 rayleighQuotient T (c \u2022 x) = rayleighQuotient T x\n[PROOFSTEP]\nhave : \u2016c\u2016 \u2260 0 := by simp [hc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : \u00acx = 0\n\u22a2 \u2016c\u2016 \u2260 0\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : \u00acx = 0\nthis : \u2016c\u2016 \u2260 0\n\u22a2 rayleighQuotient T (c \u2022 x) = rayleighQuotient T x\n[PROOFSTEP]\nhave : \u2016x\u2016 \u2260 0 := by simp [hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : \u00acx = 0\nthis : \u2016c\u2016 \u2260 0\n\u22a2 \u2016x\u2016 \u2260 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : \u00acx = 0\nthis\u271d : \u2016c\u2016 \u2260 0\nthis : \u2016x\u2016 \u2260 0\n\u22a2 rayleighQuotient T (c \u2022 x) = rayleighQuotient T x\n[PROOFSTEP]\nfield_simp [norm_smul, T.reApplyInnerSelf_smul]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\nhc : c \u2260 0\nhx : \u00acx = 0\nthis\u271d : \u2016c\u2016 \u2260 0\nthis : \u2016x\u2016 \u2260 0\n\u22a2 \u2016c\u2016 ^ 2 * reApplyInnerSelf T x * \u2016x\u2016 ^ 2 = reApplyInnerSelf T x * (\u2016c\u2016 * \u2016x\u2016) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\n\u22a2 rayleighQuotient T '' {0}\u1d9c = rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\n\u22a2 a \u2208 rayleighQuotient T '' {0}\u1d9c \u2194 a \u2208 rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\n\u22a2 a \u2208 rayleighQuotient T '' {0}\u1d9c \u2192 a \u2208 rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\nrintro \u27e8x, hx : x \u2260 0, hxT\u27e9\n[GOAL]\ncase h.mp.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\n\u22a2 a \u2208 rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\nhave : \u2016x\u2016 \u2260 0 := by simp [hx]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\n\u22a2 \u2016x\u2016 \u2260 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase h.mp.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis : \u2016x\u2016 \u2260 0\n\u22a2 a \u2208 rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\nlet c : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * r\n[GOAL]\ncase h.mp.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis : \u2016x\u2016 \u2260 0\nc : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * \u2191r\n\u22a2 a \u2208 rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\nhave : c \u2260 0 := by simp [hx, hr.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis : \u2016x\u2016 \u2260 0\nc : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * \u2191r\n\u22a2 c \u2260 0\n[PROOFSTEP]\nsimp [hx, hr.ne']\n[GOAL]\ncase h.mp.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis\u271d : \u2016x\u2016 \u2260 0\nc : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * \u2191r\nthis : c \u2260 0\n\u22a2 a \u2208 rayleighQuotient T '' sphere 0 r\n[PROOFSTEP]\nrefine' \u27e8c \u2022 x, _, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis\u271d : \u2016x\u2016 \u2260 0\nc : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * \u2191r\nthis : c \u2260 0\n\u22a2 c \u2022 x \u2208 sphere 0 r\n[PROOFSTEP]\nfield_simp [norm_smul, abs_of_pos hr]\n[GOAL]\ncase h.mp.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis\u271d : \u2016x\u2016 \u2260 0\nc : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * \u2191r\nthis : c \u2260 0\n\u22a2 rayleighQuotient T (c \u2022 x) = a\n[PROOFSTEP]\nrw [T.rayleigh_smul x this]\n[GOAL]\ncase h.mp.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2260 0\nhxT : rayleighQuotient T x = a\nthis\u271d : \u2016x\u2016 \u2260 0\nc : \ud835\udd5c := \u2191\u2016x\u2016\u207b\u00b9 * \u2191r\nthis : c \u2260 0\n\u22a2 rayleighQuotient T x = a\n[PROOFSTEP]\nexact hxT\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\n\u22a2 a \u2208 rayleighQuotient T '' sphere 0 r \u2192 a \u2208 rayleighQuotient T '' {0}\u1d9c\n[PROOFSTEP]\nrintro \u27e8x, hx, hxT\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\na : \u211d\nx : E\nhx : x \u2208 sphere 0 r\nhxT : rayleighQuotient T x = a\n\u22a2 a \u2208 rayleighQuotient T '' {0}\u1d9c\n[PROOFSTEP]\nexact \u27e8x, ne_zero_of_mem_sphere hr.ne' \u27e8x, hx\u27e9, hxT\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\n\u22a2 \u2a06 (x : \u2191{0}\u1d9c), rayleighQuotient T \u2191x = \u2a06 (x : \u2191(sphere 0 r)), rayleighQuotient T \u2191x\n[PROOFSTEP]\nsimp only [\u2190 @sSup_image' _ _ _ _ (rayleighQuotient T), T.image_rayleigh_eq_image_rayleigh_sphere hr]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nr : \u211d\nhr : 0 < r\n\u22a2 \u2a05 (x : \u2191{0}\u1d9c), rayleighQuotient T \u2191x = \u2a05 (x : \u2191(sphere 0 r)), rayleighQuotient T \u2191x\n[PROOFSTEP]\nsimp only [\u2190 @sInf_image' _ _ _ _ (rayleighQuotient T), T.image_rayleigh_eq_image_rayleigh_sphere hr]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nT : F \u2192L[\u211d] F\nhT : LinearMap.IsSymmetric \u2191T\nx\u2080 : F\n\u22a2 HasStrictFDerivAt (ContinuousLinearMap.reApplyInnerSelf T) (2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080)) x\u2080\n[PROOFSTEP]\nconvert T.hasStrictFDerivAt.inner \u211d (hasStrictFDerivAt_id x\u2080) using 1\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nT : F \u2192L[\u211d] F\nhT : LinearMap.IsSymmetric \u2191T\nx\u2080 : F\ne_7\u271d : Real.normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe\u271d : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\n\u22a2 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) =\n    ContinuousLinearMap.comp (fderivInnerClm \u211d (\u2191T x\u2080, id x\u2080)) (ContinuousLinearMap.prod T (ContinuousLinearMap.id \u211d F))\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_10.h.h.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nT : F \u2192L[\u211d] F\nhT : LinearMap.IsSymmetric \u2191T\nx\u2080 : F\ne_7\u271d : Real.normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe\u271d : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\ny : (fun x => F) x\u2080\n\u22a2 \u2191(2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080)) y =\n    \u2191(ContinuousLinearMap.comp (fderivInnerClm \u211d (\u2191T x\u2080, id x\u2080))\n          (ContinuousLinearMap.prod T (ContinuousLinearMap.id \u211d F)))\n      y\n[PROOFSTEP]\nrw [ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply, fderivInnerClm_apply,\n  ContinuousLinearMap.prod_apply, innerSL_apply, id.def, ContinuousLinearMap.id_apply, hT.apply_clm x\u2080 y,\n  real_inner_comm _ x\u2080, two_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2203 a b, (a, b) \u2260 0 \u2227 a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\n[PROOFSTEP]\nhave H : IsLocalExtrOn T.reApplyInnerSelf {x : F | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080 :=\n  by\n  convert hextr\n  ext x\n  simp [dist_eq_norm]\n    -- find Lagrange multipliers for the function `T.re_apply_inner_self` and the\n      -- hypersurface-defining function `\u03bb x, \u2016x\u2016 ^ 2`\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\n[PROOFSTEP]\nconvert hextr\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} = sphere 0 \u2016x\u2080\u2016\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nx : F\n\u22a2 x \u2208 {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} \u2194 x \u2208 sphere 0 \u2016x\u2080\u2016\n[PROOFSTEP]\nsimp [dist_eq_norm]\n  -- find Lagrange multipliers for the function `T.re_apply_inner_self` and the\n    -- hypersurface-defining function `\u03bb x, \u2016x\u2016 ^ 2`\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\n\u22a2 \u2203 a b, (a, b) \u2260 0 \u2227 a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\n[PROOFSTEP]\nobtain \u27e8a, b, h\u2081, h\u2082\u27e9 :=\n  IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt_1d H (hasStrictFDerivAt_norm_sq x\u2080)\n    (hT.isSymmetric.hasStrictFDerivAt_reApplyInnerSelf x\u2080)\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 \u2203 a b, (a, b) \u2260 0 \u2227 a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\n[PROOFSTEP]\nrefine' \u27e8a, b, h\u2081, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\n[PROOFSTEP]\napply (InnerProductSpace.toDualMap \u211d F).injective\n[GOAL]\ncase intro.intro.intro.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 \u2191(InnerProductSpace.toDualMap \u211d F) (a \u2022 x\u2080 + b \u2022 \u2191T x\u2080) = \u2191(InnerProductSpace.toDualMap \u211d F) 0\n[PROOFSTEP]\nsimp only [LinearIsometry.map_add, LinearIsometry.map_smul, LinearIsometry.map_zero]\n[GOAL]\ncase intro.intro.intro.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 \u2191(InnerProductSpace.toDualMap \u211d F) (a \u2022 x\u2080) + \u2191(InnerProductSpace.toDualMap \u211d F) (b \u2022 \u2191T x\u2080) = 0\n[PROOFSTEP]\nsimp only [map_smul\u209b\u2097, IsROrC.conj_to_real]\n[GOAL]\ncase intro.intro.intro.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 a \u2022 \u2191(InnerProductSpace.toDualMap \u211d F) x\u2080 + b \u2022 \u2191(InnerProductSpace.toDualMap \u211d F) (\u2191T x\u2080) = 0\n[PROOFSTEP]\nchange a \u2022 innerSL \u211d x\u2080 + b \u2022 innerSL \u211d (T x\u2080) = 0\n[GOAL]\ncase intro.intro.intro.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 a \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n[PROOFSTEP]\napply smul_right_injective (F \u2192L[\u211d] \u211d) (two_ne_zero : (2 : \u211d) \u2260 0)\n[GOAL]\ncase intro.intro.intro.a.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nH : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) {x | \u2016x\u2016 ^ 2 = \u2016x\u2080\u2016 ^ 2} x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 2 \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 2 \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080) = 0\n\u22a2 (fun x x_1 => x \u2022 x_1) 2 (a \u2022 \u2191(innerSL \u211d) x\u2080 + b \u2022 \u2191(innerSL \u211d) (\u2191T x\u2080)) = (fun x x_1 => x \u2022 x_1) 2 0\n[PROOFSTEP]\nsimpa only [two_smul, smul_add, add_smul, add_zero] using h\u2082\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nobtain \u27e8a, b, h\u2081, h\u2082\u27e9 := hT.linearly_dependent_of_isLocalExtrOn hextr\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nby_cases hx\u2080 : x\u2080 = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : x\u2080 = 0\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nsimp [hx\u2080]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nby_cases hb : b = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : b = 0\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nhave : a \u2260 0 := by simpa [hb] using h\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : b = 0\n\u22a2 a \u2260 0\n[PROOFSTEP]\nsimpa [hb] using h\u2081\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : b = 0\nthis : a \u2260 0\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nrefine' absurd _ hx\u2080\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : b = 0\nthis : a \u2260 0\n\u22a2 x\u2080 = 0\n[PROOFSTEP]\napply smul_right_injective F this\n[GOAL]\ncase pos.a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : b = 0\nthis : a \u2260 0\n\u22a2 (fun x x_1 => x \u2022 x_1) a x\u2080 = (fun x x_1 => x \u2022 x_1) a 0\n[PROOFSTEP]\nsimpa [hb] using h\u2082\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nlet c : \u211d := -b\u207b\u00b9 * a\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nhave hc : T x\u2080 = c \u2022 x\u2080 := by\n  have : b * (b\u207b\u00b9 * a) = a := by field_simp [mul_comm]\n  apply smul_right_injective F hb\n  simp [eq_neg_of_add_eq_zero_left h\u2082, \u2190 mul_smul, this]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\n\u22a2 \u2191T x\u2080 = c \u2022 x\u2080\n[PROOFSTEP]\nhave : b * (b\u207b\u00b9 * a) = a := by field_simp [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\n\u22a2 b * (b\u207b\u00b9 * a) = a\n[PROOFSTEP]\nfield_simp [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nthis : b * (b\u207b\u00b9 * a) = a\n\u22a2 \u2191T x\u2080 = c \u2022 x\u2080\n[PROOFSTEP]\napply smul_right_injective F hb\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nthis : b * (b\u207b\u00b9 * a) = a\n\u22a2 (fun x x_1 => x \u2022 x_1) b (\u2191T x\u2080) = (fun x x_1 => x \u2022 x_1) b (c \u2022 x\u2080)\n[PROOFSTEP]\nsimp [eq_neg_of_add_eq_zero_left h\u2082, \u2190 mul_smul, this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nhc : \u2191T x\u2080 = c \u2022 x\u2080\n\u22a2 \u2191T x\u2080 = ContinuousLinearMap.rayleighQuotient T x\u2080 \u2022 x\u2080\n[PROOFSTEP]\nconvert hc\n[GOAL]\ncase h.e'_3.h.e'_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nhc : \u2191T x\u2080 = c \u2022 x\u2080\n\u22a2 ContinuousLinearMap.rayleighQuotient T x\u2080 = c\n[PROOFSTEP]\nhave : \u2016x\u2080\u2016 \u2260 0 := by simp [hx\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nhc : \u2191T x\u2080 = c \u2022 x\u2080\n\u22a2 \u2016x\u2080\u2016 \u2260 0\n[PROOFSTEP]\nsimp [hx\u2080]\n[GOAL]\ncase h.e'_3.h.e'_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nhc : \u2191T x\u2080 = c \u2022 x\u2080\nthis : \u2016x\u2080\u2016 \u2260 0\n\u22a2 ContinuousLinearMap.rayleighQuotient T x\u2080 = c\n[PROOFSTEP]\nhave := congr_arg (fun x => \u27eax, x\u2080\u27eb_\u211d) hc\n[GOAL]\ncase h.e'_3.h.e'_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nhc : \u2191T x\u2080 = c \u2022 x\u2080\nthis\u271d : \u2016x\u2080\u2016 \u2260 0\nthis : (fun x => inner x x\u2080) (\u2191T x\u2080) = (fun x => inner x x\u2080) (c \u2022 x\u2080)\n\u22a2 ContinuousLinearMap.rayleighQuotient T x\u2080 = c\n[PROOFSTEP]\nfield_simp [inner_smul_left, real_inner_self_eq_norm_mul_norm, sq] at this \u22a2\n[GOAL]\ncase h.e'_3.h.e'_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ninst\u271d : CompleteSpace F\nT : F \u2192L[\u211d] F\nhT : IsSelfAdjoint T\nx\u2080 : F\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\na b : \u211d\nh\u2081 : (a, b) \u2260 0\nh\u2082 : a \u2022 x\u2080 + b \u2022 \u2191T x\u2080 = 0\nhx\u2080 : \u00acx\u2080 = 0\nhb : \u00acb = 0\nc : \u211d := -b\u207b\u00b9 * a\nhc : \u2191T x\u2080 = c \u2022 x\u2080\nthis\u271d : \u2016x\u2080\u2016 \u2260 0\nthis : inner (\u2191T x\u2080) x\u2080 * b = -(a * (\u2016x\u2080\u2016 * \u2016x\u2080\u2016))\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x\u2080 * b = -(a * (\u2016x\u2080\u2016 * \u2016x\u2080\u2016))\n[PROOFSTEP]\nexact this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2191T x\u2080 = \u2191(ContinuousLinearMap.rayleighQuotient T x\u2080) \u2022 x\u2080\n[PROOFSTEP]\nletI := InnerProductSpace.isROrCToReal \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nthis : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\n\u22a2 \u2191T x\u2080 = \u2191(ContinuousLinearMap.rayleighQuotient T x\u2080) \u2022 x\u2080\n[PROOFSTEP]\nlet hSA := hT.isSymmetric.restrictScalars.toSelfAdjoint.prop\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nthis : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nhSA : \u2191(LinearMap.IsSymmetric.toSelfAdjoint (_ : LinearMap.IsSymmetric (\u2191\u211d \u2191T))) \u2208 selfAdjoint (E \u2192L[\u211d] E) :=\n  Subtype.prop (LinearMap.IsSymmetric.toSelfAdjoint (LinearMap.IsSymmetric.restrictScalars (isSymmetric hT)))\n\u22a2 \u2191T x\u2080 = \u2191(ContinuousLinearMap.rayleighQuotient T x\u2080) \u2022 x\u2080\n[PROOFSTEP]\nexact hSA.eq_smul_self_of_isLocalExtrOn_real hextr\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 HasEigenvector (\u2191T) (\u2191(ContinuousLinearMap.rayleighQuotient T x\u2080)) x\u2080\n[PROOFSTEP]\nrefine' \u27e8_, hx\u2080\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 x\u2080 \u2208 eigenspace \u2191T \u2191(ContinuousLinearMap.rayleighQuotient T x\u2080)\n[PROOFSTEP]\nrw [Module.End.mem_eigenspace_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsLocalExtrOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2191\u2191T x\u2080 = \u2191(ContinuousLinearMap.rayleighQuotient T x\u2080) \u2022 x\u2080\n[PROOFSTEP]\nexact hT.eq_smul_self_of_isLocalExtrOn hextr\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 HasEigenvector (\u2191T) (\u2191(\u2a06 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x)) x\u2080\n[PROOFSTEP]\nconvert hT.hasEigenvector_of_isLocalExtrOn hx\u2080 (Or.inr hextr.localize)\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2a06 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nhave hx\u2080' : 0 < \u2016x\u2080\u2016 := by simp [hx\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 0 < \u2016x\u2080\u2016\n[PROOFSTEP]\nsimp [hx\u2080]\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\n\u22a2 \u2a06 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nhave hx\u2080'' : x\u2080 \u2208 sphere (0 : E) \u2016x\u2080\u2016 := by simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\n\u22a2 x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 \u2a06 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nrw [T.iSup_rayleigh_eq_iSup_rayleigh_sphere hx\u2080']\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 \u2a06 (x : \u2191(sphere 0 \u2016x\u2080\u2016)), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nrefine' IsMaxOn.iSup_eq hx\u2080'' _\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 IsMaxOn (ContinuousLinearMap.rayleighQuotient T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 x \u2208 {x | (fun x => ContinuousLinearMap.rayleighQuotient T x \u2264 ContinuousLinearMap.rayleighQuotient T x\u2080) x}\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.rayleighQuotient T x \u2264 ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nhave : \u2016x\u2016 = \u2016x\u2080\u2016 := by simpa using hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 \u2016x\u2016 = \u2016x\u2080\u2016\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.rayleighQuotient T x \u2264 ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.rayleighQuotient]\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x / \u2016x\u2016 ^ 2 \u2264 ContinuousLinearMap.reApplyInnerSelf T x\u2080 / \u2016x\u2080\u2016 ^ 2\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x / \u2016x\u2080\u2016 ^ 2 \u2264 ContinuousLinearMap.reApplyInnerSelf T x\u2080 / \u2016x\u2080\u2016 ^ 2\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.e'_7.h.e'_3.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x \u2264 ContinuousLinearMap.reApplyInnerSelf T x\u2080\n[PROOFSTEP]\nexact hextr hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 HasEigenvector (\u2191T) (\u2191(\u2a05 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x)) x\u2080\n[PROOFSTEP]\nconvert hT.hasEigenvector_of_isLocalExtrOn hx\u2080 (Or.inl hextr.localize)\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2a05 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nhave hx\u2080' : 0 < \u2016x\u2080\u2016 := by simp [hx\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 0 < \u2016x\u2080\u2016\n[PROOFSTEP]\nsimp [hx\u2080]\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\n\u22a2 \u2a05 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nhave hx\u2080'' : x\u2080 \u2208 sphere (0 : E) \u2016x\u2080\u2016 := by simp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\n\u22a2 x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 \u2a05 (x : { x // x \u2260 0 }), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nrw [T.iInf_rayleigh_eq_iInf_rayleigh_sphere hx\u2080']\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 \u2a05 (x : \u2191(sphere 0 \u2016x\u2080\u2016)), ContinuousLinearMap.rayleighQuotient T \u2191x = ContinuousLinearMap.rayleighQuotient T x\u2080\n[PROOFSTEP]\nrefine' IsMinOn.iInf_eq hx\u2080'' _\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 IsMinOn (ContinuousLinearMap.rayleighQuotient T) (sphere 0 \u2016x\u2080\u2016) x\u2080\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 x \u2208 {x | (fun x => ContinuousLinearMap.rayleighQuotient T x\u2080 \u2264 ContinuousLinearMap.rayleighQuotient T x) x}\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.rayleighQuotient T x\u2080 \u2264 ContinuousLinearMap.rayleighQuotient T x\n[PROOFSTEP]\nhave : \u2016x\u2016 = \u2016x\u2080\u2016 := by simpa using hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\n\u22a2 \u2016x\u2016 = \u2016x\u2080\u2016\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.rayleighQuotient T x\u2080 \u2264 ContinuousLinearMap.rayleighQuotient T x\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.rayleighQuotient]\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x\u2080 / \u2016x\u2080\u2016 ^ 2 \u2264 ContinuousLinearMap.reApplyInnerSelf T x / \u2016x\u2016 ^ 2\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h.e'_7.h.e'_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x\u2080 / \u2016x\u2080\u2016 ^ 2 \u2264 ContinuousLinearMap.reApplyInnerSelf T x / \u2016x\u2080\u2016 ^ 2\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.e'_7.h.e'_3.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nx\u2080 : E\nhx\u2080 : x\u2080 \u2260 0\nhextr : IsMinOn (ContinuousLinearMap.reApplyInnerSelf T) (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080' : 0 < \u2016x\u2080\u2016\nhx\u2080'' : x\u2080 \u2208 sphere 0 \u2016x\u2080\u2016\nx : E\nhx : x \u2208 sphere 0 \u2016x\u2080\u2016\nthis : \u2016x\u2016 = \u2016x\u2080\u2016\n\u22a2 ContinuousLinearMap.reApplyInnerSelf T x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf T x\n[PROOFSTEP]\nexact hextr hx\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhaveI := FiniteDimensional.proper_isROrC \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nlet T' := hT.toSelfAdjoint\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x : E, x \u2260 0 := exists_ne 0\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave H\u2081 : IsCompact (sphere (0 : E) \u2016x\u2016) := isCompact_sphere _ _\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave H\u2082 : (sphere (0 : E) \u2016x\u2016).Nonempty :=\n  \u27e8x, by simp\u27e9\n    -- key point: in finite dimension, a continuous function on the sphere has a max\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\n\u22a2 x \u2208 sphere 0 \u2016x\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nobtain \u27e8x\u2080, hx\u2080', hTx\u2080\u27e9 := H\u2081.exists_forall_ge H\u2082 T'.val.reApplyInnerSelf_continuous.continuousOn\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave hx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016 := by simpa using hx\u2080'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\n\u22a2 \u2016x\u2080\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimpa using hx\u2080'\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave : IsMaxOn T'.val.reApplyInnerSelf (sphere 0 \u2016x\u2080\u2016) x\u2080 := by simpa only [\u2190 hx\u2080] using hTx\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\n\u22a2 IsMaxOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n[PROOFSTEP]\nsimpa only [\u2190 hx\u2080] using hTx\u2080\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave hx\u2080_ne : x\u2080 \u2260 0 :=\n  by\n  have : \u2016x\u2080\u2016 \u2260 0 := by simp only [hx\u2080, norm_eq_zero, hx, Ne.def, not_false_iff]\n  simpa [\u2190 norm_eq_zero, Ne.def]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 x\u2080 \u2260 0\n[PROOFSTEP]\nhave : \u2016x\u2080\u2016 \u2260 0 := by simp only [hx\u2080, norm_eq_zero, hx, Ne.def, not_false_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2016x\u2080\u2016 \u2260 0\n[PROOFSTEP]\nsimp only [hx\u2080, norm_eq_zero, hx, Ne.def, not_false_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d\u00b9 : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis\u271d : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\nthis : \u2016x\u2080\u2016 \u2260 0\n\u22a2 x\u2080 \u2260 0\n[PROOFSTEP]\nsimpa [\u2190 norm_eq_zero, Ne.def]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMaxOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080_ne : x\u2080 \u2260 0\n\u22a2 HasEigenvalue T \u2191(\u2a06 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nexact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMaxOn hx\u2080_ne this)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhaveI := FiniteDimensional.proper_isROrC \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nlet T' := hT.toSelfAdjoint\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x : E, x \u2260 0 := exists_ne 0\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave H\u2081 : IsCompact (sphere (0 : E) \u2016x\u2016) := isCompact_sphere _ _\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave H\u2082 : (sphere (0 : E) \u2016x\u2016).Nonempty :=\n  \u27e8x, by simp\u27e9\n    -- key point: in finite dimension, a continuous function on the sphere has a min\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\n\u22a2 x \u2208 sphere 0 \u2016x\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nobtain \u27e8x\u2080, hx\u2080', hTx\u2080\u27e9 := H\u2081.exists_forall_le H\u2082 T'.val.reApplyInnerSelf_continuous.continuousOn\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave hx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016 := by simpa using hx\u2080'\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\n\u22a2 \u2016x\u2080\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimpa using hx\u2080'\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave : IsMinOn T'.val.reApplyInnerSelf (sphere 0 \u2016x\u2080\u2016) x\u2080 := by simpa only [\u2190 hx\u2080] using hTx\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\n\u22a2 IsMinOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n[PROOFSTEP]\nsimpa only [\u2190 hx\u2080] using hTx\u2080\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMinOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nhave hx\u2080_ne : x\u2080 \u2260 0 :=\n  by\n  have : \u2016x\u2080\u2016 \u2260 0 := by simp only [hx\u2080, norm_eq_zero, hx, Ne.def, not_false_iff]\n  simpa [\u2190 norm_eq_zero, Ne.def]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMinOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 x\u2080 \u2260 0\n[PROOFSTEP]\nhave : \u2016x\u2080\u2016 \u2260 0 := by simp only [hx\u2080, norm_eq_zero, hx, Ne.def, not_false_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMinOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\n\u22a2 \u2016x\u2080\u2016 \u2260 0\n[PROOFSTEP]\nsimp only [hx\u2080, norm_eq_zero, hx, Ne.def, not_false_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d\u00b9 : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis\u271d : IsMinOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\nthis : \u2016x\u2080\u2016 \u2260 0\n\u22a2 x\u2080 \u2260 0\n[PROOFSTEP]\nsimpa [\u2190 norm_eq_zero, Ne.def]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n_i : Nontrivial E\nT : E \u2192\u2097[\ud835\udd5c] E\nhT : IsSymmetric T\nthis\u271d : ProperSpace E\nT' : { x // x \u2208 selfAdjoint (E \u2192L[\ud835\udd5c] E) } := toSelfAdjoint hT\nx : E\nhx : x \u2260 0\nH\u2081 : IsCompact (sphere 0 \u2016x\u2016)\nH\u2082 : Set.Nonempty (sphere 0 \u2016x\u2016)\nx\u2080 : E\nhx\u2080' : x\u2080 \u2208 sphere 0 \u2016x\u2016\nhTx\u2080 :\n  \u2200 (y : E),\n    y \u2208 sphere 0 \u2016x\u2016 \u2192 ContinuousLinearMap.reApplyInnerSelf (\u2191T') x\u2080 \u2264 ContinuousLinearMap.reApplyInnerSelf (\u2191T') y\nhx\u2080 : \u2016x\u2080\u2016 = \u2016x\u2016\nthis : IsMinOn (ContinuousLinearMap.reApplyInnerSelf \u2191T') (sphere 0 \u2016x\u2080\u2016) x\u2080\nhx\u2080_ne : x\u2080 \u2260 0\n\u22a2 HasEigenvalue T \u2191(\u2a05 (x : { x // x \u2260 0 }), \u2191IsROrC.re (inner (\u2191T \u2191x) \u2191x) / \u2016\u2191x\u2016 ^ 2)\n[PROOFSTEP]\nexact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMinOn hx\u2080_ne this)\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Rayleigh", "llama_tokens": 32942, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.7154240079185319, "lm_q1q2_score": 0.5746098164447973}}
{"text": "[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x p = sum p fun e a => \u2191f a * x ^ e\n[PROOFSTEP]\nrw [eval\u2082_def]\n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np q r : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nx : S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf g : R \u2192+* S\ns t : S\n\u03c6 \u03c8 : R[X]\n\u22a2 f = g \u2192 s = t \u2192 \u03c6 = \u03c8 \u2192 eval\u2082 f s \u03c6 = eval\u2082 g t \u03c8\n[PROOFSTEP]\nrintro rfl rfl rfl\n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np q r : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nx : S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\ns : S\n\u03c6 : R[X]\n\u22a2 eval\u2082 f s \u03c6 = eval\u2082 f s \u03c6\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f 0 p = \u2191f (coeff p 0)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [eval\u2082_eq_sum, zero_pow_eq, mul_ite, mul_zero, mul_one, sum,\n  Classical.not_not, mem_support_iff, sum_ite_eq', ite_eq_left_iff, RingHom.map_zero, imp_true_iff, eq_self_iff_true]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x 0 = 0\n[PROOFSTEP]\nsimp [eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (\u2191C a) = \u2191f a\n[PROOFSTEP]\nsimp [eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x X = x\n[PROOFSTEP]\nsimp [eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\nr : R\n\u22a2 eval\u2082 f x (\u2191(monomial n) r) = \u2191f r * x ^ n\n[PROOFSTEP]\nsimp [eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\n\u22a2 eval\u2082 f x (X ^ n) = x ^ n\n[PROOFSTEP]\nrw [X_pow_eq_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\n\u22a2 eval\u2082 f x (\u2191(monomial n) 1) = x ^ n\n[PROOFSTEP]\nconvert eval\u2082_monomial f x (n := n) (r := 1)\n[GOAL]\ncase h.e'_3\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\n\u22a2 x ^ n = \u2191f 1 * x ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q\n[PROOFSTEP]\nsimp only [eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 (sum (p + q) fun e a => \u2191f a * x ^ e) = (sum p fun e a => \u2191f a * x ^ e) + sum q fun e a => \u2191f a * x ^ e\n[PROOFSTEP]\napply sum_add_index\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 \u2200 (i : \u2115), \u2191f 0 * x ^ i = 0\n[PROOFSTEP]\nsimp [add_mul]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 \u2200 (a : \u2115) (b\u2081 b\u2082 : R), \u2191f (b\u2081 + b\u2082) * x ^ a = \u2191f b\u2081 * x ^ a + \u2191f b\u2082 * x ^ a\n[PROOFSTEP]\nsimp [add_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x 1 = 1\n[PROOFSTEP]\nrw [\u2190 C_1, eval\u2082_C, f.map_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (bit0 p) = bit0 (eval\u2082 f x p)\n[PROOFSTEP]\nrw [bit0, eval\u2082_add, bit0]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (bit1 p) = bit1 (eval\u2082 f x p)\n[PROOFSTEP]\nrw [bit1, eval\u2082_add, eval\u2082_bit0, eval\u2082_one, bit1]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\ng : R \u2192+* S\np : R[X]\nx : S\ns : R\n\u22a2 eval\u2082 g x (s \u2022 p) = \u2191g s * eval\u2082 g x p\n[PROOFSTEP]\nhave A : p.natDegree < p.natDegree.succ := Nat.lt_succ_self _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\ng : R \u2192+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\n\u22a2 eval\u2082 g x (s \u2022 p) = \u2191g s * eval\u2082 g x p\n[PROOFSTEP]\nhave B : (s \u2022 p).natDegree < p.natDegree.succ := (natDegree_smul_le _ _).trans_lt A\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\ng : R \u2192+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\nB : natDegree (s \u2022 p) < Nat.succ (natDegree p)\n\u22a2 eval\u2082 g x (s \u2022 p) = \u2191g s * eval\u2082 g x p\n[PROOFSTEP]\nrw [eval\u2082_eq_sum, eval\u2082_eq_sum, sum_over_range' _ _ _ A, sum_over_range' _ _ _ B]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\ng : R \u2192+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\nB : natDegree (s \u2022 p) < Nat.succ (natDegree p)\n\u22a2 \u2211 a in range (Nat.succ (natDegree p)), \u2191g (coeff (s \u2022 p) a) * x ^ a =\n    \u2191g s * \u2211 a in range (Nat.succ (natDegree p)), \u2191g (coeff p a) * x ^ a\n[PROOFSTEP]\nsimp [mul_sum, mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\ng : R \u2192+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\nB : natDegree (s \u2022 p) < Nat.succ (natDegree p)\n\u22a2 \u2200 (n : \u2115), \u2191g 0 * x ^ n = 0\n[PROOFSTEP]\nsimp [mul_sum, mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\ng : R \u2192+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\nB : natDegree (s \u2022 p) < Nat.succ (natDegree p)\n\u22a2 \u2200 (n : \u2115), \u2191g 0 * x ^ n = 0\n[PROOFSTEP]\nsimp [mul_sum, mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\np q : R[X]\nhp : eval\u2082 C X p = p\nhq : eval\u2082 C X q = q\n\u22a2 eval\u2082 C X (p + q) = p + q\n[PROOFSTEP]\nsimp [hp, hq]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx\u271d : S\nn : \u2115\nx : R\n\u22a2 eval\u2082 C X (\u2191(monomial n) x) = \u2191(monomial n) x\n[PROOFSTEP]\nrw [eval\u2082_monomial, \u2190 smul_X_eq_monomial, C_mul']\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\n\u22a2 eval\u2082 f x \u2191n = \u2191n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x \u2191Nat.zero = \u2191Nat.zero\n[PROOFSTEP]\nsimp only [eval\u2082_zero, Nat.cast_zero, Nat.zero_eq]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\nih : eval\u2082 f x \u2191n = \u2191n\n\u22a2 eval\u2082 f x \u2191(Nat.succ n) = \u2191(Nat.succ n)\n[PROOFSTEP]\nrw [n.cast_succ, eval\u2082_add, ih, eval\u2082_one, n.cast_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\np : T[X]\ng : \u2115 \u2192 T \u2192 R[X]\nx : S\n\u22a2 eval\u2082 f x (sum p g) = sum p fun n a => eval\u2082 f x (g n a)\n[PROOFSTEP]\nlet T : R[X] \u2192+ S :=\n  { toFun := eval\u2082 f x\n    map_zero' := eval\u2082_zero _ _\n    map_add' := fun p q => eval\u2082_add _ _ }\n[GOAL]\nR : Type u\nS : Type v\nT\u271d : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\u271d\np : T\u271d[X]\ng : \u2115 \u2192 T\u271d \u2192 R[X]\nx : S\nT : R[X] \u2192+ S :=\n  { toZeroHom := { toFun := eval\u2082 f x, map_zero' := (_ : eval\u2082 f x 0 = 0) },\n    map_add' := (_ : \u2200 (p q : R[X]), eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q) }\n\u22a2 eval\u2082 f x (sum p g) = sum p fun n a => eval\u2082 f x (g n a)\n[PROOFSTEP]\nhave A : \u2200 y, eval\u2082 f x y = T y := fun y => rfl\n[GOAL]\nR : Type u\nS : Type v\nT\u271d : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\u271d\np : T\u271d[X]\ng : \u2115 \u2192 T\u271d \u2192 R[X]\nx : S\nT : R[X] \u2192+ S :=\n  { toZeroHom := { toFun := eval\u2082 f x, map_zero' := (_ : eval\u2082 f x 0 = 0) },\n    map_add' := (_ : \u2200 (p q : R[X]), eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q) }\nA : \u2200 (y : R[X]), eval\u2082 f x y = \u2191T y\n\u22a2 eval\u2082 f x (sum p g) = sum p fun n a => eval\u2082 f x (g n a)\n[PROOFSTEP]\nsimp only [A]\n[GOAL]\nR : Type u\nS : Type v\nT\u271d : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\u271d\np : T\u271d[X]\ng : \u2115 \u2192 T\u271d \u2192 R[X]\nx : S\nT : R[X] \u2192+ S :=\n  { toZeroHom := { toFun := eval\u2082 f x, map_zero' := (_ : eval\u2082 f x 0 = 0) },\n    map_add' := (_ : \u2200 (p q : R[X]), eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q) }\nA : \u2200 (y : R[X]), eval\u2082 f x y = \u2191T y\n\u22a2 \u2191{ toZeroHom := { toFun := eval\u2082 f x, map_zero' := (_ : eval\u2082 f x 0 = 0) },\n          map_add' := (_ : \u2200 (p q : R[X]), eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q) }\n      (sum p g) =\n    sum p fun n a =>\n      \u2191{ toZeroHom := { toFun := eval\u2082 f x, map_zero' := (_ : eval\u2082 f x 0 = 0) },\n            map_add' := (_ : \u2200 (p q : R[X]), eval\u2082 f x (p + q) = eval\u2082 f x p + eval\u2082 f x q) }\n        (g n a)\n[PROOFSTEP]\nrw [sum, T.map_sum, sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : AddMonoidAlgebra R \u2115\n\u22a2 eval\u2082 f x { toFinsupp := p } = \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) p\n[PROOFSTEP]\nsimp only [eval\u2082_eq_sum, sum, toFinsupp_sum, support, coeff]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : AddMonoidAlgebra R \u2115\n\u22a2 \u2211 x_1 in p.support, \u2191f (\u2191p x_1) * x ^ x_1 = \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) p\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nhf : \u2200 (k : \u2115), Commute (\u2191f (coeff q k)) x\n\u22a2 eval\u2082 f x (p * q) = eval\u2082 f x p * eval\u2082 f x q\n[PROOFSTEP]\nrcases p with \u27e8p\u27e9\n[GOAL]\ncase ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nq r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nhf : \u2200 (k : \u2115), Commute (\u2191f (coeff q k)) x\np : AddMonoidAlgebra R \u2115\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * q) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x q\n[PROOFSTEP]\nrcases q with \u27e8q\u27e9\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : AddMonoidAlgebra R \u2115\nhf : \u2200 (k : \u2115), Commute (\u2191f (coeff { toFinsupp := q } k)) x\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * { toFinsupp := q }) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x { toFinsupp := q }\n[PROOFSTEP]\nsimp only [coeff] at hf \n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : AddMonoidAlgebra R \u2115\nhf : \u2200 (k : \u2115), Commute (\u2191f (\u2191q k)) x\n\u22a2 eval\u2082 f x ({ toFinsupp := p } * { toFinsupp := q }) = eval\u2082 f x { toFinsupp := p } * eval\u2082 f x { toFinsupp := q }\n[PROOFSTEP]\nsimp only [\u2190 ofFinsupp_mul, eval\u2082_ofFinsupp]\n[GOAL]\ncase ofFinsupp.ofFinsupp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\nr : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\np q : AddMonoidAlgebra R \u2115\nhf : \u2200 (k : \u2115), Commute (\u2191f (\u2191q k)) x\n\u22a2 \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) (p * q) = \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) p * \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) q\n[PROOFSTEP]\nexact liftNC_mul _ _ p q fun {k n} _hn => (hf k).pow_right n\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\n\u22a2 eval\u2082 f x (p * X) = eval\u2082 f x p * x\n[PROOFSTEP]\nrefine' _root_.trans (eval\u2082_mul_noncomm _ _ fun k => _) (by rw [eval\u2082_X])\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\n\u22a2 eval\u2082 f x p * eval\u2082 f x X = eval\u2082 f x p * x\n[PROOFSTEP]\nrw [eval\u2082_X]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nk : \u2115\n\u22a2 Commute (\u2191f (coeff X k)) x\n[PROOFSTEP]\nrcases em (k = 1) with (rfl | hk)\n[GOAL]\ncase inl\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\n\u22a2 Commute (\u2191f (coeff X 1)) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nk : \u2115\nhk : \u00ack = 1\n\u22a2 Commute (\u2191f (coeff X k)) x\n[PROOFSTEP]\nsimp [coeff_X_of_ne_one hk]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\n\u22a2 eval\u2082 f x (X * p) = eval\u2082 f x p * x\n[PROOFSTEP]\nrw [X_mul, eval\u2082_mul_X]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nh : Commute (\u2191f a) x\n\u22a2 eval\u2082 f x (p * \u2191C a) = eval\u2082 f x p * \u2191f a\n[PROOFSTEP]\nrw [eval\u2082_mul_noncomm, eval\u2082_C]\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nh : Commute (\u2191f a) x\n\u22a2 \u2200 (k : \u2115), Commute (\u2191f (coeff (\u2191C a) k)) x\n[PROOFSTEP]\nintro k\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nh : Commute (\u2191f a) x\nk : \u2115\n\u22a2 Commute (\u2191f (coeff (\u2191C a) k)) x\n[PROOFSTEP]\nby_cases hk : k = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nh : Commute (\u2191f a) x\nk : \u2115\nhk : k = 0\n\u22a2 Commute (\u2191f (coeff (\u2191C a) k)) x\n[PROOFSTEP]\nsimp only [hk, h, coeff_C_zero, coeff_C_ne_zero]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nh : Commute (\u2191f a) x\nk : \u2115\nhk : \u00ack = 0\n\u22a2 Commute (\u2191f (coeff (\u2191C a) k)) x\n[PROOFSTEP]\nsimp only [coeff_C_ne_zero hk, RingHom.map_zero, Commute.zero_left]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nps : List R[X]\nhf : \u2200 (p : R[X]), p \u2208 ps \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x\n\u22a2 eval\u2082 f x (List.prod ps) = List.prod (List.map (eval\u2082 f x) ps)\n[PROOFSTEP]\ninduction' ps using List.reverseRecOn with ps p ihp\n[GOAL]\ncase H0\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nps : List R[X]\nhf\u271d : \u2200 (p : R[X]), p \u2208 ps \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x\nhf : \u2200 (p : R[X]), p \u2208 [] \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x\n\u22a2 eval\u2082 f x (List.prod []) = List.prod (List.map (eval\u2082 f x) [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H1\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nps\u271d : List R[X]\nhf\u271d : \u2200 (p : R[X]), p \u2208 ps\u271d \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x\nps : List R[X]\np : R[X]\nihp :\n  (\u2200 (p : R[X]), p \u2208 ps \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x) \u2192\n    eval\u2082 f x (List.prod ps) = List.prod (List.map (eval\u2082 f x) ps)\nhf : \u2200 (p_1 : R[X]), p_1 \u2208 ps ++ [p] \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p_1 k)) x\n\u22a2 eval\u2082 f x (List.prod (ps ++ [p])) = List.prod (List.map (eval\u2082 f x) (ps ++ [p]))\n[PROOFSTEP]\nsimp only [List.forall_mem_append, List.forall_mem_singleton] at hf \n[GOAL]\ncase H1\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nx : S\ninst\u271d : Semiring T\nps\u271d : List R[X]\nhf\u271d : \u2200 (p : R[X]), p \u2208 ps\u271d \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x\nps : List R[X]\np : R[X]\nihp :\n  (\u2200 (p : R[X]), p \u2208 ps \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x) \u2192\n    eval\u2082 f x (List.prod ps) = List.prod (List.map (eval\u2082 f x) ps)\nhf : (\u2200 (x_1 : R[X]), x_1 \u2208 ps \u2192 \u2200 (k : \u2115), Commute (\u2191f (coeff x_1 k)) x) \u2227 \u2200 (k : \u2115), Commute (\u2191f (coeff p k)) x\n\u22a2 eval\u2082 f x (List.prod (ps ++ [p])) = List.prod (List.map (eval\u2082 f x) (ps ++ [p]))\n[PROOFSTEP]\nsimp [eval\u2082_mul_noncomm _ _ hf.2, ihp hf.1]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 (fun i => eval\u2082 f x (\u2191(monomial i) (coeff p i))) = fun i => \u2191f (coeff p i) * x ^ i\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\nf : R \u2192+* S\np : R[X]\nn : \u2115\nhn : natDegree p < n\nx : S\n\u22a2 eval\u2082 f x p = \u2211 i in range n, \u2191f (coeff p i) * x ^ i\n[PROOFSTEP]\nrw [eval\u2082_eq_sum, p.sum_over_range' _ _ hn]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\nf : R \u2192+* S\np : R[X]\nn : \u2115\nhn : natDegree p < n\nx : S\n\u22a2 \u2200 (n : \u2115), \u2191f 0 * x ^ n = 0\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\nf : R \u2192+* S\np : R[X]\nn : \u2115\nhn : natDegree p < n\nx : S\ni : \u2115\n\u22a2 \u2191f 0 * x ^ i = 0\n[PROOFSTEP]\nrw [f.map_zero, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q\u271d r : R[X]\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nx : S\nq : R[X]\nhp : eval\u2082 f x p = 0\n\u22a2 eval\u2082 f x (p * q) = 0\n[PROOFSTEP]\nrw [eval\u2082_mul f x]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q\u271d r : R[X]\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nx : S\nq : R[X]\nhp : eval\u2082 f x p = 0\n\u22a2 eval\u2082 f x p * eval\u2082 f x q = 0\n[PROOFSTEP]\nexact mul_eq_zero_of_left hp (q.eval\u2082 f x)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nx : S\np : R[X]\nhq : eval\u2082 f x q = 0\n\u22a2 eval\u2082 f x (p * q) = 0\n[PROOFSTEP]\nrw [eval\u2082_mul f x]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nx : S\np : R[X]\nhq : eval\u2082 f x q = 0\n\u22a2 eval\u2082 f x p * eval\u2082 f x q = 0\n[PROOFSTEP]\nexact mul_eq_zero_of_right (p.eval\u2082 f x) hq\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 eval x p = sum p fun e a => a * x ^ e\n[PROOFSTEP]\nrw [eval, eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 (sum p fun e a => \u2191(RingHom.id R) a * x ^ e) = sum p fun e a => a * x ^ e\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx\u271d : R\np : R[X]\nx : R\n\u22a2 eval x p = \u2211 i in range (natDegree p + 1), coeff p i * x ^ i\n[PROOFSTEP]\nrw [eval_eq_sum, sum_over_range]\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx\u271d : R\np : R[X]\nx : R\n\u22a2 \u2200 (n : \u2115), 0 * x ^ n = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx\u271d : R\np : R[X]\nn : \u2115\nhn : natDegree p < n\nx : R\n\u22a2 eval x p = \u2211 i in range n, coeff p i * x ^ i\n[PROOFSTEP]\nrw [eval_eq_sum, p.sum_over_range' _ _ hn]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx\u271d : R\np : R[X]\nn : \u2115\nhn : natDegree p < n\nx : R\n\u22a2 \u2200 (n : \u2115), 0 * x ^ n = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\nx : R\nS : Type u_1\ninst\u271d : Semiring S\nf : R \u2192+* S\nr : R\n\u22a2 eval\u2082 f (\u2191f r) p = \u2191f (eval r p)\n[PROOFSTEP]\nrw [eval\u2082_eq_sum, eval_eq_sum, sum, sum, f.map_sum]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\nx : R\nS : Type u_1\ninst\u271d : Semiring S\nf : R \u2192+* S\nr : R\n\u22a2 \u2211 n in support p, \u2191f (coeff p n) * \u2191f r ^ n = \u2211 x in support p, \u2191f (coeff p x * r ^ x)\n[PROOFSTEP]\nsimp only [f.map_mul, f.map_pow]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 eval\u2082 f 1 p = \u2191f (eval 1 p)\n[PROOFSTEP]\nconvert eval\u2082_at_apply (p := p) f 1\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 1 = \u2191f 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n\u22a2 eval\u2082 f (\u2191n) p = \u2191f (eval (\u2191n) p)\n[PROOFSTEP]\nconvert eval\u2082_at_apply (p := p) f n\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n\u22a2 \u2191n = \u2191f \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\n\u22a2 eval x \u2191n = \u2191n\n[PROOFSTEP]\nsimp only [\u2190 C_eq_nat_cast, eval_C]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np\u271d q r : R[X]\nx\u271d : R\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\np : R[X]\nx : R\n\u22a2 eval x (s \u2022 p) = s \u2022 eval x p\n[PROOFSTEP]\nrw [\u2190 smul_one_smul R s p, eval, eval\u2082_smul, RingHom.id_apply, smul_one_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 eval x (\u2191C a * p) = a * eval x p\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q ph qh => simp only [mul_add, eval_add, ph, qh]\n| h_monomial n b => simp only [mul_assoc, C_mul_monomial, eval_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 eval x (\u2191C a * p) = a * eval x p\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q ph qh => simp only [mul_add, eval_add, ph, qh]\n| h_monomial n b => simp only [mul_assoc, C_mul_monomial, eval_monomial]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nx : R\np q : R[X]\nph : eval x (\u2191C a * p) = a * eval x p\nqh : eval x (\u2191C a * q) = a * eval x q\n\u22a2 eval x (\u2191C a * (p + q)) = a * eval x (p + q)\n[PROOFSTEP]\n\n| h_add p q ph qh => simp only [mul_add, eval_add, ph, qh]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nx : R\np q : R[X]\nph : eval x (\u2191C a * p) = a * eval x p\nqh : eval x (\u2191C a * q) = a * eval x q\n\u22a2 eval x (\u2191C a * (p + q)) = a * eval x (p + q)\n[PROOFSTEP]\nsimp only [mul_add, eval_add, ph, qh]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\nb : R\n\u22a2 eval x (\u2191C a * \u2191(monomial n) b) = a * eval x (\u2191(monomial n) b)\n[PROOFSTEP]\n\n| h_monomial n b => simp only [mul_assoc, C_mul_monomial, eval_monomial]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\nb : R\n\u22a2 eval x (\u2191C a * \u2191(monomial n) b) = a * eval x (\u2191(monomial n) b)\n[PROOFSTEP]\nsimp only [mul_assoc, C_mul_monomial, eval_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\n\u22a2 eval (1 + y) (\u2191(monomial d) (\u2191d + 1)) - eval y (\u2191(monomial d) (\u2191d + 1)) =\n    \u2211 x_1 in range (d + 1), \u2191(Nat.choose (d + 1) x_1) * (\u2191x_1 * y ^ (x_1 - 1))\n[PROOFSTEP]\nhave cast_succ : (d + 1 : S) = ((d.succ : \u2115) : S) := by simp only [Nat.cast_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\n\u22a2 \u2191d + 1 = \u2191(Nat.succ d)\n[PROOFSTEP]\nsimp only [Nat.cast_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n\u22a2 eval (1 + y) (\u2191(monomial d) (\u2191d + 1)) - eval y (\u2191(monomial d) (\u2191d + 1)) =\n    \u2211 x_1 in range (d + 1), \u2191(Nat.choose (d + 1) x_1) * (\u2191x_1 * y ^ (x_1 - 1))\n[PROOFSTEP]\nrw [cast_succ, eval_monomial, eval_monomial, add_comm, add_pow]\n  -- Porting note: `apply_congr` hadn't been ported yet, so `congr` & `ext` is used.\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n\u22a2 \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m) - \u2191(Nat.succ d) * y ^ d =\n    \u2211 x_1 in range (d + 1), \u2191(Nat.choose (d + 1) x_1) * (\u2191x_1 * y ^ (x_1 - 1))\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  \u00b7 congr\n    \u00b7skip\n    \u00b7 congr\n      \u00b7skip\n      \u00b7 ext\n        rw [one_pow, mul_one, mul_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m) - \u2191(Nat.succ d) * y ^ d\n[PROOFSTEP]\n  congr\n  \u00b7 congr\n    \u00b7skip\n    \u00b7 congr\n      \u00b7skip\n      \u00b7 ext\n        rw [one_pow, mul_one, mul_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m) - \u2191(Nat.succ d) * y ^ d\n[PROOFSTEP]\n  congr\n  \u00b7 congr\n    \u00b7skip\n    \u00b7 congr\n      \u00b7skip\n      \u00b7 ext\n        rw [one_pow, mul_one, mul_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m) - \u2191(Nat.succ d) * y ^ d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * y ^ d\n[PROOFSTEP]\n\u00b7 congr\n  \u00b7skip\n  \u00b7 congr\n    \u00b7skip\n    \u00b7 ext\n      rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7 congr\n    \u00b7skip\n    \u00b7 ext\n      rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7 congr\n    \u00b7skip\n    \u00b7 ext\n      rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d) * \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d)\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2191(Nat.succ d)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n\u00b7 congr\n  \u00b7skip\n  \u00b7 ext\n    rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7 ext\n    rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7 ext\n    rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| \u2211 m in range (d + 1), y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a.s\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| range (d + 1)\ncase a.a.f\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| fun m => y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.a.s\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| range (d + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a.s\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| range (d + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a.s\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| range (d + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a.f\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| fun m => y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n\u00b7 ext\n  rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a.a.f\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| fun m => y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n  ext\n  rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a.a.f\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| fun m => y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\n  ext\n  rw [one_pow, mul_one, mul_comm]\n[GOAL]\ncase a.a.f\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n| fun m => y ^ m * 1 ^ (d - m) * \u2191(Nat.choose d m)\n[PROOFSTEP]\next\n[GOAL]\ncase a.a.f.h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\nx\u271d : \u2115\n| y ^ x\u271d * 1 ^ (d - x\u271d) * \u2191(Nat.choose d x\u271d)\n[PROOFSTEP]\nrw [one_pow, mul_one, mul_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n\u22a2 \u2191(Nat.succ d) * \u2211 x in range (d + 1), \u2191(Nat.choose d x) * y ^ x - \u2191(Nat.succ d) * y ^ d =\n    \u2211 x_1 in range (d + 1), \u2191(Nat.choose (d + 1) x_1) * (\u2191x_1 * y ^ (x_1 - 1))\n[PROOFSTEP]\nrw [sum_range_succ, mul_add, Nat.choose_self, Nat.cast_one, one_mul, add_sub_cancel, mul_sum, sum_range_succ',\n  Nat.cast_zero, zero_mul, mul_zero, add_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\n\u22a2 \u2211 x in range d, \u2191(Nat.succ d) * (\u2191(Nat.choose d x) * y ^ x) =\n    \u2211 k in range d, \u2191(Nat.choose (d + 1) (k + 1)) * (\u2191(k + 1) * y ^ (k + 1 - 1))\n[PROOFSTEP]\nrefine sum_congr rfl fun y _hy => ?_\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : CommRing S\nd : \u2115\ny\u271d : S\ncast_succ : \u2191d + 1 = \u2191(Nat.succ d)\ny : \u2115\n_hy : y \u2208 range d\n\u22a2 \u2191(Nat.succ d) * (\u2191(Nat.choose d y) * y\u271d ^ y) = \u2191(Nat.choose (d + 1) (y + 1)) * (\u2191(y + 1) * y\u271d ^ (y + 1 - 1))\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 mul_assoc, \u2190 Nat.cast_mul, Nat.succ_mul_choose_eq, Nat.cast_mul, Nat.add_sub_cancel]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\n\u22a2 eval x (\u2191n * p) = \u2191n * eval x p\n[PROOFSTEP]\nrw [\u2190 C_eq_nat_cast, eval_C_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 eval x (p * X) = eval x p * x\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q ph qh => simp only [add_mul, eval_add, ph, qh]\n| h_monomial n a =>\n  simp only [\u2190 monomial_one_one_eq_X, monomial_mul_monomial, eval_monomial, mul_one, pow_succ', mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 eval x (p * X) = eval x p * x\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q ph qh => simp only [add_mul, eval_add, ph, qh]\n| h_monomial n a =>\n  simp only [\u2190 monomial_one_one_eq_X, monomial_mul_monomial, eval_monomial, mul_one, pow_succ', mul_assoc]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nx : R\np q : R[X]\nph : eval x (p * X) = eval x p * x\nqh : eval x (q * X) = eval x q * x\n\u22a2 eval x ((p + q) * X) = eval x (p + q) * x\n[PROOFSTEP]\n\n| h_add p q ph qh => simp only [add_mul, eval_add, ph, qh]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r : R[X]\nx : R\np q : R[X]\nph : eval x (p * X) = eval x p * x\nqh : eval x (q * X) = eval x q * x\n\u22a2 eval x ((p + q) * X) = eval x (p + q) * x\n[PROOFSTEP]\nsimp only [add_mul, eval_add, ph, qh]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\na : R\n\u22a2 eval x (\u2191(monomial n) a * X) = eval x (\u2191(monomial n) a) * x\n[PROOFSTEP]\n\n| h_monomial n a =>\n  simp only [\u2190 monomial_one_one_eq_X, monomial_mul_monomial, eval_monomial, mul_one, pow_succ', mul_assoc]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\na : R\n\u22a2 eval x (\u2191(monomial n) a * X) = eval x (\u2191(monomial n) a) * x\n[PROOFSTEP]\nsimp only [\u2190 monomial_one_one_eq_X, monomial_mul_monomial, eval_monomial, mul_one, pow_succ', mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nk : \u2115\n\u22a2 eval x (p * X ^ k) = eval x p * x ^ k\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\n\u22a2 eval x (p * X ^ Nat.zero) = eval x p * x ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nk : \u2115\nih : eval x (p * X ^ k) = eval x p * x ^ k\n\u22a2 eval x (p * X ^ Nat.succ k) = eval x p * x ^ Nat.succ k\n[PROOFSTEP]\nsimp [pow_succ', \u2190 mul_assoc, ih]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : DecidableEq R\n\u22a2 Decidable (IsRoot p a)\n[PROOFSTEP]\nunfold IsRoot\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nx : R\ninst\u271d : DecidableEq R\n\u22a2 Decidable (eval a p = 0)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\n\u22a2 coeff p 0 = coeff p 0 * 0 ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\n\u22a2 coeff p 0 * 0 ^ 0 = eval 0 p\n[PROOFSTEP]\nsymm\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\n\u22a2 eval 0 p = coeff p 0 * 0 ^ 0\n[PROOFSTEP]\nrw [eval_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\n\u22a2 (sum p fun e a => a * 0 ^ e) = coeff p 0 * 0 ^ 0\n[PROOFSTEP]\nexact Finset.sum_eq_single _ (fun b _ hb => by simp [zero_pow (Nat.pos_of_ne_zero hb)]) (by simp)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\nb : \u2115\nx\u271d : b \u2208 support p\nhb : b \u2260 0\n\u22a2 (fun e a => a * 0 ^ e) b (coeff p b) = 0\n[PROOFSTEP]\nsimp [zero_pow (Nat.pos_of_ne_zero hb)]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\n\u22a2 \u00ac0 \u2208 support p \u2192 (fun e a => a * 0 ^ e) 0 (coeff p 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q r : R[X]\nx : R\np : R[X]\nhp : coeff p 0 = 0\n\u22a2 IsRoot p 0\n[PROOFSTEP]\nrwa [coeff_zero_eq_eval_zero] at hp \n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np\u271d q\u271d r : R\u271d[X]\nx\u271d : R\u271d\nR : Type u_1\ninst\u271d : CommSemiring R\np q : R[X]\nx : R\nh : IsRoot p x\nhpq : p \u2223 q\n\u22a2 IsRoot q x\n[PROOFSTEP]\nrwa [IsRoot, eval, eval\u2082_eq_zero_of_dvd_of_eval\u2082_eq_zero _ _ hpq]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r\u271d : R[X]\nx r a : R\nhr : r \u2260 0\n\u22a2 \u00acIsRoot (\u2191C r) a\n[PROOFSTEP]\nsimpa using hr\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp p q = sum p fun e a => \u2191C a * q ^ e\n[PROOFSTEP]\nrw [comp, eval\u2082_eq_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp p X = p\n[PROOFSTEP]\nsimp only [comp, eval\u2082_def, C_mul_X_pow_eq_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 (sum p fun e a => \u2191(monomial e) a) = p\n[PROOFSTEP]\nexact sum_monomial_eq _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp p (\u2191C a) = \u2191C (eval a p)\n[PROOFSTEP]\nsimp [comp, (C : R \u2192+* _).map_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\n\u22a2 comp (\u2191n) p = \u2191n\n[PROOFSTEP]\nrw [\u2190 C_eq_nat_cast, C_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp p 0 = \u2191C (eval 0 p)\n[PROOFSTEP]\nrw [\u2190 C_0, comp_C]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp 0 p = 0\n[PROOFSTEP]\nrw [\u2190 C_0, C_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp p 1 = \u2191C (eval 1 p)\n[PROOFSTEP]\nrw [\u2190 C_1, comp_C]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp 1 p = 1\n[PROOFSTEP]\nrw [\u2190 C_1, C_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (p * X) r = comp p r * r\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, add_mul, add_comp]\n| h_monomial n b => simp only [pow_succ', mul_assoc, monomial_mul_X, monomial_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (p * X) r = comp p r * r\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, add_mul, add_comp]\n| h_monomial n b => simp only [pow_succ', mul_assoc, monomial_mul_X, monomial_comp]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r p q : R[X]\nhp : comp (p * X) r = comp p r * r\nhq : comp (q * X) r = comp q r * r\n\u22a2 comp ((p + q) * X) r = comp (p + q) r * r\n[PROOFSTEP]\n\n| h_add p q hp hq => simp only [hp, hq, add_mul, add_comp]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r p q : R[X]\nhp : comp (p * X) r = comp p r * r\nhq : comp (q * X) r = comp q r * r\n\u22a2 comp ((p + q) * X) r = comp (p + q) r * r\n[PROOFSTEP]\nsimp only [hp, hq, add_mul, add_comp]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\nb : R\n\u22a2 comp (\u2191(monomial n) b * X) r = comp (\u2191(monomial n) b) r * r\n[PROOFSTEP]\n\n| h_monomial n b => simp only [pow_succ', mul_assoc, monomial_mul_X, monomial_comp]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\nb : R\n\u22a2 comp (\u2191(monomial n) b * X) r = comp (\u2191(monomial n) b) r * r\n[PROOFSTEP]\nsimp only [pow_succ', mul_assoc, monomial_mul_X, monomial_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nk : \u2115\n\u22a2 comp (X ^ k) p = p ^ k\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (X ^ Nat.zero) p = p ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nk : \u2115\nih : comp (X ^ k) p = p ^ k\n\u22a2 comp (X ^ Nat.succ k) p = p ^ Nat.succ k\n[PROOFSTEP]\nsimp [pow_succ', mul_X_comp, ih]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nk : \u2115\n\u22a2 comp (p * X ^ k) r = comp p r * r ^ k\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (p * X ^ Nat.zero) r = comp p r * r ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nk : \u2115\nih : comp (p * X ^ k) r = comp p r * r ^ k\n\u22a2 comp (p * X ^ Nat.succ k) r = comp p r * r ^ Nat.succ k\n[PROOFSTEP]\nsimp [ih, pow_succ', \u2190 mul_assoc, mul_X_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (\u2191C a * p) r = \u2191C a * comp p r\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp [hp, hq, mul_add]\n| h_monomial n b => simp [mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (\u2191C a * p) r = \u2191C a * comp p r\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp [hp, hq, mul_add]\n| h_monomial n b => simp [mul_assoc]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r p q : R[X]\nhp : comp (\u2191C a * p) r = \u2191C a * comp p r\nhq : comp (\u2191C a * q) r = \u2191C a * comp q r\n\u22a2 comp (\u2191C a * (p + q)) r = \u2191C a * comp (p + q) r\n[PROOFSTEP]\n\n| h_add p q hp hq => simp [hp, hq, mul_add]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r p q : R[X]\nhp : comp (\u2191C a * p) r = \u2191C a * comp p r\nhq : comp (\u2191C a * q) r = \u2191C a * comp q r\n\u22a2 comp (\u2191C a * (p + q)) r = \u2191C a * comp (p + q) r\n[PROOFSTEP]\nsimp [hp, hq, mul_add]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\nb : R\n\u22a2 comp (\u2191C a * \u2191(monomial n) b) r = \u2191C a * comp (\u2191(monomial n) b) r\n[PROOFSTEP]\n\n| h_monomial n b => simp [mul_assoc]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\nb : R\n\u22a2 comp (\u2191C a * \u2191(monomial n) b) r = \u2191C a * comp (\u2191(monomial n) b) r\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\n\u22a2 comp (\u2191n * p) r = \u2191n * comp p r\n[PROOFSTEP]\nrw [\u2190 C_eq_nat_cast, C_mul_comp, C_eq_nat_cast]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (bit0 p) q = bit0 (comp p q)\n[PROOFSTEP]\nsimp only [bit0, add_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (bit1 p) q = bit1 (comp p q)\n[PROOFSTEP]\nsimp only [bit1, add_comp, bit0_comp, one_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\np q : R[X]\n\u22a2 comp (s \u2022 p) q = s \u2022 comp p q\n[PROOFSTEP]\nrw [\u2190 smul_one_smul R s p, comp, comp, eval\u2082_smul, \u2190 smul_eq_C_mul, smul_assoc, one_smul]\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 : R[X]\n\u22a2 comp (comp \u03c6 \u03c8) \u03c7 = comp \u03c6 (comp \u03c8 \u03c7)\n[PROOFSTEP]\nrefine Polynomial.induction_on \u03c6 ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 : R[X]\n\u22a2 \u2200 (a : R), comp (comp (\u2191C a) \u03c8) \u03c7 = comp (\u2191C a) (comp \u03c8 \u03c7)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine_1\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 : R[X]\na\u271d : R\n\u22a2 comp (comp (\u2191C a\u271d) \u03c8) \u03c7 = comp (\u2191C a\u271d) (comp \u03c8 \u03c7)\n[PROOFSTEP]\nsimp_all only [add_comp, mul_comp, C_comp, X_comp, pow_succ', \u2190 mul_assoc]\n[GOAL]\ncase refine_2\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 : R[X]\n\u22a2 \u2200 (p q : R[X]),\n    comp (comp p \u03c8) \u03c7 = comp p (comp \u03c8 \u03c7) \u2192\n      comp (comp q \u03c8) \u03c7 = comp q (comp \u03c8 \u03c7) \u2192 comp (comp (p + q) \u03c8) \u03c7 = comp (p + q) (comp \u03c8 \u03c7)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine_2\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 p\u271d q\u271d : R[X]\na\u271d\u00b9 : comp (comp p\u271d \u03c8) \u03c7 = comp p\u271d (comp \u03c8 \u03c7)\na\u271d : comp (comp q\u271d \u03c8) \u03c7 = comp q\u271d (comp \u03c8 \u03c7)\n\u22a2 comp (comp (p\u271d + q\u271d) \u03c8) \u03c7 = comp (p\u271d + q\u271d) (comp \u03c8 \u03c7)\n[PROOFSTEP]\nsimp_all only [add_comp, mul_comp, C_comp, X_comp, pow_succ', \u2190 mul_assoc]\n[GOAL]\ncase refine_3\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 : R[X]\n\u22a2 \u2200 (n : \u2115) (a : R),\n    comp (comp (\u2191C a * X ^ n) \u03c8) \u03c7 = comp (\u2191C a * X ^ n) (comp \u03c8 \u03c7) \u2192\n      comp (comp (\u2191C a * X ^ (n + 1)) \u03c8) \u03c7 = comp (\u2191C a * X ^ (n + 1)) (comp \u03c8 \u03c7)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine_3\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c6 \u03c8 \u03c7 : R[X]\nn\u271d : \u2115\na\u271d\u00b9 : R\na\u271d : comp (comp (\u2191C a\u271d\u00b9 * X ^ n\u271d) \u03c8) \u03c7 = comp (\u2191C a\u271d\u00b9 * X ^ n\u271d) (comp \u03c8 \u03c7)\n\u22a2 comp (comp (\u2191C a\u271d\u00b9 * X ^ (n\u271d + 1)) \u03c8) \u03c7 = comp (\u2191C a\u271d\u00b9 * X ^ (n\u271d + 1)) (comp \u03c8 \u03c7)\n[PROOFSTEP]\nsimp_all only [add_comp, mul_comp, C_comp, X_comp, pow_succ', \u2190 mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 coeff (comp p q) (natDegree p * natDegree q) = leadingCoeff p * leadingCoeff q ^ natDegree p\n[PROOFSTEP]\nrw [comp, eval\u2082_def, coeff_sum]\n  -- Porting note: `convert` \u2192 `refine`\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 (sum p fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) = leadingCoeff p * leadingCoeff q ^ natDegree p\n[PROOFSTEP]\nrefine Eq.trans (Finset.sum_eq_single p.natDegree ?h\u2080 ?h\u2081) ?h\u2082\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u2200 (b : \u2115),\n    b \u2208 support p \u2192 b \u2260 natDegree p \u2192 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) b (coeff p b) = 0\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u00acnatDegree p \u2208 support p \u2192\n    (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) = 0\ncase h\u2082\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) =\n    leadingCoeff p * leadingCoeff q ^ natDegree p\n[PROOFSTEP]\ncase h\u2082 => simp only [coeff_natDegree, coeff_C_mul, coeff_pow_mul_natDegree]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) =\n    leadingCoeff p * leadingCoeff q ^ natDegree p\n[PROOFSTEP]\ncase h\u2082 => simp only [coeff_natDegree, coeff_C_mul, coeff_pow_mul_natDegree]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) =\n    leadingCoeff p * leadingCoeff q ^ natDegree p\n[PROOFSTEP]\nsimp only [coeff_natDegree, coeff_C_mul, coeff_pow_mul_natDegree]\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u2200 (b : \u2115),\n    b \u2208 support p \u2192 b \u2260 natDegree p \u2192 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) b (coeff p b) = 0\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u00acnatDegree p \u2208 support p \u2192\n    (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) = 0\n[PROOFSTEP]\ncase h\u2080 =>\n  intro b hbs hbp\n  refine' coeff_eq_zero_of_natDegree_lt (natDegree_mul_le.trans_lt _)\n  rw [natDegree_C, zero_add]\n  refine' natDegree_pow_le.trans_lt ((mul_lt_mul_right (pos_iff_ne_zero.mpr hqd0)).mpr _)\n  exact lt_of_le_of_ne (le_natDegree_of_mem_supp _ hbs) hbp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u2200 (b : \u2115),\n    b \u2208 support p \u2192 b \u2260 natDegree p \u2192 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) b (coeff p b) = 0\n[PROOFSTEP]\ncase h\u2080 =>\n  intro b hbs hbp\n  refine' coeff_eq_zero_of_natDegree_lt (natDegree_mul_le.trans_lt _)\n  rw [natDegree_C, zero_add]\n  refine' natDegree_pow_le.trans_lt ((mul_lt_mul_right (pos_iff_ne_zero.mpr hqd0)).mpr _)\n  exact lt_of_le_of_ne (le_natDegree_of_mem_supp _ hbs) hbp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u2200 (b : \u2115),\n    b \u2208 support p \u2192 b \u2260 natDegree p \u2192 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) b (coeff p b) = 0\n[PROOFSTEP]\nintro b hbs hbp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\nb : \u2115\nhbs : b \u2208 support p\nhbp : b \u2260 natDegree p\n\u22a2 (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) b (coeff p b) = 0\n[PROOFSTEP]\nrefine' coeff_eq_zero_of_natDegree_lt (natDegree_mul_le.trans_lt _)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\nb : \u2115\nhbs : b \u2208 support p\nhbp : b \u2260 natDegree p\n\u22a2 natDegree (\u2191C (coeff p b)) + natDegree (q ^ b) < natDegree p * natDegree q\n[PROOFSTEP]\nrw [natDegree_C, zero_add]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\nb : \u2115\nhbs : b \u2208 support p\nhbp : b \u2260 natDegree p\n\u22a2 natDegree (q ^ b) < natDegree p * natDegree q\n[PROOFSTEP]\nrefine' natDegree_pow_le.trans_lt ((mul_lt_mul_right (pos_iff_ne_zero.mpr hqd0)).mpr _)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\nb : \u2115\nhbs : b \u2208 support p\nhbp : b \u2260 natDegree p\n\u22a2 b < natDegree p\n[PROOFSTEP]\nexact lt_of_le_of_ne (le_natDegree_of_mem_supp _ hbs) hbp\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u00acnatDegree p \u2208 support p \u2192\n    (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) = 0\n[PROOFSTEP]\ncase h\u2081 => simp (config := { contextual := true })\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u00acnatDegree p \u2208 support p \u2192\n    (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) = 0\n[PROOFSTEP]\ncase h\u2081 => simp (config := { contextual := true })\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhqd0 : natDegree q \u2260 0\n\u22a2 \u00acnatDegree p \u2208 support p \u2192\n    (fun a b => coeff (\u2191C b * q ^ a) (natDegree p * natDegree q)) (natDegree p) (coeff p (natDegree p)) = 0\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\na : R\n\u22a2 map f (\u2191(monomial n) a) = \u2191(monomial n) (\u2191f a)\n[PROOFSTEP]\ndsimp only [map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\na : R\n\u22a2 eval\u2082 (RingHom.comp C f) X (\u2191(monomial n) a) = \u2191(monomial n) (\u2191f a)\n[PROOFSTEP]\nrw [eval\u2082_monomial, \u2190 C_mul_X_pow_eq_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\na : R\n\u22a2 \u2191(RingHom.comp C f) a * X ^ n = \u2191C (\u2191f a) * X ^ n\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 map f (p * q) = map f p * map f q\n[PROOFSTEP]\nrw [map, eval\u2082_mul_noncomm]\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 \u2200 (k : \u2115), Commute (\u2191(RingHom.comp C f) (coeff q k)) X\n[PROOFSTEP]\nexact fun k => (commute_X _).symm\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nr : R\n\u22a2 map f (r \u2022 p) = \u2191f r \u2022 map f p\n[PROOFSTEP]\nrw [map, eval\u2082_smul, RingHom.comp_apply, C_mul']\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\nn : \u2115\ninst\u271d : Nat.AtLeastTwo n\n\u22a2 map f \u2191n = \u2191n\n[PROOFSTEP]\nrw [Polynomial.map_nat_cast]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n\u22a2 coeff (map f p) n = \u2191f (coeff p n)\n[PROOFSTEP]\nrw [map, eval\u2082_def, coeff_sum, sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n\u22a2 \u2211 n_1 in support p, coeff (\u2191(RingHom.comp C f) (coeff p n_1) * X ^ n_1) n = \u2191f (coeff p n)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 sum_C_mul_X_pow_eq p, coeff_sum, sum, map_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n| \u2191f (coeff p n)\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq p, coeff_sum, sum, map_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n| \u2191f (coeff p n)\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq p, coeff_sum, sum, map_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n| \u2191f (coeff p n)\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq p, coeff_sum, sum, map_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn : \u2115\n\u22a2 \u2211 n_1 in support p, coeff (\u2191(RingHom.comp C f) (coeff p n_1) * X ^ n_1) n =\n    \u2211 x in support p, \u2191f (coeff (\u2191C (coeff p x) * X ^ x) n)\n[PROOFSTEP]\nrefine'\n  Finset.sum_congr rfl fun x _hx =>\n    _\n      -- Porting note: Was `simp [Function.comp, coeff_C_mul_X_pow, f.map_mul]`.\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn x : \u2115\n_hx : x \u2208 support p\n\u22a2 coeff (\u2191(RingHom.comp C f) (coeff p x) * X ^ x) n = \u2191f (coeff (\u2191C (coeff p x) * X ^ x) n)\n[PROOFSTEP]\nsimp [Function.comp, coeff_C_mul_X_pow, -map_mul, -coeff_C_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn x : \u2115\n_hx : x \u2208 support p\n\u22a2 (if n = x then \u2191f (coeff p x) else 0) = \u2191f (if n = x then coeff p x else 0)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn x : \u2115\n_hx : x \u2208 support p\nh\u271d : n = x\n\u22a2 \u2191f (coeff p x) = \u2191f (coeff p x)\n[PROOFSTEP]\nsimp [f.map_zero]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nn x : \u2115\n_hx : x \u2208 support p\nh\u271d : \u00acn = x\n\u22a2 0 = \u2191f 0\n[PROOFSTEP]\nsimp [f.map_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\ne : R \u2243+* S\n\u22a2 RingHom.comp \u2191(mapRingHom \u2191(RingEquiv.symm e)) \u2191(mapRingHom \u2191e) = RingHom.id R[X]\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\ne : R \u2243+* S\nx\u271d : R\nn\u271d : \u2115\n\u22a2 coeff (\u2191(RingHom.comp (RingHom.comp \u2191(mapRingHom \u2191(RingEquiv.symm e)) \u2191(mapRingHom \u2191e)) C) x\u271d) n\u271d =\n    coeff (\u2191(RingHom.comp (RingHom.id R[X]) C) x\u271d) n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\ne : R \u2243+* S\nn\u271d : \u2115\n\u22a2 coeff (\u2191(RingHom.comp \u2191(mapRingHom \u2191(RingEquiv.symm e)) \u2191(mapRingHom \u2191e)) X) n\u271d = coeff (\u2191(RingHom.id R[X]) X) n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\ne : R \u2243+* S\n\u22a2 RingHom.comp \u2191(mapRingHom \u2191e) \u2191(mapRingHom \u2191(RingEquiv.symm e)) = RingHom.id S[X]\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.a.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\ne : R \u2243+* S\nx\u271d : S\nn\u271d : \u2115\n\u22a2 coeff (\u2191(RingHom.comp (RingHom.comp \u2191(mapRingHom \u2191e) \u2191(mapRingHom \u2191(RingEquiv.symm e))) C) x\u271d) n\u271d =\n    coeff (\u2191(RingHom.comp (RingHom.id S[X]) C) x\u271d) n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082.a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\ne : R \u2243+* S\nn\u271d : \u2115\n\u22a2 coeff (\u2191(RingHom.comp \u2191(mapRingHom \u2191e) \u2191(mapRingHom \u2191(RingEquiv.symm e))) X) n\u271d = coeff (\u2191(RingHom.id S[X]) X) n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Semiring T\ng : S \u2192+* T\np : R[X]\n\u22a2 \u2200 (n : \u2115), coeff (map g (map f p)) n = coeff (map (RingHom.comp g f) p) n\n[PROOFSTEP]\nsimp [coeff_map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 map (RingHom.id R) p = p\n[PROOFSTEP]\nsimp [Polynomial.ext_iff, coeff_map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x p = eval x (map f p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp [hp, hq]\n| h_monomial n r => simp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x p = eval x (map f p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp [hp, hq]\n| h_monomial n r => simp\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\np q : R[X]\nhp : eval\u2082 f x p = eval x (map f p)\nhq : eval\u2082 f x q = eval x (map f q)\n\u22a2 eval\u2082 f x (p + q) = eval x (map f (p + q))\n[PROOFSTEP]\n\n| h_add p q hp hq => simp [hp, hq]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\np q : R[X]\nhp : eval\u2082 f x p = eval x (map f p)\nhq : eval\u2082 f x q = eval x (map f q)\n\u22a2 eval\u2082 f x (p + q) = eval x (map f (p + q))\n[PROOFSTEP]\nsimp [hp, hq]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\nr : R\n\u22a2 eval\u2082 f x (\u2191(monomial n) r) = eval x (map f (\u2191(monomial n) r))\n[PROOFSTEP]\n\n| h_monomial n r => simp\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nx : S\nn : \u2115\nr : R\n\u22a2 eval\u2082 f x (\u2191(monomial n) r) = eval x (map f (\u2191(monomial n) r))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm\u271d n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np q : R[X]\nh : map f p = map f q\nm : \u2115\n\u22a2 \u2191f (coeff p m) = \u2191f (coeff q m)\n[PROOFSTEP]\nrw [\u2190 coeff_map f, \u2190 coeff_map f, h]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d\u00b9 q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\np\u271d p q : S[X]\nhp : \u2203 a, map f a = p\nhq : \u2203 a, map f a = q\np' : R[X]\nhp' : map f p' = p\nq' : R[X]\nhq' : map f q' = q\n\u22a2 map f (p' + q') = p + q\n[PROOFSTEP]\nrw [Polynomial.map_add f, hp', hq']\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\np : S[X]\nn : \u2115\ns : S\nr : R\nhr : \u2191f r = s\n\u22a2 map f (\u2191(monomial n) r) = \u2191(monomial n) s\n[PROOFSTEP]\nrw [map_monomial f, hr]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\n\u22a2 degree (map f p) \u2264 degree p\n[PROOFSTEP]\nrefine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm\u271d n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nm : \u2115\nhm : degree p < \u2191m\n\u22a2 coeff (map f p) m = 0\n[PROOFSTEP]\nrw [degree_lt_iff_coeff_zero] at hm \n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm\u271d n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nm : \u2115\nhm : \u2200 (m_1 : \u2115), m \u2264 m_1 \u2192 coeff p m_1 = 0\n\u22a2 coeff (map f p) m = 0\n[PROOFSTEP]\nsimp [hm m le_rfl]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\nhfp : map f p = 0\nx : R\n\u22a2 \u2191f x = \u2191f x * \u2191f (leadingCoeff p)\n[PROOFSTEP]\nsimp only [mul_one, hp.leadingCoeff, f.map_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\nhfp : map f p = 0\nx : R\n\u22a2 \u2191f x * coeff (map f p) (natDegree p) = 0\n[PROOFSTEP]\nsimp only [hfp, mul_zero, coeff_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\nh : \u2200 (x : R), \u2191f x = 0\nn : \u2115\n\u22a2 coeff (map f p) n = coeff 0 n\n[PROOFSTEP]\nsimp only [h, coeff_map, coeff_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\n\u22a2 degree p \u2264 degree (map f p)\n[PROOFSTEP]\nhave hp0 : p \u2260 0 := leadingCoeff_ne_zero.mp fun hp0 => hf (_root_.trans (congr_arg _ hp0) f.map_zero)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\nhp0 : p \u2260 0\n\u22a2 degree p \u2264 degree (map f p)\n[PROOFSTEP]\nrw [degree_eq_natDegree hp0]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\nhp0 : p \u2260 0\n\u22a2 \u2191(natDegree p) \u2264 degree (map f p)\n[PROOFSTEP]\nrefine' le_degree_of_ne_zero _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\nhp0 : p \u2260 0\n\u22a2 coeff (map f p) (natDegree p) \u2260 0\n[PROOFSTEP]\nrw [coeff_map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\nhp0 : p \u2260 0\n\u22a2 \u2191f (coeff p (natDegree p)) \u2260 0\n[PROOFSTEP]\nexact hf\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\n\u22a2 leadingCoeff (map f p) = \u2191f (leadingCoeff p)\n[PROOFSTEP]\nunfold leadingCoeff\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nhf : \u2191f (leadingCoeff p) \u2260 0\n\u22a2 coeff (map f p) (natDegree (map f p)) = \u2191f (coeff p (natDegree p))\n[PROOFSTEP]\nrw [coeff_map, natDegree_map_of_leadingCoeff_ne_zero f hf]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 RingHom.rangeS (mapRingHom f) \u2194 \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 RingHom.rangeS (mapRingHom f) \u2192 \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\n[PROOFSTEP]\nrintro \u27e8p, rfl\u27e9 n\n[GOAL]\ncase mp.intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nn : \u2115\n\u22a2 coeff (\u2191(mapRingHom f) p) n \u2208 RingHom.rangeS f\n[PROOFSTEP]\nrw [coe_mapRingHom, coeff_map]\n[GOAL]\ncase mp.intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nn : \u2115\n\u22a2 \u2191f (coeff p n) \u2208 RingHom.rangeS f\n[PROOFSTEP]\nexact Set.mem_range_self _\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 (\u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f) \u2192 p \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\n\u22a2 p \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nrw [p.as_sum_range_C_mul_X_pow]\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\n\u22a2 \u2211 i in range (natDegree p + 1), \u2191C (coeff p i) * X ^ i \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nrefine' (mapRingHom f).rangeS.sum_mem _\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\n\u22a2 \u2200 (c : \u2115), c \u2208 range (natDegree p + 1) \u2192 \u2191C (coeff p c) * X ^ c \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nintro i _hi\n[GOAL]\ncase mpr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\ni : \u2115\n_hi : i \u2208 range (natDegree p + 1)\n\u22a2 \u2191C (coeff p i) * X ^ i \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nrcases h i with \u27e8c, hc\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\ni : \u2115\n_hi : i \u2208 range (natDegree p + 1)\nc : R\nhc : \u2191f c = coeff p i\n\u22a2 \u2191C (coeff p i) * X ^ i \u2208 RingHom.rangeS (mapRingHom f)\n[PROOFSTEP]\nuse C c * X ^ i\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\nh : \u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f\ni : \u2115\n_hi : i \u2208 range (natDegree p + 1)\nc : R\nhc : \u2191f c = coeff p i\n\u22a2 \u2191(mapRingHom f) (\u2191C c * X ^ i) = \u2191C (coeff p i) * X ^ i\n[PROOFSTEP]\nrw [coe_mapRingHom, Polynomial.map_mul, map_C, hc, Polynomial.map_pow, map_X]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf : R \u2192+* S\ninst\u271d : Semiring T\ng : S \u2192+* T\nx : T\n\u22a2 eval\u2082 g x (map f p) = eval\u2082 (RingHom.comp g f) x p\n[PROOFSTEP]\nrw [eval\u2082_eq_eval_map, eval\u2082_eq_eval_map, map_map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np q : R[X]\n\u22a2 \u2200 (a : R), map f (comp (\u2191C a) q) = comp (map f (\u2191C a)) (map f q)\n[PROOFSTEP]\nsimp only [map_C, forall_const, C_comp, eq_self_iff_true]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np q : R[X]\n\u22a2 \u2200 (p q_1 : R[X]),\n    map f (comp p q) = comp (map f p) (map f q) \u2192\n      map f (comp q_1 q) = comp (map f q_1) (map f q) \u2192 map f (comp (p + q_1) q) = comp (map f (p + q_1)) (map f q)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [Polynomial.map_add, add_comp, forall_const, imp_true_iff,\n  eq_self_iff_true]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np q : R[X]\n\u22a2 \u2200 (n : \u2115) (a : R),\n    map f (comp (\u2191C a * X ^ n) q) = comp (map f (\u2191C a * X ^ n)) (map f q) \u2192\n      map f (comp (\u2191C a * X ^ (n + 1)) q) = comp (map f (\u2191C a * X ^ (n + 1))) (map f q)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [pow_succ', \u2190 mul_assoc, comp, forall_const, eval\u2082_mul_X, imp_true_iff,\n  eq_self_iff_true, map_X, Polynomial.map_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np : R[X]\n\u22a2 eval 0 (map f p) = \u2191f (eval 0 p)\n[PROOFSTEP]\nsimp [\u2190 coeff_zero_eq_eval_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np : R[X]\n\u22a2 eval 1 (map f p) = \u2191f (eval 1 p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n| h_monomial n r => simp only [one_pow, mul_one, eval_monomial, map_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np : R[X]\n\u22a2 eval 1 (map f p) = \u2191f (eval 1 p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n| h_monomial n r => simp only [one_pow, mul_one, eval_monomial, map_monomial]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np q : R[X]\nhp : eval 1 (map f p) = \u2191f (eval 1 p)\nhq : eval 1 (map f q) = \u2191f (eval 1 q)\n\u22a2 eval 1 (map f (p + q)) = \u2191f (eval 1 (p + q))\n[PROOFSTEP]\n\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np q : R[X]\nhp : eval 1 (map f p) = \u2191f (eval 1 p)\nhq : eval 1 (map f q) = \u2191f (eval 1 q)\n\u22a2 eval 1 (map f (p + q)) = \u2191f (eval 1 (p + q))\n[PROOFSTEP]\nsimp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn : \u2115\nr : R\n\u22a2 eval 1 (map f (\u2191(monomial n) r)) = \u2191f (eval 1 (\u2191(monomial n) r))\n[PROOFSTEP]\n\n| h_monomial n r => simp only [one_pow, mul_one, eval_monomial, map_monomial]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn : \u2115\nr : R\n\u22a2 eval 1 (map f (\u2191(monomial n) r)) = \u2191f (eval 1 (\u2191(monomial n) r))\n[PROOFSTEP]\nsimp only [one_pow, mul_one, eval_monomial, map_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np : R[X]\nn : \u2115\n\u22a2 eval (\u2191n) (map f p) = \u2191f (eval (\u2191n) p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n| h_monomial n r => simp only [map_natCast f, eval_monomial, map_monomial, f.map_pow, f.map_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\np : R[X]\nn : \u2115\n\u22a2 eval (\u2191n) (map f p) = \u2191f (eval (\u2191n) p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n| h_monomial n r => simp only [map_natCast f, eval_monomial, map_monomial, f.map_pow, f.map_mul]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn : \u2115\np q : R[X]\nhp : eval (\u2191n) (map f p) = \u2191f (eval (\u2191n) p)\nhq : eval (\u2191n) (map f q) = \u2191f (eval (\u2191n) q)\n\u22a2 eval (\u2191n) (map f (p + q)) = \u2191f (eval (\u2191n) (p + q))\n[PROOFSTEP]\n\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn : \u2115\np q : R[X]\nhp : eval (\u2191n) (map f p) = \u2191f (eval (\u2191n) p)\nhq : eval (\u2191n) (map f q) = \u2191f (eval (\u2191n) q)\n\u22a2 eval (\u2191n) (map f (p + q)) = \u2191f (eval (\u2191n) (p + q))\n[PROOFSTEP]\nsimp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn\u271d n : \u2115\nr : R\n\u22a2 eval (\u2191n\u271d) (map f (\u2191(monomial n) r)) = \u2191f (eval (\u2191n\u271d) (\u2191(monomial n) r))\n[PROOFSTEP]\n\n| h_monomial n r => simp only [map_natCast f, eval_monomial, map_monomial, f.map_pow, f.map_mul]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn\u271d n : \u2115\nr : R\n\u22a2 eval (\u2191n\u271d) (map f (\u2191(monomial n) r)) = \u2191f (eval (\u2191n\u271d) (\u2191(monomial n) r))\n[PROOFSTEP]\nsimp only [map_natCast f, eval_monomial, map_monomial, f.map_pow, f.map_mul]\n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\np : R[X]\ni : \u2124\n\u22a2 eval (\u2191i) (map f p) = \u2191f (eval (\u2191i) p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n| h_monomial n r => simp only [map_intCast, eval_monomial, map_monomial, map_pow, map_mul]\n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\np : R[X]\ni : \u2124\n\u22a2 eval (\u2191i) (map f p) = \u2191f (eval (\u2191i) p)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n| h_monomial n r => simp only [map_intCast, eval_monomial, map_monomial, map_pow, map_mul]\n[GOAL]\ncase h_add\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np\u271d q\u271d r : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\ni : \u2124\np q : R[X]\nhp : eval (\u2191i) (map f p) = \u2191f (eval (\u2191i) p)\nhq : eval (\u2191i) (map f q) = \u2191f (eval (\u2191i) q)\n\u22a2 eval (\u2191i) (map f (p + q)) = \u2191f (eval (\u2191i) (p + q))\n[PROOFSTEP]\n\n| h_add p q hp hq => simp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n[GOAL]\ncase h_add\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np\u271d q\u271d r : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\ni : \u2124\np q : R[X]\nhp : eval (\u2191i) (map f p) = \u2191f (eval (\u2191i) p)\nhq : eval (\u2191i) (map f q) = \u2191f (eval (\u2191i) q)\n\u22a2 eval (\u2191i) (map f (p + q)) = \u2191f (eval (\u2191i) (p + q))\n[PROOFSTEP]\nsimp only [hp, hq, Polynomial.map_add, RingHom.map_add, eval_add]\n[GOAL]\ncase h_monomial\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n\u271d : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np q r\u271d : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\ni : \u2124\nn : \u2115\nr : R\n\u22a2 eval (\u2191i) (map f (\u2191(monomial n) r)) = \u2191f (eval (\u2191i) (\u2191(monomial n) r))\n[PROOFSTEP]\n\n| h_monomial n r => simp only [map_intCast, eval_monomial, map_monomial, map_pow, map_mul]\n[GOAL]\ncase h_monomial\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n\u271d : \u2115\ninst\u271d\u00b3 : Semiring R\u271d\np q r\u271d : R\u271d[X]\ninst\u271d\u00b2 : Semiring S\u271d\nf\u271d : R\u271d \u2192+* S\u271d\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nf : R \u2192+* S\ni : \u2124\nn : \u2115\nr : R\n\u22a2 eval (\u2191i) (map f (\u2191(monomial n) r)) = \u2191f (eval (\u2191i) (\u2191(monomial n) r))\n[PROOFSTEP]\nsimp only [map_intCast, eval_monomial, map_monomial, map_pow, map_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Semiring T\nf : R \u2192+* S\ng : S \u2192+* T\nx : S\n\u22a2 \u2191g (eval\u2082 f x p) = eval\u2082 (RingHom.comp g f) (\u2191g x) p\n[PROOFSTEP]\nrw [\u2190 eval\u2082_map, eval\u2082_at_apply, eval_map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nx\u271d : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (comp p q) = eval\u2082 f (eval\u2082 f x q) p\n[PROOFSTEP]\nrw [comp, p.as_sum_range]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nx\u271d : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (eval\u2082 C q (\u2211 i in range (natDegree p + 1), \u2191(monomial i) (coeff p i))) =\n    eval\u2082 f (eval\u2082 f x q) (\u2211 i in range (natDegree p + 1), \u2191(monomial i) (coeff p i))\n[PROOFSTEP]\nsimp [eval\u2082_finset_sum, eval\u2082_pow]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nk : \u2115\nt : S\n\u22a2 eval\u2082 f t ((comp p)^[k] q) = (fun x => eval\u2082 f x p)^[k] (eval\u2082 f t q)\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nt : S\n\u22a2 eval\u2082 f t ((comp p)^[Nat.zero] q) = (fun x => eval\u2082 f x p)^[Nat.zero] (eval\u2082 f t q)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nt : S\nk : \u2115\nIH : eval\u2082 f t ((comp p)^[k] q) = (fun x => eval\u2082 f x p)^[k] (eval\u2082 f t q)\n\u22a2 eval\u2082 f t ((comp p)^[Nat.succ k] q) = (fun x => eval\u2082 f x p)^[Nat.succ k] (eval\u2082 f t q)\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', Function.iterate_succ_apply', eval\u2082_comp, IH]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\n\u22a2 eval x (comp p q) = eval (eval x q) p\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add r s hr hs => simp [add_comp, hr, hs]\n| h_monomial n a => simp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\n\u22a2 eval x (comp p q) = eval (eval x q) p\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_add r s hr hs => simp [add_comp, hr, hs]\n| h_monomial n a => simp\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nr s : R[X]\nhr : eval x (comp r q) = eval (eval x q) r\nhs : eval x (comp s q) = eval (eval x q) s\n\u22a2 eval x (comp (r + s) q) = eval (eval x q) (r + s)\n[PROOFSTEP]\n\n| h_add r s hr hs => simp [add_comp, hr, hs]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nr s : R[X]\nhr : eval x (comp r q) = eval (eval x q) r\nhs : eval x (comp s q) = eval (eval x q) s\n\u22a2 eval x (comp (r + s) q) = eval (eval x q) (r + s)\n[PROOFSTEP]\nsimp [add_comp, hr, hs]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nn : \u2115\na : R\n\u22a2 eval x (comp (\u2191(monomial n) a) q) = eval (eval x q) (\u2191(monomial n) a)\n[PROOFSTEP]\n\n| h_monomial n a => simp\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\nn : \u2115\na : R\n\u22a2 eval x (comp (\u2191(monomial n) a) q) = eval (eval x q) (\u2191(monomial n) a)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q\u271d : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\np q : R[X]\nH : IsRoot q a\n\u22a2 IsRoot (p * q) a\n[PROOFSTEP]\nrw [IsRoot, eval_mul, IsRoot.def.1 H, mul_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\np\u271d q\u271d : R[X]\nx : R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\np q : R[X]\nH : IsRoot p a\n\u22a2 IsRoot (p * q) a\n[PROOFSTEP]\nrw [IsRoot, eval_mul, IsRoot.def.1 H, zero_mul]\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9\u271d : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b3 : CommSemiring R\u271d\np\u271d q : R\u271d[X]\nx\u271d : R\u271d\ninst\u271d\u00b2 : CommSemiring S\nf : R\u271d \u2192+* S\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b9 : Type u_1\ns : Finset \u03b9\np : \u03b9 \u2192 R[X]\nx : R\n\u22a2 IsRoot (\u220f j in s, p j) x \u2194 \u2203 i, i \u2208 s \u2227 IsRoot (p i) x\n[PROOFSTEP]\nsimp only [IsRoot, eval_prod, Finset.prod_eq_zero_iff]\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n\u271d : \u2115\ninst\u271d\u00b2 : CommSemiring R\u271d\np q : R\u271d[X]\nx\u271d : R\u271d\ninst\u271d\u00b9 : CommSemiring S\nf : R\u271d \u2192+* S\nR : Type u_1\ninst\u271d : CommSemiring R\nn : \u2115\nx : R\n\u22a2 eval x (\u2211 i in range n, X ^ i) = \u2211 i in range n, x ^ i\n[PROOFSTEP]\nsimp [eval_finset_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\n\u22a2 support (map f p) \u2286 support p\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nx : \u2115\n\u22a2 x \u2208 support (map f p) \u2192 x \u2208 support p\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\nx : \u2115\n\u22a2 \u00acx \u2208 support p \u2192 \u00acx \u2208 support (map f p)\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nhf : Function.Injective \u2191f\n\u22a2 support (map f p) = support p\n[PROOFSTEP]\nsimp_rw [Finset.ext_iff, mem_support_iff, coeff_map, \u2190 map_zero f, hf.ne_iff, forall_const]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf\u271d f : R \u2192+* S\nx : R\np : R[X]\nh : IsRoot p x\n\u22a2 IsRoot (Polynomial.map f p) (\u2191f x)\n[PROOFSTEP]\nrw [IsRoot, eval_map, eval\u2082_hom, h.eq_zero, f.map_zero]\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b2 : CommSemiring R\u271d\ninst\u271d\u00b9 : CommSemiring S\nf\u271d : R\u271d \u2192+* S\nR : Type u_1\ninst\u271d : CommRing R\nf : R \u2192+* S\nx : R\np : R[X]\nh : IsRoot (Polynomial.map f p) (\u2191f x)\nhf : Function.Injective \u2191f\n\u22a2 IsRoot p x\n[PROOFSTEP]\nrwa [IsRoot, \u2190 (injective_iff_map_eq_zero' f).mp hf, \u2190 eval\u2082_hom, \u2190 eval_map]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\np q r : R[X]\nn : \u2124\nx : R\n\u22a2 eval x \u2191n = \u2191n\n[PROOFSTEP]\nsimp only [\u2190 C_eq_int_cast, eval_C]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Ring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (-p) = -eval\u2082 f x p\n[PROOFSTEP]\nrw [eq_neg_iff_add_eq_zero, \u2190 eval\u2082_add, add_left_neg, eval\u2082_zero]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Ring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Ring S\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f x (p - q) = eval\u2082 f x p - eval\u2082 f x q\n[PROOFSTEP]\nrw [sub_eq_add_neg, eval\u2082_add, eval\u2082_neg, sub_eq_add_neg]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q r : R[X]\n\u22a2 IsRoot (X - \u2191C a) b \u2194 a = b\n[PROOFSTEP]\nrw [IsRoot.def, eval_sub, eval_X, eval_C, sub_eq_zero, eq_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q r : R[X]\ni : \u2124\n\u22a2 comp (\u2191i) p = \u2191i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase ofNat\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q r : R[X]\na\u271d : \u2115\n\u22a2 comp (\u2191(Int.ofNat a\u271d)) p = \u2191(Int.ofNat a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase negSucc\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q r : R[X]\na\u271d : \u2115\n\u22a2 comp (\u2191(Int.negSucc a\u271d)) p = \u2191(Int.negSucc a\u271d)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Eval", "llama_tokens": 47739, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.7154240018510026, "lm_q1q2_score": 0.5746098115715168}}
{"text": "[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 \u2200 (a b : { v // v \u2260 0 }), Setoid.r a b \u2192 Submodule.span K {\u2191a} = Submodule.span K {\u2191b}\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9 \u27e8x, rfl : x \u2022 b = a\u27e9\n[GOAL]\ncase mk.mk.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\nb : V\nhb : b \u2260 0\nx : K\u02e3\nha : x \u2022 b \u2260 0\n\u22a2 Submodule.span K {\u2191{ val := x \u2022 b, property := ha }} = Submodule.span K {\u2191{ val := b, property := hb }}\n[PROOFSTEP]\nexact Submodule.span_singleton_group_smul_eq _ x _\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\n\u22a2 mk K v hv = mk K w hw \u2194 \u2203 a, a \u2022 w = v\n[PROOFSTEP]\nrw [mk_eq_mk_iff K v w hv hw]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\n\u22a2 (\u2203 a, a \u2022 w = v) \u2194 \u2203 a, a \u2022 w = v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\n\u22a2 (\u2203 a, a \u2022 w = v) \u2192 \u2203 a, a \u2022 w = v\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9\n[GOAL]\ncase mp.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\na : K\u02e3\nha : a \u2022 w = v\n\u22a2 \u2203 a, a \u2022 w = v\n[PROOFSTEP]\nexact \u27e8a, ha\u27e9\n[GOAL]\ncase mpr\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\n\u22a2 (\u2203 a, a \u2022 w = v) \u2192 \u2203 a, a \u2022 w = v\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\na : K\nha : a \u2022 w = v\n\u22a2 \u2203 a, a \u2022 w = v\n[PROOFSTEP]\nrefine' \u27e8Units.mk0 a fun c => hv.symm _, ha\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv w : V\nhv : v \u2260 0\nhw : w \u2260 0\na : K\nha : a \u2022 w = v\nc : a = 0\n\u22a2 0 = v\n[PROOFSTEP]\nrwa [c, zero_smul] at ha \n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 Projectivization.submodule v = Submodule.span K {Projectivization.rep v}\n[PROOFSTEP]\nconv_lhs => rw [\u2190 v.mk_rep]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n| Projectivization.submodule v\n[PROOFSTEP]\nrw [\u2190 v.mk_rep]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n| Projectivization.submodule v\n[PROOFSTEP]\nrw [\u2190 v.mk_rep]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n| Projectivization.submodule v\n[PROOFSTEP]\nrw [\u2190 v.mk_rep]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 finrank K { x // x \u2208 Projectivization.submodule v } = 1\n[PROOFSTEP]\nrw [submodule_eq]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 finrank K { x // x \u2208 Submodule.span K {Projectivization.rep v} } = 1\n[PROOFSTEP]\nexact finrank_span_singleton v.rep_nonzero\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 FiniteDimensional K { x // x \u2208 Projectivization.submodule v }\n[PROOFSTEP]\nrw [\u2190 v.mk_rep]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 FiniteDimensional K\n    { x // x \u2208 Projectivization.submodule (mk K (Projectivization.rep v) (_ : Projectivization.rep v \u2260 0)) }\n[PROOFSTEP]\nchange FiniteDimensional K (K \u2219 v.rep)\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : \u2119 K V\n\u22a2 FiniteDimensional K { x // x \u2208 Submodule.span K {Projectivization.rep v} }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu v : \u2119 K V\nh : Projectivization.submodule u = Projectivization.submodule v\n\u22a2 u = v\n[PROOFSTEP]\ninduction' u using ind with u hu\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu\u271d v : \u2119 K V\nh\u271d : Projectivization.submodule u\u271d = Projectivization.submodule v\nu : V\nhu : u \u2260 0\nh : Projectivization.submodule (mk K u hu) = Projectivization.submodule v\n\u22a2 mk K u hu = v\n[PROOFSTEP]\ninduction' v using ind with v hv\n[GOAL]\ncase h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu\u271d v\u271d : \u2119 K V\nh\u271d\u00b2 : Projectivization.submodule u\u271d = Projectivization.submodule v\u271d\nu : V\nhu : u \u2260 0\nh\u271d\u00b9 : Projectivization.submodule (mk K u hu) = Projectivization.submodule v\u271d\nv : V\nhv : v \u2260 0\nh\u271d : Projectivization.submodule u\u271d = Projectivization.submodule (mk K v hv)\nh : Projectivization.submodule (mk K u hu) = Projectivization.submodule (mk K v hv)\n\u22a2 mk K u hu = mk K v hv\n[PROOFSTEP]\nrw [submodule_mk, submodule_mk, Submodule.span_singleton_eq_span_singleton] at h \n[GOAL]\ncase h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nu\u271d v\u271d : \u2119 K V\nh\u271d\u00b2 : Projectivization.submodule u\u271d = Projectivization.submodule v\u271d\nu : V\nhu : u \u2260 0\nh\u271d\u00b9 : Projectivization.submodule (mk K u hu) = Projectivization.submodule v\u271d\nv : V\nhv : v \u2260 0\nh\u271d : Projectivization.submodule u\u271d = Projectivization.submodule (mk K v hv)\nh : \u2203 z, z \u2022 u = v\n\u22a2 mk K u hu = mk K v hv\n[PROOFSTEP]\nexact ((mk_eq_mk_iff K v u hv hu).2 h).symm\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\n\u22a2 H \u2208 Set.range Projectivization.submodule \u2194 finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\n[PROOFSTEP]\nrefine \u27e8fun \u27e8v, hv\u27e9 \u21a6 hv \u25b8 v.finrank_submodule, fun h \u21a6 ?_\u27e9\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\nh : finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\n\u22a2 H \u2208 Set.range Projectivization.submodule\n[PROOFSTEP]\nrcases finrank_eq_one_iff'.1 h with \u27e8v : H, hv\u2080, hv : \u2200 w : H, _\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\nh : finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\nv : { x // x \u2208 H }\nhv\u2080 : v \u2260 0\nhv : \u2200 (w : { x // x \u2208 H }), \u2203 c, c \u2022 v = w\n\u22a2 H \u2208 Set.range Projectivization.submodule\n[PROOFSTEP]\nuse mk K (v : V) (Subtype.coe_injective.ne hv\u2080)\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\nh : finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\nv : { x // x \u2208 H }\nhv\u2080 : v \u2260 0\nhv : \u2200 (w : { x // x \u2208 H }), \u2203 c, c \u2022 v = w\n\u22a2 Projectivization.submodule (mk K \u2191v (_ : \u2191v \u2260 \u21910)) = H\n[PROOFSTEP]\nrw [submodule_mk, SetLike.ext'_iff, Submodule.span_singleton_eq_range]\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\nh : finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\nv : { x // x \u2208 H }\nhv\u2080 : v \u2260 0\nhv : \u2200 (w : { x // x \u2208 H }), \u2203 c, c \u2022 v = w\n\u22a2 (Set.range fun x => x \u2022 \u2191v) = \u2191H\n[PROOFSTEP]\nrefine (Set.range_subset_iff.2 fun _ \u21a6 H.smul_mem _ v.2).antisymm fun x hx \u21a6 ?_\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\nh : finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\nv : { x // x \u2208 H }\nhv\u2080 : v \u2260 0\nhv : \u2200 (w : { x // x \u2208 H }), \u2203 c, c \u2022 v = w\nx : V\nhx : x \u2208 \u2191H\n\u22a2 x \u2208 Set.range fun x => x \u2022 \u2191v\n[PROOFSTEP]\nrcases hv \u27e8x, hx\u27e9 with \u27e8c, hc\u27e9\n[GOAL]\ncase h.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nH : Submodule K V\nh : finrank K { x // x \u2208 \u2191(Equiv.refl (Submodule K V)) H } = 1\nv : { x // x \u2208 H }\nhv\u2080 : v \u2260 0\nhv : \u2200 (w : { x // x \u2208 H }), \u2203 c, c \u2022 v = w\nx : V\nhx : x \u2208 \u2191H\nc : K\nhc : c \u2022 v = { val := x, property := hx }\n\u22a2 x \u2208 Set.range fun x => x \u2022 \u2191v\n[PROOFSTEP]\nexact \u27e8c, congr_arg Subtype.val hc\u27e9\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\n\u03c3 : K \u2192+* L\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nv : { v // v \u2260 0 }\nc : \u2191f \u2191v = 0\n\u22a2 \u2191f \u2191v = \u2191f 0\n[PROOFSTEP]\nsimp [c]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\n\u03c3 : K \u2192+* L\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\n\u22a2 (Setoid.r \u21d2 Setoid.r) (fun v => { val := \u2191f \u2191v, property := (_ : \u2191f \u2191v = 0 \u2192 False) }) fun v =>\n    { val := \u2191f \u2191v, property := (_ : \u2191f \u2191v = 0 \u2192 False) }\n[PROOFSTEP]\nrintro \u27e8u, hu\u27e9 \u27e8v, hv\u27e9 \u27e8a, ha\u27e9\n[GOAL]\ncase mk.mk.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\n\u03c3 : K \u2192+* L\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\na : K\u02e3\nha : (fun m => m \u2022 \u2191{ val := v, property := hv }) a = \u2191{ val := u, property := hu }\n\u22a2 Setoid.r ((fun v => { val := \u2191f \u2191v, property := (_ : \u2191f \u2191v = 0 \u2192 False) }) { val := u, property := hu })\n    ((fun v => { val := \u2191f \u2191v, property := (_ : \u2191f \u2191v = 0 \u2192 False) }) { val := v, property := hv })\n[PROOFSTEP]\nuse Units.map \u03c3.toMonoidHom a\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\n\u03c3 : K \u2192+* L\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\na : K\u02e3\nha : (fun m => m \u2022 \u2191{ val := v, property := hv }) a = \u2191{ val := u, property := hu }\n\u22a2 (fun m => m \u2022 \u2191((fun v => { val := \u2191f \u2191v, property := (_ : \u2191f \u2191v = 0 \u2192 False) }) { val := v, property := hv }))\n      (\u2191(Units.map \u2191\u03c3) a) =\n    \u2191((fun v => { val := \u2191f \u2191v, property := (_ : \u2191f \u2191v = 0 \u2192 False) }) { val := u, property := hu })\n[PROOFSTEP]\ndsimp at ha \u22a2\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\n\u03c3 : K \u2192+* L\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu : V\nhu : u \u2260 0\nv : V\nhv : v \u2260 0\na : K\u02e3\nha : a \u2022 v = u\n\u22a2 \u2191(Units.map \u2191\u03c3) a \u2022 \u2191f v = \u2191f u\n[PROOFSTEP]\nerw [\u2190 f.map_smul\u209b\u2097, ha]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : DivisionRing L\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* K\ninst\u271d : RingHomInvPair \u03c3 \u03c4\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu v : \u2119 K V\nh : map f hf u = map f hf v\n\u22a2 u = v\n[PROOFSTEP]\ninduction' u using ind with u hu\n[GOAL]\ncase h\nK : Type u_1\nV : Type u_2\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : DivisionRing L\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* K\ninst\u271d : RingHomInvPair \u03c3 \u03c4\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu\u271d v : \u2119 K V\nh\u271d : map f hf u\u271d = map f hf v\nu : V\nhu : u \u2260 0\nh : map f hf (mk K u hu) = map f hf v\n\u22a2 mk K u hu = v\n[PROOFSTEP]\ninduction' v using ind with v hv\n[GOAL]\ncase h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : DivisionRing L\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* K\ninst\u271d : RingHomInvPair \u03c3 \u03c4\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu\u271d v\u271d : \u2119 K V\nh\u271d\u00b2 : map f hf u\u271d = map f hf v\u271d\nu : V\nhu : u \u2260 0\nh\u271d\u00b9 : map f hf (mk K u hu) = map f hf v\u271d\nv : V\nhv : v \u2260 0\nh\u271d : map f hf u\u271d = map f hf (mk K v hv)\nh : map f hf (mk K u hu) = map f hf (mk K v hv)\n\u22a2 mk K u hu = mk K v hv\n[PROOFSTEP]\nsimp only [map_mk, mk_eq_mk_iff'] at h \u22a2\n[GOAL]\ncase h.h\nK : Type u_1\nV : Type u_2\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : DivisionRing L\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* K\ninst\u271d : RingHomInvPair \u03c3 \u03c4\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu\u271d v\u271d : \u2119 K V\nh\u271d\u00b2 : map f hf u\u271d = map f hf v\u271d\nu : V\nhu : u \u2260 0\nh\u271d\u00b9 : map f hf (mk K u hu) = map f hf v\u271d\nv : V\nhv : v \u2260 0\nh\u271d : map f hf u\u271d = map f hf (mk K v hv)\nh : \u2203 a, a \u2022 \u2191f v = \u2191f u\n\u22a2 \u2203 a, a \u2022 v = u\n[PROOFSTEP]\nrcases h with \u27e8a, ha\u27e9\n[GOAL]\ncase h.h.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : DivisionRing L\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* K\ninst\u271d : RingHomInvPair \u03c3 \u03c4\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu\u271d v\u271d : \u2119 K V\nh\u271d\u00b9 : map f hf u\u271d = map f hf v\u271d\nu : V\nhu : u \u2260 0\nh\u271d : map f hf (mk K u hu) = map f hf v\u271d\nv : V\nhv : v \u2260 0\nh : map f hf u\u271d = map f hf (mk K v hv)\na : L\nha : a \u2022 \u2191f v = \u2191f u\n\u22a2 \u2203 a, a \u2022 v = u\n[PROOFSTEP]\nrefine \u27e8\u03c4 a, hf ?_\u27e9\n[GOAL]\ncase h.h.intro\nK : Type u_1\nV : Type u_2\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b3 : DivisionRing L\ninst\u271d\u00b2 : AddCommGroup W\ninst\u271d\u00b9 : Module L W\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* K\ninst\u271d : RingHomInvPair \u03c3 \u03c4\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\nu\u271d v\u271d : \u2119 K V\nh\u271d\u00b9 : map f hf u\u271d = map f hf v\u271d\nu : V\nhu : u \u2260 0\nh\u271d : map f hf (mk K u hu) = map f hf v\u271d\nv : V\nhv : v \u2260 0\nh : map f hf u\u271d = map f hf (mk K v hv)\na : L\nha : a \u2022 \u2191f v = \u2191f u\n\u22a2 \u2191f (\u2191\u03c4 a \u2022 v) = \u2191f u\n[PROOFSTEP]\nrwa [f.map_smul\u209b\u2097, RingHomInvPair.comp_apply_eq\u2082]\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\n\u22a2 map LinearMap.id (_ : Function.Injective \u2191(LinearEquiv.refl K V)) = id\n[PROOFSTEP]\next \u27e8v\u27e9\n[GOAL]\ncase h.mk\nK : Type u_1\nV : Type u_2\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u00b2 : DivisionRing L\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module L W\nx\u271d : \u2119 K V\nv : { v // v \u2260 0 }\n\u22a2 map LinearMap.id (_ : Function.Injective \u2191(LinearEquiv.refl K V)) (Quot.mk Setoid.r v) = id (Quot.mk Setoid.r v)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\nV : Type u_2\ninst\u271d\u2079 : DivisionRing K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u2076 : DivisionRing L\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module L W\nF : Type u_5\nU : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : AddCommGroup U\ninst\u271d\u00b9 : Module F U\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* F\n\u03b3 : K \u2192+* F\ninst\u271d : RingHomCompTriple \u03c3 \u03c4 \u03b3\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\ng : W \u2192\u209b\u2097[\u03c4] U\nhg : Function.Injective \u2191g\n\u22a2 map (LinearMap.comp g f) (_ : Function.Injective (\u2191g \u2218 fun x => \u2191f x)) = map g hg \u2218 map f hf\n[PROOFSTEP]\next \u27e8v\u27e9\n[GOAL]\ncase h.mk\nK : Type u_1\nV : Type u_2\ninst\u271d\u2079 : DivisionRing K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nL : Type u_3\nW : Type u_4\ninst\u271d\u2076 : DivisionRing L\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module L W\nF : Type u_5\nU : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : AddCommGroup U\ninst\u271d\u00b9 : Module F U\n\u03c3 : K \u2192+* L\n\u03c4 : L \u2192+* F\n\u03b3 : K \u2192+* F\ninst\u271d : RingHomCompTriple \u03c3 \u03c4 \u03b3\nf : V \u2192\u209b\u2097[\u03c3] W\nhf : Function.Injective \u2191f\ng : W \u2192\u209b\u2097[\u03c4] U\nhg : Function.Injective \u2191g\nx\u271d : \u2119 K V\nv : { v // v \u2260 0 }\n\u22a2 map (LinearMap.comp g f) (_ : Function.Injective (\u2191g \u2218 fun x => \u2191f x)) (Quot.mk Setoid.r v) =\n    (map g hg \u2218 map f hf) (Quot.mk Setoid.r v)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.ProjectiveSpace.Basic", "llama_tokens": 8024, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737869342624, "lm_q2_score": 0.7154240079185319, "lm_q1q2_score": 0.574609809703615}}
{"text": "[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) (p + q) =\n    (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) p + (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) q\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (sum (p + q) fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) =\n    (sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) + sum q fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)\n[PROOFSTEP]\nrw [sum_add_index]\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2200 (i : \u2115), \u2191C (0 * \u2191i) * X ^ (i - 1) = 0\n[PROOFSTEP]\nsimp only [add_mul, forall_const, RingHom.map_add, eq_self_iff_true, zero_mul, RingHom.map_zero]\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2200 (a : \u2115) (b\u2081 b\u2082 : R), \u2191C ((b\u2081 + b\u2082) * \u2191a) * X ^ (a - 1) = \u2191C (b\u2081 * \u2191a) * X ^ (a - 1) + \u2191C (b\u2082 * \u2191a) * X ^ (a - 1)\n[PROOFSTEP]\nsimp only [add_mul, forall_const, RingHom.map_add, eq_self_iff_true, zero_mul, RingHom.map_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\np : R[X]\n\u22a2 AddHom.toFun\n      { toFun := fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1),\n        map_add' :=\n          (_ :\n            \u2200 (p q : R[X]),\n              (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) (p + q) =\n                (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) p +\n                  (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) q) }\n      (a \u2022 p) =\n    \u2191(RingHom.id R) a \u2022\n      AddHom.toFun\n        { toFun := fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1),\n          map_add' :=\n            (_ :\n              \u2200 (p q : R[X]),\n                (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) (p + q) =\n                  (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) p +\n                    (fun p => sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) q) }\n        p\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\np : R[X]\n\u22a2 (sum (a \u2022 p) fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) = a \u2022 sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)\n[PROOFSTEP]\nrw [sum_smul_index]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\np : R[X]\n\u22a2 (sum p fun n a_1 => \u2191C (a * a_1 * \u2191n) * X ^ (n - 1)) = a \u2022 sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)\n[PROOFSTEP]\nsimp only [mul_sum, \u2190 C_mul', mul_assoc, coeff_C_mul, RingHom.map_mul, forall_const, zero_mul, RingHom.map_zero, sum]\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\np : R[X]\n\u22a2 \u2200 (i : \u2115), \u2191C (0 * \u2191i) * X ^ (i - 1) = 0\n[PROOFSTEP]\nsimp only [mul_sum, \u2190 C_mul', mul_assoc, coeff_C_mul, RingHom.map_mul, forall_const, zero_mul, RingHom.map_zero, sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 coeff (\u2191derivative p) n = coeff p (n + 1) * (\u2191n + 1)\n[PROOFSTEP]\nrw [derivative_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 coeff (sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) n = coeff p (n + 1) * (\u2191n + 1)\n[PROOFSTEP]\nsimp only [coeff_X_pow, coeff_sum, coeff_C_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 (sum p fun a b => b * \u2191a * if n = a - 1 then 1 else 0) = coeff p (n + 1) * (\u2191n + 1)\n[PROOFSTEP]\nrw [sum, Finset.sum_eq_single (n + 1)]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 (coeff p (n + 1) * \u2191(n + 1) * if n = n + 1 - 1 then 1 else 0) = coeff p (n + 1) * (\u2191n + 1)\ncase h\u2080\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u2200 (b : \u2115), b \u2208 support p \u2192 b \u2260 n + 1 \u2192 (coeff p b * \u2191b * if n = b - 1 then 1 else 0) = 0\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u00acn + 1 \u2208 support p \u2192 (coeff p (n + 1) * \u2191(n + 1) * if n = n + 1 - 1 then 1 else 0) = 0\n[PROOFSTEP]\nsimp only [Nat.add_succ_sub_one, add_zero, mul_one, if_true, eq_self_iff_true]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 coeff p (n + 1) * \u2191(n + 1) = coeff p (n + 1) * (\u2191n + 1)\ncase h\u2080\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u2200 (b : \u2115), b \u2208 support p \u2192 b \u2260 n + 1 \u2192 (coeff p b * \u2191b * if n = b - 1 then 1 else 0) = 0\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u00acn + 1 \u2208 support p \u2192 (coeff p (n + 1) * \u2191(n + 1) * if n = n + 1 - 1 then 1 else 0) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u2200 (b : \u2115), b \u2208 support p \u2192 b \u2260 n + 1 \u2192 (coeff p b * \u2191b * if n = b - 1 then 1 else 0) = 0\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h\u2080\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b\u271d : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn b : \u2115\n\u22a2 b \u2208 support p \u2192 b \u2260 n + 1 \u2192 (coeff p b * \u2191b * if n = b - 1 then 1 else 0) = 0\n[PROOFSTEP]\ncases b\n[GOAL]\ncase h\u2080.zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 Nat.zero \u2208 support p \u2192 Nat.zero \u2260 n + 1 \u2192 (coeff p Nat.zero * \u2191Nat.zero * if n = Nat.zero - 1 then 1 else 0) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2080.zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\na\u271d\u00b9 : Nat.zero \u2208 support p\na\u271d : Nat.zero \u2260 n + 1\n\u22a2 (coeff p Nat.zero * \u2191Nat.zero * if n = Nat.zero - 1 then 1 else 0) = 0\n[PROOFSTEP]\nrw [Nat.cast_zero, mul_zero, zero_mul]\n[GOAL]\ncase h\u2080.succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d : Semiring R\np : R[X]\nn n\u271d : \u2115\n\u22a2 Nat.succ n\u271d \u2208 support p \u2192\n    Nat.succ n\u271d \u2260 n + 1 \u2192 (coeff p (Nat.succ n\u271d) * \u2191(Nat.succ n\u271d) * if n = Nat.succ n\u271d - 1 then 1 else 0) = 0\n[PROOFSTEP]\nintro _ H\n[GOAL]\ncase h\u2080.succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d : Semiring R\np : R[X]\nn n\u271d : \u2115\na\u271d : Nat.succ n\u271d \u2208 support p\nH : Nat.succ n\u271d \u2260 n + 1\n\u22a2 (coeff p (Nat.succ n\u271d) * \u2191(Nat.succ n\u271d) * if n = Nat.succ n\u271d - 1 then 1 else 0) = 0\n[PROOFSTEP]\nrw [Nat.succ_sub_one, if_neg (mt (congr_arg Nat.succ) H.symm), mul_zero]\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u00acn + 1 \u2208 support p \u2192 (coeff p (n + 1) * \u2191(n + 1) * if n = n + 1 - 1 then 1 else 0) = 0\n[PROOFSTEP]\nrw [if_pos (add_tsub_cancel_right n 1).symm, mul_one, Nat.cast_add, Nat.cast_one, mem_support_iff]\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\n\u22a2 \u00accoeff p (n + 1) \u2260 0 \u2192 coeff p (n + 1) * (\u2191n + 1) = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\nh : \u00accoeff p (n + 1) \u2260 0\n\u22a2 coeff p (n + 1) * (\u2191n + 1) = 0\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase h\u2081\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\nh : coeff p (n + 1) = 0\n\u22a2 coeff p (n + 1) * (\u2191n + 1) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nk : \u2115\n\u22a2 (\u2191derivative)^[k] 0 = 0\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\n\u22a2 (\u2191derivative)^[Nat.zero] 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nk : \u2115\nih : (\u2191derivative)^[k] 0 = 0\n\u22a2 (\u2191derivative)^[Nat.succ k] 0 = 0\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\na : R\nn : \u2115\n\u22a2 \u2191derivative (\u2191(monomial n) a) = \u2191(monomial (n - 1)) (a * \u2191n)\n[PROOFSTEP]\nrw [derivative_apply, sum_monomial_index, C_mul_X_pow_eq_monomial]\n[GOAL]\ncase hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\na : R\nn : \u2115\n\u22a2 \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\n\u22a2 \u2191derivative (\u2191C a * X) = \u2191C a\n[PROOFSTEP]\nsimp [C_mul_X_eq_monomial, derivative_monomial, Nat.cast_one, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\na : R\nn : \u2115\n\u22a2 \u2191derivative (\u2191C a * X ^ n) = \u2191C (a * \u2191n) * X ^ (n - 1)\n[PROOFSTEP]\nrw [C_mul_X_pow_eq_monomial, C_mul_X_pow_eq_monomial, derivative_monomial]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\n\u22a2 \u2191derivative (\u2191C a * X ^ 2) = \u2191C (a * 2) * X\n[PROOFSTEP]\nrw [derivative_C_mul_X_pow, Nat.cast_two, pow_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 \u2191derivative (X ^ n) = \u2191C \u2191n * X ^ (n - 1)\n[PROOFSTEP]\nconvert derivative_C_mul_X_pow (1 : R) n\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 X ^ n = \u2191C 1 * X ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_6\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 \u2191n = 1 * \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\n\u22a2 \u2191derivative (X ^ 2) = \u2191C 2 * X\n[PROOFSTEP]\nrw [derivative_X_pow, Nat.cast_two, pow_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\n\u22a2 \u2191derivative (\u2191C a) = 0\n[PROOFSTEP]\nsimp [derivative_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nhp : natDegree p = 0\n\u22a2 \u2191derivative p = 0\n[PROOFSTEP]\nrw [eq_C_of_natDegree_eq_zero hp, derivative_C]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\n\u22a2 \u2191(monomial (1 - 1)) (1 * \u21911) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R[X]\n\u22a2 \u2191derivative (bit0 a) = bit0 (\u2191derivative a)\n[PROOFSTEP]\nsimp [bit0]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R[X]\n\u22a2 \u2191derivative (bit1 a) = bit0 (\u2191derivative a)\n[PROOFSTEP]\nsimp [bit1]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nc : R\n\u22a2 \u2191derivative (X + \u2191C c) = 1\n[PROOFSTEP]\nrw [derivative_add, derivative_X, derivative_C, add_zero]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Semiring R\nS : Type u_1\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\np : R[X]\nk : \u2115\n\u22a2 (\u2191derivative)^[k] (s \u2022 p) = s \u2022 (\u2191derivative)^[k] p\n[PROOFSTEP]\ninduction' k with k ih generalizing p\n[GOAL]\ncase zero\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Semiring R\nS : Type u_1\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\np\u271d p : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (s \u2022 p) = s \u2022 (\u2191derivative)^[Nat.zero] p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Semiring R\nS : Type u_1\ninst\u271d\u00b2 : Monoid S\ninst\u271d\u00b9 : DistribMulAction S R\ninst\u271d : IsScalarTower S R R\ns : S\np\u271d : R[X]\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative)^[k] (s \u2022 p) = s \u2022 (\u2191derivative)^[k] p\np : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (s \u2022 p) = s \u2022 (\u2191derivative)^[Nat.succ k] p\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : Semiring R\na : R\np : R[X]\nk : \u2115\n\u22a2 (\u2191derivative)^[k] (\u2191C a * p) = \u2191C a * (\u2191derivative)^[k] p\n[PROOFSTEP]\nsimp_rw [\u2190 smul_eq_C_mul, iterate_derivative_smul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np : R[X]\nn : \u2115\nh : n \u2208 support (\u2191derivative p)\nh1 : coeff p (n + 1) = 0\n\u22a2 coeff (\u2191derivative p) n = 0\n[PROOFSTEP]\nrw [coeff_derivative, h1, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nH : p = 0\n\u22a2 degree (\u2191derivative p) = degree p\n[PROOFSTEP]\nrw [H, derivative_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nhp : natDegree p \u2260 0\n\u22a2 natDegree (\u2191derivative p) < natDegree p\n[PROOFSTEP]\ncases' eq_or_ne (derivative p) 0 with hp' hp'\n[GOAL]\ncase inl\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nhp : natDegree p \u2260 0\nhp' : \u2191derivative p = 0\n\u22a2 natDegree (\u2191derivative p) < natDegree p\n[PROOFSTEP]\nrw [hp', Polynomial.natDegree_zero]\n[GOAL]\ncase inl\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nhp : natDegree p \u2260 0\nhp' : \u2191derivative p = 0\n\u22a2 0 < natDegree p\n[PROOFSTEP]\nexact hp.bot_lt\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nhp : natDegree p \u2260 0\nhp' : \u2191derivative p \u2260 0\n\u22a2 natDegree (\u2191derivative p) < natDegree p\n[PROOFSTEP]\nrw [natDegree_lt_natDegree_iff hp']\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nhp : natDegree p \u2260 0\nhp' : \u2191derivative p \u2260 0\n\u22a2 degree (\u2191derivative p) < degree p\n[PROOFSTEP]\nexact degree_derivative_lt fun h => hp (h.symm \u25b8 natDegree_zero)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\n\u22a2 natDegree (\u2191derivative p) \u2264 natDegree p - 1\n[PROOFSTEP]\nby_cases p0 : p.natDegree = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\np0 : natDegree p = 0\n\u22a2 natDegree (\u2191derivative p) \u2264 natDegree p - 1\n[PROOFSTEP]\nsimp [p0, derivative_of_natDegree_zero]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\np0 : \u00acnatDegree p = 0\n\u22a2 natDegree (\u2191derivative p) \u2264 natDegree p - 1\n[PROOFSTEP]\nexact Nat.le_pred_of_lt (natDegree_derivative_lt p0)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 \u2191derivative \u2191n = 0\n[PROOFSTEP]\nrw [\u2190 map_natCast C n]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\n\u22a2 \u2191derivative (\u2191C \u2191n) = 0\n[PROOFSTEP]\nexact derivative_C\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nx : \u2115\nhx : natDegree p < x\n\u22a2 (\u2191derivative)^[x] p = 0\n[PROOFSTEP]\ninduction' h : p.natDegree using Nat.strong_induction_on with _ ih generalizing p x\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx\u271d\u00b9 : \u2115\nhx\u271d : natDegree p\u271d < x\u271d\u00b9\nx\u271d : \u2115\nh\u271d : natDegree p\u271d = x\u271d\nn\u271d : \u2115\nih : \u2200 (m : \u2115), m < n\u271d \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\np : R[X]\nx : \u2115\nhx : natDegree p < x\nh : natDegree p = n\u271d\n\u22a2 (\u2191derivative)^[x] p = 0\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase h\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx\u271d\u00b9 : \u2115\nhx\u271d : natDegree p\u271d < x\u271d\u00b9\nx\u271d : \u2115\nh : natDegree p\u271d = x\u271d\np : R[X]\nx : \u2115\nhx : natDegree p < x\nih : \u2200 (m : \u2115), m < natDegree p \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\n\u22a2 (\u2191derivative)^[x] p = 0\n[PROOFSTEP]\nobtain \u27e8t, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero (pos_of_gt hx).ne'\n[GOAL]\ncase h.intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx : \u2115\nhx\u271d : natDegree p\u271d < x\nx\u271d : \u2115\nh : natDegree p\u271d = x\u271d\np : R[X]\nih : \u2200 (m : \u2115), m < natDegree p \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\nt : \u2115\nhx : natDegree p < Nat.succ t\n\u22a2 (\u2191derivative)^[Nat.succ t] p = 0\n[PROOFSTEP]\nrw [Function.iterate_succ_apply]\n[GOAL]\ncase h.intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx : \u2115\nhx\u271d : natDegree p\u271d < x\nx\u271d : \u2115\nh : natDegree p\u271d = x\u271d\np : R[X]\nih : \u2200 (m : \u2115), m < natDegree p \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\nt : \u2115\nhx : natDegree p < Nat.succ t\n\u22a2 (\u2191derivative)^[t] (\u2191derivative p) = 0\n[PROOFSTEP]\nby_cases hp : p.natDegree = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx : \u2115\nhx\u271d : natDegree p\u271d < x\nx\u271d : \u2115\nh : natDegree p\u271d = x\u271d\np : R[X]\nih : \u2200 (m : \u2115), m < natDegree p \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\nt : \u2115\nhx : natDegree p < Nat.succ t\nhp : natDegree p = 0\n\u22a2 (\u2191derivative)^[t] (\u2191derivative p) = 0\n[PROOFSTEP]\nrw [derivative_of_natDegree_zero hp, iterate_derivative_zero]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx : \u2115\nhx\u271d : natDegree p\u271d < x\nx\u271d : \u2115\nh : natDegree p\u271d = x\u271d\np : R[X]\nih : \u2200 (m : \u2115), m < natDegree p \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\nt : \u2115\nhx : natDegree p < Nat.succ t\nhp : \u00acnatDegree p = 0\n\u22a2 (\u2191derivative)^[t] (\u2191derivative p) = 0\n[PROOFSTEP]\nhave := natDegree_derivative_lt hp\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np\u271d : R[X]\nx : \u2115\nhx\u271d : natDegree p\u271d < x\nx\u271d : \u2115\nh : natDegree p\u271d = x\u271d\np : R[X]\nih : \u2200 (m : \u2115), m < natDegree p \u2192 \u2200 {p : R[X]} {x : \u2115}, natDegree p < x \u2192 natDegree p = m \u2192 (\u2191derivative)^[x] p = 0\nt : \u2115\nhx : natDegree p < Nat.succ t\nhp : \u00acnatDegree p = 0\nthis : natDegree (\u2191derivative p) < natDegree p\n\u22a2 (\u2191derivative)^[t] (\u2191derivative p) = 0\n[PROOFSTEP]\nexact ih _ this (this.trans_le <| Nat.le_of_lt_succ hx) rfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\n\u22a2 natDegree f = 0\n[PROOFSTEP]\nrcases eq_or_ne f 0 with (rfl | hf)\n[GOAL]\ncase inl\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nh : \u2191derivative 0 = 0\n\u22a2 natDegree 0 = 0\n[PROOFSTEP]\nexact natDegree_zero\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\n\u22a2 natDegree f = 0\n[PROOFSTEP]\nrw [natDegree_eq_zero_iff_degree_le_zero]\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\n\u22a2 degree f \u2264 0\n[PROOFSTEP]\nby_contra' f_nat_degree_pos\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\nf_nat_degree_pos : 0 < degree f\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 natDegree_pos_iff_degree_pos] at f_nat_degree_pos \n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\n\u22a2 False\n[PROOFSTEP]\nlet m := f.natDegree - 1\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\n\u22a2 False\n[PROOFSTEP]\nhave hm : m + 1 = f.natDegree := tsub_add_cancel_of_le f_nat_degree_pos\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\nhm : m + 1 = natDegree f\n\u22a2 False\n[PROOFSTEP]\nhave h2 := coeff_derivative f m\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2191derivative f = 0\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\nhm : m + 1 = natDegree f\nh2 : coeff (\u2191derivative f) m = coeff f (m + 1) * (\u2191m + 1)\n\u22a2 False\n[PROOFSTEP]\nrw [Polynomial.ext_iff] at h \n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2200 (n : \u2115), coeff (\u2191derivative f) n = coeff 0 n\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\nhm : m + 1 = natDegree f\nh2 : coeff (\u2191derivative f) m = coeff f (m + 1) * (\u2191m + 1)\n\u22a2 False\n[PROOFSTEP]\nrw [h m, coeff_zero, \u2190 Nat.cast_add_one, \u2190 nsmul_eq_mul', eq_comm, smul_eq_zero] at h2 \n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2200 (n : \u2115), coeff (\u2191derivative f) n = coeff 0 n\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\nhm : m + 1 = natDegree f\nh2 : m + 1 = 0 \u2228 coeff f (m + 1) = 0\n\u22a2 False\n[PROOFSTEP]\nreplace h2 := h2.resolve_left m.succ_ne_zero\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2200 (n : \u2115), coeff (\u2191derivative f) n = coeff 0 n\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\nhm : m + 1 = natDegree f\nh2 : coeff f (m + 1) = 0\n\u22a2 False\n[PROOFSTEP]\nrw [hm, \u2190 leadingCoeff, leadingCoeff_eq_zero] at h2 \n[GOAL]\ncase inr\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\nf : R[X]\nh : \u2200 (n : \u2115), coeff (\u2191derivative f) n = coeff 0 n\nhf : f \u2260 0\nf_nat_degree_pos : 0 < natDegree f\nm : \u2115 := natDegree f - 1\nhm : m + 1 = natDegree f\nh2 : f = 0\n\u22a2 False\n[PROOFSTEP]\nexact hf h2\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2191derivative (f * g) = sum f fun n a => sum g fun m b => (n + m) \u2022 (\u2191C (a * b) * X ^ (n + m - 1))\n[PROOFSTEP]\nrw [mul_eq_sum_sum, derivative_sum]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 b in support f, \u2191derivative (sum g fun j a => \u2191(monomial (b + j)) (coeff f b * a)) =\n    sum f fun n a => sum g fun m b => (n + m) \u2022 (\u2191C (a * b) * X ^ (n + m - 1))\n[PROOFSTEP]\ntrans\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 b in support f, \u2191derivative (sum g fun j a => \u2191(monomial (b + j)) (coeff f b * a)) = ?m.380316\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2200 (x : \u2115), x \u2208 support f \u2192 \u2191derivative (sum g fun j a => \u2191(monomial (x + j)) (coeff f x * a)) = ?m.380366 x\nR : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d : Semiring R f g : R[X] \u22a2 \u2115 \u2192 R[X]\n[PROOFSTEP]\nintro x _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nx : \u2115\na\u271d : x \u2208 support f\n\u22a2 \u2191derivative (sum g fun j a => \u2191(monomial (x + j)) (coeff f x * a)) = ?m.380366 x\nR : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d : Semiring R f g : R[X] \u22a2 \u2115 \u2192 R[X]\n[PROOFSTEP]\nexact derivative_sum\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 x in support f, \u2211 b in support g, \u2191derivative ((fun j a => \u2191(monomial (x + j)) (coeff f x * a)) b (coeff g b)) =\n    sum f fun n a => sum g fun m b => (n + m) \u2022 (\u2191C (a * b) * X ^ (n + m - 1))\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2200 (x : \u2115),\n    x \u2208 support f \u2192\n      \u2211 b in support g, \u2191derivative ((fun j a => \u2191(monomial (x + j)) (coeff f x * a)) b (coeff g b)) =\n        (fun n a => sum g fun m b => (n + m) \u2022 (\u2191C (a * b) * X ^ (n + m - 1))) x (coeff f x)\n[PROOFSTEP]\nintro n _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d : n \u2208 support f\n\u22a2 \u2211 b in support g, \u2191derivative ((fun j a => \u2191(monomial (n + j)) (coeff f n * a)) b (coeff g b)) =\n    (fun n a => sum g fun m b => (n + m) \u2022 (\u2191C (a * b) * X ^ (n + m - 1))) n (coeff f n)\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d : n \u2208 support f\n\u22a2 \u2200 (x : \u2115),\n    x \u2208 support g \u2192\n      \u2191derivative ((fun j a => \u2191(monomial (n + j)) (coeff f n * a)) x (coeff g x)) =\n        (fun m b => (n + m) \u2022 (\u2191C (coeff f n * b) * X ^ (n + m - 1))) x (coeff g x)\n[PROOFSTEP]\nintro m _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d\u00b9 : n \u2208 support f\nm : \u2115\na\u271d : m \u2208 support g\n\u22a2 \u2191derivative ((fun j a => \u2191(monomial (n + j)) (coeff f n * a)) m (coeff g m)) =\n    (fun m b => (n + m) \u2022 (\u2191C (coeff f n * b) * X ^ (n + m - 1))) m (coeff g m)\n[PROOFSTEP]\ntrans\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d\u00b9 : n \u2208 support f\nm : \u2115\na\u271d : m \u2208 support g\n\u22a2 \u2191derivative ((fun j a => \u2191(monomial (n + j)) (coeff f n * a)) m (coeff g m)) = ?m.380816\n[PROOFSTEP]\nexact congr_arg _ C_mul_X_pow_eq_monomial.symm\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d\u00b9 : n \u2208 support f\nm : \u2115\na\u271d : m \u2208 support g\n\u22a2 \u2191derivative (\u2191C (coeff f n * coeff g m) * X ^ (n + m)) =\n    (fun m b => (n + m) \u2022 (\u2191C (coeff f n * b) * X ^ (n + m - 1))) m (coeff g m)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d\u00b9 : n \u2208 support f\nm : \u2115\na\u271d : m \u2208 support g\n\u22a2 \u2191derivative (\u2191C (coeff f n * coeff g m) * X ^ (n + m)) = (n + m) \u2022 (\u2191C (coeff f n * coeff g m) * X ^ (n + m - 1))\n[PROOFSTEP]\nrw [\u2190 smul_mul_assoc, smul_C, nsmul_eq_mul']\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\na\u271d\u00b9 : n \u2208 support f\nm : \u2115\na\u271d : m \u2208 support g\n\u22a2 \u2191derivative (\u2191C (coeff f n * coeff g m) * X ^ (n + m)) = \u2191C (coeff f n * coeff g m * \u2191(n + m)) * X ^ (n + m - 1)\n[PROOFSTEP]\nexact derivative_C_mul_X_pow _ _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn : \u2115\nhn : n \u2208 support f\nm : \u2115\nhm : m \u2208 support g\n\u22a2 (fun m b => (n + m) \u2022 (\u2191C (coeff f n * b) * X ^ (n + m - 1))) m (coeff g m) =\n    (fun m b => n \u2022 (\u2191C (coeff f n) * X ^ (n - 1)) * (\u2191C b * X ^ m) + \u2191C (coeff f n) * X ^ n * m \u2022 (\u2191C b * X ^ (m - 1)))\n      m (coeff g m)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nm : \u2115\nhm : m \u2208 support g\nhn : Nat.zero \u2208 support f\n\u22a2 (fun m b => (Nat.zero + m) \u2022 (\u2191C (coeff f Nat.zero * b) * X ^ (Nat.zero + m - 1))) m (coeff g m) =\n    (fun m b =>\n        Nat.zero \u2022 (\u2191C (coeff f Nat.zero) * X ^ (Nat.zero - 1)) * (\u2191C b * X ^ m) +\n          \u2191C (coeff f Nat.zero) * X ^ Nat.zero * m \u2022 (\u2191C b * X ^ (m - 1)))\n      m (coeff g m)\n[PROOFSTEP]\ncases m\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nm : \u2115\nhm : m \u2208 support g\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2208 support f\n\u22a2 (fun m b => (Nat.succ n\u271d + m) \u2022 (\u2191C (coeff f (Nat.succ n\u271d) * b) * X ^ (Nat.succ n\u271d + m - 1))) m (coeff g m) =\n    (fun m b =>\n        Nat.succ n\u271d \u2022 (\u2191C (coeff f (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d - 1)) * (\u2191C b * X ^ m) +\n          \u2191C (coeff f (Nat.succ n\u271d)) * X ^ Nat.succ n\u271d * m \u2022 (\u2191C b * X ^ (m - 1)))\n      m (coeff g m)\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero.zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nhn : Nat.zero \u2208 support f\nhm : Nat.zero \u2208 support g\n\u22a2 (fun m b => (Nat.zero + m) \u2022 (\u2191C (coeff f Nat.zero * b) * X ^ (Nat.zero + m - 1))) Nat.zero (coeff g Nat.zero) =\n    (fun m b =>\n        Nat.zero \u2022 (\u2191C (coeff f Nat.zero) * X ^ (Nat.zero - 1)) * (\u2191C b * X ^ m) +\n          \u2191C (coeff f Nat.zero) * X ^ Nat.zero * m \u2022 (\u2191C b * X ^ (m - 1)))\n      Nat.zero (coeff g Nat.zero)\n[PROOFSTEP]\nsimp_rw [add_smul, mul_smul_comm, smul_mul_assoc, X_pow_mul_assoc, \u2190 mul_assoc, \u2190 C_mul, mul_assoc, \u2190 pow_add]\n[GOAL]\ncase zero.succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nhn : Nat.zero \u2208 support f\nn\u271d : \u2115\nhm : Nat.succ n\u271d \u2208 support g\n\u22a2 (fun m b => (Nat.zero + m) \u2022 (\u2191C (coeff f Nat.zero * b) * X ^ (Nat.zero + m - 1))) (Nat.succ n\u271d)\n      (coeff g (Nat.succ n\u271d)) =\n    (fun m b =>\n        Nat.zero \u2022 (\u2191C (coeff f Nat.zero) * X ^ (Nat.zero - 1)) * (\u2191C b * X ^ m) +\n          \u2191C (coeff f Nat.zero) * X ^ Nat.zero * m \u2022 (\u2191C b * X ^ (m - 1)))\n      (Nat.succ n\u271d) (coeff g (Nat.succ n\u271d))\n[PROOFSTEP]\nsimp_rw [add_smul, mul_smul_comm, smul_mul_assoc, X_pow_mul_assoc, \u2190 mul_assoc, \u2190 C_mul, mul_assoc, \u2190 pow_add]\n[GOAL]\ncase succ.zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2208 support f\nhm : Nat.zero \u2208 support g\n\u22a2 (fun m b => (Nat.succ n\u271d + m) \u2022 (\u2191C (coeff f (Nat.succ n\u271d) * b) * X ^ (Nat.succ n\u271d + m - 1))) Nat.zero\n      (coeff g Nat.zero) =\n    (fun m b =>\n        Nat.succ n\u271d \u2022 (\u2191C (coeff f (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d - 1)) * (\u2191C b * X ^ m) +\n          \u2191C (coeff f (Nat.succ n\u271d)) * X ^ Nat.succ n\u271d * m \u2022 (\u2191C b * X ^ (m - 1)))\n      Nat.zero (coeff g Nat.zero)\n[PROOFSTEP]\nsimp_rw [add_smul, mul_smul_comm, smul_mul_assoc, X_pow_mul_assoc, \u2190 mul_assoc, \u2190 C_mul, mul_assoc, \u2190 pow_add]\n[GOAL]\ncase succ.succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn\u271d\u00b9 : \u2115\nhn : Nat.succ n\u271d\u00b9 \u2208 support f\nn\u271d : \u2115\nhm : Nat.succ n\u271d \u2208 support g\n\u22a2 (fun m b => (Nat.succ n\u271d\u00b9 + m) \u2022 (\u2191C (coeff f (Nat.succ n\u271d\u00b9) * b) * X ^ (Nat.succ n\u271d\u00b9 + m - 1))) (Nat.succ n\u271d)\n      (coeff g (Nat.succ n\u271d)) =\n    (fun m b =>\n        Nat.succ n\u271d\u00b9 \u2022 (\u2191C (coeff f (Nat.succ n\u271d\u00b9)) * X ^ (Nat.succ n\u271d\u00b9 - 1)) * (\u2191C b * X ^ m) +\n          \u2191C (coeff f (Nat.succ n\u271d\u00b9)) * X ^ Nat.succ n\u271d\u00b9 * m \u2022 (\u2191C b * X ^ (m - 1)))\n      (Nat.succ n\u271d) (coeff g (Nat.succ n\u271d))\n[PROOFSTEP]\nsimp_rw [add_smul, mul_smul_comm, smul_mul_assoc, X_pow_mul_assoc, \u2190 mul_assoc, \u2190 C_mul, mul_assoc, \u2190 pow_add]\n[GOAL]\ncase zero.succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nhn : Nat.zero \u2208 support f\nn\u271d : \u2115\nhm : Nat.succ n\u271d \u2208 support g\n\u22a2 Nat.zero \u2022 (\u2191C (coeff f Nat.zero * coeff g (Nat.succ n\u271d)) * X ^ (Nat.zero + Nat.succ n\u271d - 1)) +\n      Nat.succ n\u271d \u2022 (\u2191C (coeff f Nat.zero * coeff g (Nat.succ n\u271d)) * X ^ (Nat.zero + Nat.succ n\u271d - 1)) =\n    Nat.zero \u2022 (\u2191C (coeff f Nat.zero * coeff g (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d + (Nat.zero - 1))) +\n      Nat.succ n\u271d \u2022 (\u2191C (coeff f Nat.zero * coeff g (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d - 1 + Nat.zero))\n[PROOFSTEP]\nsimp [Nat.add_succ, Nat.succ_add, Nat.succ_sub_one, zero_smul, add_comm]\n[GOAL]\ncase succ.zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn\u271d : \u2115\nhn : Nat.succ n\u271d \u2208 support f\nhm : Nat.zero \u2208 support g\n\u22a2 Nat.succ n\u271d \u2022 (\u2191C (coeff f (Nat.succ n\u271d) * coeff g Nat.zero) * X ^ (Nat.succ n\u271d + Nat.zero - 1)) +\n      Nat.zero \u2022 (\u2191C (coeff f (Nat.succ n\u271d) * coeff g Nat.zero) * X ^ (Nat.succ n\u271d + Nat.zero - 1)) =\n    Nat.succ n\u271d \u2022 (\u2191C (coeff f (Nat.succ n\u271d) * coeff g Nat.zero) * X ^ (Nat.zero + (Nat.succ n\u271d - 1))) +\n      Nat.zero \u2022 (\u2191C (coeff f (Nat.succ n\u271d) * coeff g Nat.zero) * X ^ (Nat.zero - 1 + Nat.succ n\u271d))\n[PROOFSTEP]\nsimp [Nat.add_succ, Nat.succ_add, Nat.succ_sub_one, zero_smul, add_comm]\n[GOAL]\ncase succ.succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\nn\u271d\u00b9 : \u2115\nhn : Nat.succ n\u271d\u00b9 \u2208 support f\nn\u271d : \u2115\nhm : Nat.succ n\u271d \u2208 support g\n\u22a2 Nat.succ n\u271d\u00b9 \u2022 (\u2191C (coeff f (Nat.succ n\u271d\u00b9) * coeff g (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d\u00b9 + Nat.succ n\u271d - 1)) +\n      Nat.succ n\u271d \u2022 (\u2191C (coeff f (Nat.succ n\u271d\u00b9) * coeff g (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d\u00b9 + Nat.succ n\u271d - 1)) =\n    Nat.succ n\u271d\u00b9 \u2022 (\u2191C (coeff f (Nat.succ n\u271d\u00b9) * coeff g (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d + (Nat.succ n\u271d\u00b9 - 1))) +\n      Nat.succ n\u271d \u2022 (\u2191C (coeff f (Nat.succ n\u271d\u00b9) * coeff g (Nat.succ n\u271d)) * X ^ (Nat.succ n\u271d - 1 + Nat.succ n\u271d\u00b9))\n[PROOFSTEP]\nsimp [Nat.add_succ, Nat.succ_add, Nat.succ_sub_one, zero_smul, add_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 (sum f fun n a =>\n      sum g fun m b => n \u2022 (\u2191C a * X ^ (n - 1)) * (\u2191C b * X ^ m) + \u2191C a * X ^ n * m \u2022 (\u2191C b * X ^ (m - 1))) =\n    \u2191derivative f * g + f * \u2191derivative g\n[PROOFSTEP]\nconv =>\n  rhs\n  congr\n  \u00b7rw [\u2190 sum_C_mul_X_pow_eq g]\n  \u00b7rw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| (sum f fun n a =>\n      sum g fun m b => n \u2022 (\u2191C a * X ^ (n - 1)) * (\u2191C b * X ^ m) + \u2191C a * X ^ n * m \u2022 (\u2191C b * X ^ (m - 1))) =\n    \u2191derivative f * g + f * \u2191derivative g\n[PROOFSTEP]\n  rhs\n  congr\n  \u00b7rw [\u2190 sum_C_mul_X_pow_eq g]\n  \u00b7rw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| (sum f fun n a =>\n      sum g fun m b => n \u2022 (\u2191C a * X ^ (n - 1)) * (\u2191C b * X ^ m) + \u2191C a * X ^ n * m \u2022 (\u2191C b * X ^ (m - 1))) =\n    \u2191derivative f * g + f * \u2191derivative g\n[PROOFSTEP]\n  rhs\n  congr\n  \u00b7rw [\u2190 sum_C_mul_X_pow_eq g]\n  \u00b7rw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| (sum f fun n a =>\n      sum g fun m b => n \u2022 (\u2191C a * X ^ (n - 1)) * (\u2191C b * X ^ m) + \u2191C a * X ^ n * m \u2022 (\u2191C b * X ^ (m - 1))) =\n    \u2191derivative f * g + f * \u2191derivative g\n[PROOFSTEP]\nrhs\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| \u2191derivative f * g + f * \u2191derivative g\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| \u2191derivative f * g\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| f * \u2191derivative g\n[PROOFSTEP]\n\u00b7rw [\u2190 sum_C_mul_X_pow_eq g]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| \u2191derivative f * g\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq g]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| \u2191derivative f * g\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq g]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| \u2191derivative f * g\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq g]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| f * \u2191derivative g\n[PROOFSTEP]\n\u00b7rw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| f * \u2191derivative g\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| f * \u2191derivative g\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n| f * \u2191derivative g\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq f]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 (sum f fun n a =>\n      sum g fun m b => n \u2022 (\u2191C a * X ^ (n - 1)) * (\u2191C b * X ^ m) + \u2191C a * X ^ n * m \u2022 (\u2191C b * X ^ (m - 1))) =\n    (\u2191derivative f * sum g fun n a => \u2191C a * X ^ n) + (sum f fun n a => \u2191C a * X ^ n) * \u2191derivative g\n[PROOFSTEP]\nsimp only [sum, sum_add_distrib, Finset.mul_sum, Finset.sum_mul, derivative_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 x in support f, \u2211 x_1 in support g, x \u2022 (\u2191C (coeff f x) * X ^ (x - 1)) * (\u2191C (coeff g x_1) * X ^ x_1) +\n      \u2211 x in support f, \u2211 x_1 in support g, \u2191C (coeff f x) * X ^ x * x_1 \u2022 (\u2191C (coeff g x_1) * X ^ (x_1 - 1)) =\n    \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1 * \u2191x_1) * X ^ (x_1 - 1) * (\u2191C (coeff g x) * X ^ x) +\n      \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1) * X ^ x_1 * (\u2191C (coeff g x * \u2191x) * X ^ (x - 1))\n[PROOFSTEP]\nsimp_rw [\u2190 smul_mul_assoc, smul_C, nsmul_eq_mul']\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 x in support f, \u2211 x_1 in support g, \u2191C (coeff f x * \u2191x) * X ^ (x - 1) * (\u2191C (coeff g x_1) * X ^ x_1) +\n      \u2211 x in support f, \u2211 x_1 in support g, \u2191C (coeff f x) * X ^ x * (\u2191C (coeff g x_1 * \u2191x_1) * X ^ (x_1 - 1)) =\n    \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1 * \u2191x_1) * X ^ (x_1 - 1) * (\u2191C (coeff g x) * X ^ x) +\n      \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1) * X ^ x_1 * (\u2191C (coeff g x * \u2191x) * X ^ (x - 1))\n[PROOFSTEP]\nrw [Finset.sum_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 y in support g, \u2211 x in support f, \u2191C (coeff f x * \u2191x) * X ^ (x - 1) * (\u2191C (coeff g y) * X ^ y) +\n      \u2211 x in support f, \u2211 x_1 in support g, \u2191C (coeff f x) * X ^ x * (\u2191C (coeff g x_1 * \u2191x_1) * X ^ (x_1 - 1)) =\n    \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1 * \u2191x_1) * X ^ (x_1 - 1) * (\u2191C (coeff g x) * X ^ x) +\n      \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1) * X ^ x_1 * (\u2191C (coeff g x * \u2191x) * X ^ (x - 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nf g : R[X]\n\u22a2 \u2211 x in support f, \u2211 x_1 in support g, \u2191C (coeff f x) * X ^ x * (\u2191C (coeff g x_1 * \u2191x_1) * X ^ (x_1 - 1)) =\n    \u2211 x in support g, \u2211 x_1 in support f, \u2191C (coeff f x_1) * X ^ x_1 * (\u2191C (coeff g x * \u2191x) * X ^ (x - 1))\n[PROOFSTEP]\nrw [Finset.sum_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nx : R\n\u22a2 eval x (\u2191derivative p) = sum p fun n a => a * \u2191n * x ^ (n - 1)\n[PROOFSTEP]\nsimp_rw [derivative_apply, eval_sum, eval_mul_X_pow, eval_C]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\n\u22a2 \u2191derivative (map f p) = map f (\u2191derivative p)\n[PROOFSTEP]\nlet n := max p.natDegree (map f p).natDegree\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2191derivative (map f p) = map f (\u2191derivative p)\n[PROOFSTEP]\nrw [derivative_apply, derivative_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 (sum (map f p) fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) = map f (sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1))\n[PROOFSTEP]\nrw [sum_over_range' _ _ (n + 1) ((le_max_left _ _).trans_lt (lt_add_one _))]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 (sum (map f p) fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) = map f (\u2211 a in range (n + 1), \u2191C (coeff p a * \u2191a) * X ^ (a - 1))\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nrw [sum_over_range' _ _ (n + 1) ((le_max_right _ _).trans_lt (lt_add_one _))]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2211 a in range (n + 1), \u2191C (coeff (map f p) a * \u2191a) * X ^ (a - 1) =\n    map f (\u2211 a in range (n + 1), \u2191C (coeff p a * \u2191a) * X ^ (a - 1))\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nsimp only [Polynomial.map_sum, Polynomial.map_mul, Polynomial.map_C, map_mul, coeff_map, map_natCast,\n  Polynomial.map_nat_cast, Polynomial.map_pow, map_X]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nall_goals intro n; rw [zero_mul, C_0, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn\u271d : \u2115 := max (natDegree p) (natDegree (map f p))\nn : \u2115\n\u22a2 \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nrw [zero_mul, C_0, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn : \u2115 := max (natDegree p) (natDegree (map f p))\n\u22a2 \u2200 (n : \u2115), \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nn\u271d : \u2115 := max (natDegree p) (natDegree (map f p))\nn : \u2115\n\u22a2 \u2191C (0 * \u2191n) * X ^ (n - 1) = 0\n[PROOFSTEP]\nrw [zero_mul, C_0, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nk : \u2115\n\u22a2 (\u2191derivative)^[k] (map f p) = map f ((\u2191derivative)^[k] p)\n[PROOFSTEP]\ninduction' k with k ih generalizing p\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np\u271d : R[X]\nf : R \u2192+* S\np : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (map f p) = map f ((\u2191derivative)^[Nat.zero] p)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np\u271d : R[X]\nf : R \u2192+* S\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative)^[k] (map f p) = map f ((\u2191derivative)^[k] p)\np : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (map f p) = map f ((\u2191derivative)^[Nat.succ k] p)\n[PROOFSTEP]\nsimp only [ih, Function.iterate_succ, Polynomial.derivative_map, Function.comp_apply]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\nf : R[X]\n\u22a2 \u2191derivative (\u2191n * f) = \u2191n * \u2191derivative f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn k : \u2115\nf : R[X]\n\u22a2 (\u2191derivative)^[k] (\u2191n * f) = \u2191n * (\u2191derivative)^[k] f\n[PROOFSTEP]\ninduction' k with k ih generalizing f\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\nf\u271d f : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (\u2191n * f) = \u2191n * (\u2191derivative)^[Nat.zero] f\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\nf\u271d : R[X]\nk : \u2115\nih : \u2200 {f : R[X]}, (\u2191derivative)^[k] (\u2191n * f) = \u2191n * (\u2191derivative)^[k] f\nf : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (\u2191n * f) = \u2191n * (\u2191derivative)^[Nat.succ k] f\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 n \u2208 support (\u2191derivative p) \u2194 n + 1 \u2208 support p\n[PROOFSTEP]\nsuffices \u00acp.coeff (n + 1) * (n + 1 : \u2115) = 0 \u2194 coeff p (n + 1) \u2260 0 by\n  simpa only [mem_support_iff, coeff_derivative, Ne.def, Nat.cast_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nthis : \u00accoeff p (n + 1) * \u2191(n + 1) = 0 \u2194 coeff p (n + 1) \u2260 0\n\u22a2 n \u2208 support (\u2191derivative p) \u2194 n + 1 \u2208 support p\n[PROOFSTEP]\nsimpa only [mem_support_iff, coeff_derivative, Ne.def, Nat.cast_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 \u00accoeff p (n + 1) * \u2191(n + 1) = 0 \u2194 coeff p (n + 1) \u2260 0\n[PROOFSTEP]\nrw [\u2190 nsmul_eq_mul', smul_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 \u00ac(n + 1 = 0 \u2228 coeff p (n + 1) = 0) \u2194 coeff p (n + 1) \u2260 0\n[PROOFSTEP]\nsimp only [Nat.succ_ne_zero, false_or_iff]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 degree (\u2191derivative p) = \u2191(natDegree p - 1)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 degree (\u2191derivative p) \u2264 \u2191(natDegree p - 1)\n[PROOFSTEP]\nrw [derivative_apply]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 degree (sum p fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) \u2264 \u2191(natDegree p - 1)\n[PROOFSTEP]\napply le_trans (degree_sum_le _ _) (Finset.sup_le _)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 \u2200 (b : \u2115), b \u2208 support p \u2192 degree ((fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) b (coeff p b)) \u2264 \u2191(natDegree p - 1)\n[PROOFSTEP]\nintro n hn\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\nn : \u2115\nhn : n \u2208 support p\n\u22a2 degree ((fun n a => \u2191C (a * \u2191n) * X ^ (n - 1)) n (coeff p n)) \u2264 \u2191(natDegree p - 1)\n[PROOFSTEP]\nsimp only [Nat.cast_withBot]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\nn : \u2115\nhn : n \u2208 support p\n\u22a2 degree (\u2191C (coeff p n * \u2191n) * X ^ (n - 1)) \u2264 \u2191(natDegree p - 1)\n[PROOFSTEP]\napply le_trans (degree_C_mul_X_pow_le _ _) (WithBot.coe_le_coe.2 (tsub_le_tsub_right _ _))\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\nn : \u2115\nhn : n \u2208 support p\n\u22a2 n \u2264 natDegree p\n[PROOFSTEP]\napply le_natDegree_of_mem_supp _ hn\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 \u2191(natDegree p - 1) \u2264 degree (\u2191derivative p)\n[PROOFSTEP]\nrefine' le_sup _\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 natDegree p - 1 \u2208 support (\u2191derivative p)\n[PROOFSTEP]\nrw [mem_support_derivative, tsub_add_cancel_of_le, mem_support_iff]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 coeff p (natDegree p) \u2260 0\n[PROOFSTEP]\nshow \u00acleadingCoeff p = 0\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 \u00acleadingCoeff p = 0\n[PROOFSTEP]\nrw [leadingCoeff_eq_zero]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 \u00acp = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\nh : p = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h, natDegree_zero] at hp \n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < 0\nh : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl 0 (lt_of_le_of_lt (zero_le _) hp)\n[GOAL]\ncase a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nhp : 0 < natDegree p\n\u22a2 1 \u2264 natDegree p\n[PROOFSTEP]\nexact hp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nk : \u2115\np : R[X]\n\u22a2 \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\n\u22a2 \u2200 (m : \u2115),\n    coeff ((\u2191derivative)^[Nat.zero] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ Nat.zero), i) \u2022 coeff p (m + Nat.zero)\n[PROOFSTEP]\nsimp [add_zero, forall_const, one_smul, Ico_self, eq_self_iff_true, Function.iterate_zero_apply, prod_empty]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\n\u22a2 \u2200 (m : \u2115),\n    coeff ((\u2191derivative)^[Nat.succ k] p) m =\n      (\u220f i in Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)), i) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\nintro m\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 coeff ((\u2191derivative)^[Nat.succ k] p) m =\n    (\u220f i in Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)), i) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', coeff_derivative, ih (m + 1), \u2190 Nat.cast_add_one, \u2190 nsmul_eq_mul', smul_smul,\n  mul_comm]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 ((\u220f i in Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k), i) * (m + 1)) \u2022 coeff p (m + 1 + k) =\n    (\u220f i in Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)), i) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\napply congr_arg\u2082\n[GOAL]\ncase succ.hx\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 (\u220f i in Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k), i) * (m + 1) =\n    \u220f i in Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)), i\n[PROOFSTEP]\nhave set_eq : Ico m.succ (m + k.succ.succ) = Ico (m + 1).succ (m + 1 + k.succ) \u222a {m + 1} :=\n  by\n  simp_rw [\u2190 Nat.Ico_succ_singleton, union_comm, Nat.succ_eq_add_one, add_comm (k + 1), add_assoc]\n  rw [Ico_union_Ico_eq_Ico] <;> simp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)) = Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k) \u222a {m + 1}\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.Ico_succ_singleton, union_comm, Nat.succ_eq_add_one, add_comm (k + 1), add_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 Ico (m + 1) (m + (1 + (k + 1))) = Ico (m + 1) (m + (1 + 1)) \u222a Ico (m + (1 + 1)) (m + (1 + (k + 1)))\n[PROOFSTEP]\nrw [Ico_union_Ico_eq_Ico]\n[GOAL]\ncase hab\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 m + 1 \u2264 m + (1 + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hbc\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 m + (1 + 1) \u2264 m + (1 + (k + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.hx\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\nset_eq : Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)) = Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k) \u222a {m + 1}\n\u22a2 (\u220f i in Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k), i) * (m + 1) =\n    \u220f i in Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)), i\n[PROOFSTEP]\nrw [set_eq, prod_union, prod_singleton]\n[GOAL]\ncase succ.hx\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\nset_eq : Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)) = Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k) \u222a {m + 1}\n\u22a2 Disjoint (Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k)) {m + 1}\n[PROOFSTEP]\nrw [disjoint_singleton_right, mem_Ico]\n[GOAL]\ncase succ.hx\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\nset_eq : Ico (Nat.succ m) (m + Nat.succ (Nat.succ k)) = Ico (Nat.succ (m + 1)) (m + 1 + Nat.succ k) \u222a {m + 1}\n\u22a2 \u00ac(Nat.succ (m + 1) \u2264 m + 1 \u2227 m + 1 < m + 1 + Nat.succ k)\n[PROOFSTEP]\nexact fun h => (Nat.lt_succ_self _).not_le h.1\n[GOAL]\ncase succ.hy\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in Ico (Nat.succ m) (m + Nat.succ k), i) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 coeff p (m + 1 + k) = coeff p (m + Nat.succ k)\n[PROOFSTEP]\nexact congr_arg _ (Nat.succ_add m k)\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\nk : \u2115\np : R[X]\n\u22a2 \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in range k, (m + k - i)) \u2022 coeff p (m + k)\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\n\u22a2 \u2200 (m : \u2115), coeff ((\u2191derivative)^[Nat.zero] p) m = (\u220f i in range Nat.zero, (m + Nat.zero - i)) \u2022 coeff p (m + Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in range k, (m + k - i)) \u2022 coeff p (m + k)\n\u22a2 \u2200 (m : \u2115),\n    coeff ((\u2191derivative)^[Nat.succ k] p) m =\n      (\u220f i in range (Nat.succ k), (m + Nat.succ k - i)) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\nintro m\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in range k, (m + k - i)) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 coeff ((\u2191derivative)^[Nat.succ k] p) m = (\u220f i in range (Nat.succ k), (m + Nat.succ k - i)) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\ncalc\n  (derivative^[k + 1] p).coeff m = (\u220f i in range k, (m + k.succ - i)) \u2022 p.coeff (m + k.succ) * (m + 1) := by\n    rw [Function.iterate_succ_apply', coeff_derivative, ih m.succ, Nat.succ_add, Nat.add_succ]\n  _ = ((\u220f i in range k, (m + k.succ - i)) * (m + 1)) \u2022 p.coeff (m + k.succ) := by\n    rw [\u2190 Nat.cast_add_one, \u2190 nsmul_eq_mul', smul_smul, mul_comm]\n  _ = (\u220f i in range k.succ, (m + k.succ - i)) \u2022 p.coeff (m + k.succ) := by\n    rw [prod_range_succ, add_tsub_assoc_of_le k.le_succ, Nat.succ_sub le_rfl, tsub_self]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in range k, (m + k - i)) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 coeff ((\u2191derivative)^[k + 1] p) m = (\u220f i in range k, (m + Nat.succ k - i)) \u2022 coeff p (m + Nat.succ k) * (\u2191m + 1)\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', coeff_derivative, ih m.succ, Nat.succ_add, Nat.add_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in range k, (m + k - i)) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 (\u220f i in range k, (m + Nat.succ k - i)) \u2022 coeff p (m + Nat.succ k) * (\u2191m + 1) =\n    ((\u220f i in range k, (m + Nat.succ k - i)) * (m + 1)) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add_one, \u2190 nsmul_eq_mul', smul_smul, mul_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np : R[X]\nk : \u2115\nih : \u2200 (m : \u2115), coeff ((\u2191derivative)^[k] p) m = (\u220f i in range k, (m + k - i)) \u2022 coeff p (m + k)\nm : \u2115\n\u22a2 ((\u220f i in range k, (m + Nat.succ k - i)) * (m + 1)) \u2022 coeff p (m + Nat.succ k) =\n    (\u220f i in range (Nat.succ k), (m + Nat.succ k - i)) \u2022 coeff p (m + Nat.succ k)\n[PROOFSTEP]\nrw [prod_range_succ, add_tsub_assoc_of_le k.le_succ, Nat.succ_sub le_rfl, tsub_self]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\np q : R[X]\n\u22a2 (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (p * q) =\n    \u2211 k in range (Nat.succ Nat.zero), Nat.choose Nat.zero k \u2022 ((\u2191derivative)^[Nat.zero - k] p * (\u2191derivative)^[k] q)\n[PROOFSTEP]\nsimp [Finset.range]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 (\u2191derivative)^[Nat.succ n] (p * q) =\n    \u2211 k in range (Nat.succ (Nat.succ n)),\n      Nat.choose (Nat.succ n) k \u2022 ((\u2191derivative)^[Nat.succ n - k] p * (\u2191derivative)^[k] q)\n[PROOFSTEP]\ncalc\n  derivative^[n + 1] (p * q) =\n      derivative (\u2211 k : \u2115 in range n.succ, n.choose k \u2022 (derivative^[n - k] p * derivative^[k] q)) :=\n    by rw [Function.iterate_succ_apply', IH]\n  _ =\n      (\u2211 k : \u2115 in range n.succ, n.choose k \u2022 (derivative^[n - k + 1] p * derivative^[k] q)) +\n        \u2211 k : \u2115 in range n.succ, n.choose k \u2022 (derivative^[n - k] p * derivative^[k + 1] q) :=\n    by\n    simp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply', smul_add, sum_add_distrib]\n  _ =\n      (\u2211 k : \u2115 in range n.succ, n.choose k.succ \u2022 (derivative^[n - k] p * derivative^[k + 1] q)) +\n          1 \u2022 (derivative^[n + 1] p * derivative^[0] q) +\n        \u2211 k : \u2115 in range n.succ, n.choose k \u2022 (derivative^[n - k] p * derivative^[k + 1] q) :=\n    ?_\n  _ =\n      ((\u2211 k : \u2115 in range n.succ, n.choose k \u2022 (derivative^[n - k] p * derivative^[k + 1] q)) +\n          \u2211 k : \u2115 in range n.succ, n.choose k.succ \u2022 (derivative^[n - k] p * derivative^[k + 1] q)) +\n        1 \u2022 (derivative^[n + 1] p * derivative^[0] q) :=\n    by rw [add_comm, add_assoc]\n  _ =\n      (\u2211 i : \u2115 in range n.succ, (n + 1).choose (i + 1) \u2022 (derivative^[n + 1 - (i + 1)] p * derivative^[i + 1] q)) +\n        1 \u2022 (derivative^[n + 1] p * derivative^[0] q) :=\n    by simp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]\n  _ = \u2211 k : \u2115 in range n.succ.succ, n.succ.choose k \u2022 (derivative^[n.succ - k] p * derivative^[k] q) := by\n    rw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 (\u2191derivative)^[n + 1] (p * q) =\n    \u2191derivative (\u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q))\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', IH]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2191derivative (\u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k + 1] p * (\u2191derivative)^[k] q) +\n      \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q)\n[PROOFSTEP]\nsimp_rw [derivative_sum, derivative_smul, derivative_mul, Function.iterate_succ_apply', smul_add, sum_add_distrib]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 k in range (Nat.succ n), Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n        1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q) +\n      \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n        \u2211 k in range (Nat.succ n), Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n      1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q)\n[PROOFSTEP]\nrw [add_comm, add_assoc]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n        \u2211 k in range (Nat.succ n), Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n      1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q) =\n    \u2211 i in range (Nat.succ n),\n        Nat.choose (n + 1) (i + 1) \u2022 ((\u2191derivative)^[n + 1 - (i + 1)] p * (\u2191derivative)^[i + 1] q) +\n      1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q)\n[PROOFSTEP]\nsimp_rw [Nat.choose_succ_succ, Nat.succ_sub_succ, add_smul, sum_add_distrib]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 i in range (Nat.succ n),\n        Nat.choose (n + 1) (i + 1) \u2022 ((\u2191derivative)^[n + 1 - (i + 1)] p * (\u2191derivative)^[i + 1] q) +\n      1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q) =\n    \u2211 k in range (Nat.succ (Nat.succ n)),\n      Nat.choose (Nat.succ n) k \u2022 ((\u2191derivative)^[Nat.succ n - k] p * (\u2191derivative)^[k] q)\n[PROOFSTEP]\nrw [sum_range_succ' _ n.succ, Nat.choose_zero_right, tsub_zero]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k + 1] p * (\u2191derivative)^[k] q) +\n      \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n        1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q) +\n      \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase succ.e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k + 1] p * (\u2191derivative)^[k] q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q) +\n      1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q)\n[PROOFSTEP]\nrefine' (sum_range_succ' _ _).trans (congr_arg\u2082 (\u00b7 + \u00b7) _ _)\n[GOAL]\ncase succ.e_a.refine'_1\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 k in range n, Nat.choose n (k + 1) \u2022 ((\u2191derivative)^[n - (k + 1) + 1] p * (\u2191derivative)^[k + 1] q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q)\n[PROOFSTEP]\nrw [sum_range_succ, Nat.choose_succ_self, zero_smul, add_zero]\n[GOAL]\ncase succ.e_a.refine'_1\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 \u2211 k in range n, Nat.choose n (k + 1) \u2022 ((\u2191derivative)^[n - (k + 1) + 1] p * (\u2191derivative)^[k + 1] q) =\n    \u2211 x in range n, Nat.choose n (Nat.succ x) \u2022 ((\u2191derivative)^[n - x] p * (\u2191derivative)^[x + 1] q)\n[PROOFSTEP]\nrefine' sum_congr rfl fun k hk => _\n[GOAL]\ncase succ.e_a.refine'_1\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\nk : \u2115\nhk : k \u2208 range n\n\u22a2 Nat.choose n (k + 1) \u2022 ((\u2191derivative)^[n - (k + 1) + 1] p * (\u2191derivative)^[k + 1] q) =\n    Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q)\n[PROOFSTEP]\nrw [mem_range] at hk \n[GOAL]\ncase succ.e_a.refine'_1\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\nk : \u2115\nhk : k < n\n\u22a2 Nat.choose n (k + 1) \u2022 ((\u2191derivative)^[n - (k + 1) + 1] p * (\u2191derivative)^[k + 1] q) =\n    Nat.choose n (Nat.succ k) \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k + 1] q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase succ.e_a.refine'_1.e_a.e_a.e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\nk : \u2115\nhk : k < n\n\u22a2 n - (k + 1) + 1 = n - k\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub (Nat.succ_le_of_lt hk), Nat.succ_sub_succ]\n[GOAL]\ncase succ.e_a.refine'_2\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nIH :\n  (\u2191derivative)^[n] (p * q) =\n    \u2211 k in range (Nat.succ n), Nat.choose n k \u2022 ((\u2191derivative)^[n - k] p * (\u2191derivative)^[k] q)\n\u22a2 Nat.choose n 0 \u2022 ((\u2191derivative)^[n - 0 + 1] p * (\u2191derivative)^[0] q) =\n    1 \u2022 ((\u2191derivative)^[n + 1] p * (\u2191derivative)^[0] q)\n[PROOFSTEP]\nrw [Nat.choose_zero_right, tsub_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\np : R[X]\nn : \u2115\n\u22a2 \u2191derivative (p ^ (Nat.zero + 1)) = \u2191C (\u2191Nat.zero + 1) * p ^ Nat.zero * \u2191derivative p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d : CommSemiring R\np : R[X]\nn\u271d n : \u2115\nih : \u2191derivative (p ^ (n + 1)) = \u2191C (\u2191n + 1) * p ^ n * \u2191derivative p\n\u22a2 \u2191derivative (p ^ (Nat.succ n + 1)) = \u2191C (\u2191(Nat.succ n) + 1) * p ^ Nat.succ n * \u2191derivative p\n[PROOFSTEP]\nrw [pow_succ', derivative_mul, ih, Nat.add_one, mul_right_comm, C_add, add_mul, add_mul, pow_succ', \u2190 mul_assoc, C_1,\n  one_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d : CommSemiring R\np : R[X]\nn\u271d n : \u2115\nih : \u2191derivative (p ^ (n + 1)) = \u2191C (\u2191n + 1) * p ^ n * \u2191derivative p\n\u22a2 (\u2191C \u2191n * p ^ n * p + p ^ n * p) * \u2191derivative p + p ^ n * p * \u2191derivative p =\n    \u2191C (\u2191(Nat.succ n) + 1) * p ^ n * p * \u2191derivative p\n[PROOFSTEP]\nsimp [add_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\np : R[X]\nn : \u2115\n\u22a2 \u2191derivative (p ^ Nat.zero) = \u2191C \u2191Nat.zero * p ^ (Nat.zero - 1) * \u2191derivative p\n[PROOFSTEP]\nrw [pow_zero, derivative_one, Nat.cast_zero, C_0, zero_mul, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d\u00b9 : \u2115\ninst\u271d : CommSemiring R\np : R[X]\nn\u271d n : \u2115\n\u22a2 \u2191derivative (p ^ Nat.succ n) = \u2191C \u2191(Nat.succ n) * p ^ (Nat.succ n - 1) * \u2191derivative p\n[PROOFSTEP]\nrw [p.derivative_pow_succ n, n.succ_sub_one, n.cast_succ]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np : R[X]\n\u22a2 \u2191derivative (p ^ 2) = \u2191C 2 * p * \u2191derivative p\n[PROOFSTEP]\nrw [derivative_pow_succ, Nat.cast_one, one_add_one_eq_two, pow_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn m : \u2115\nc : R\nhm : m \u2260 0\n\u22a2 \u2191n \u2223 eval c ((\u2191derivative)^[m] (f ^ n))\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hm\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn : \u2115\nc : R\nm : \u2115\nhm : Nat.succ m \u2260 0\n\u22a2 \u2191n \u2223 eval c ((\u2191derivative)^[Nat.succ m] (f ^ n))\n[PROOFSTEP]\nrw [Function.iterate_succ_apply, derivative_pow, mul_assoc, C_eq_nat_cast, iterate_derivative_nat_cast_mul, eval_mul,\n  eval_nat_cast]\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn : \u2115\nc : R\nm : \u2115\nhm : Nat.succ m \u2260 0\n\u22a2 \u2191n \u2223 \u2191n * eval c ((\u2191derivative)^[m] (f ^ (n - 1) * \u2191derivative f))\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn k : \u2115\n\u22a2 (\u2191derivative)^[k] (X ^ n) = \u2191(Nat.descFactorial n k) * X ^ (n - k)\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn : \u2115\n\u22a2 (\u2191derivative)^[Nat.zero] (X ^ n) = \u2191(Nat.descFactorial n Nat.zero) * X ^ (n - Nat.zero)\n[PROOFSTEP]\nerw [Function.iterate_zero_apply, tsub_zero, Nat.descFactorial_zero, Nat.cast_one, one_mul]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn k : \u2115\nih : (\u2191derivative)^[k] (X ^ n) = \u2191(Nat.descFactorial n k) * X ^ (n - k)\n\u22a2 (\u2191derivative)^[Nat.succ k] (X ^ n) = \u2191(Nat.descFactorial n (Nat.succ k)) * X ^ (n - Nat.succ k)\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', ih, derivative_nat_cast_mul, derivative_X_pow, C_eq_nat_cast, Nat.succ_eq_add_one,\n  Nat.descFactorial_succ, Nat.sub_sub, Nat.cast_mul]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn k : \u2115\nih : (\u2191derivative)^[k] (X ^ n) = \u2191(Nat.descFactorial n k) * X ^ (n - k)\n\u22a2 \u2191(Nat.descFactorial n k) * (\u2191(n - k) * X ^ (n - (k + 1))) = \u2191(n - k) * \u2191(Nat.descFactorial n k) * X ^ (n - (k + 1))\n[PROOFSTEP]\nsimp [mul_comm, mul_assoc, mul_left_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn k : \u2115\n\u22a2 (\u2191derivative)^[k] (X ^ n) = \u2191C \u2191(Nat.descFactorial n k) * X ^ (n - k)\n[PROOFSTEP]\nrw [iterate_derivative_X_pow_eq_nat_cast_mul n k, C_eq_nat_cast]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn k : \u2115\n\u22a2 (\u2191derivative)^[k] (X ^ n) = \u2191(Nat.descFactorial n k) \u2022 X ^ (n - k)\n[PROOFSTEP]\nrw [iterate_derivative_X_pow_eq_C_mul n k, smul_eq_C_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\nc : R\nm : \u2115\n\u22a2 \u2191derivative ((X + \u2191C c) ^ m) = \u2191C \u2191m * (X + \u2191C c) ^ (m - 1)\n[PROOFSTEP]\nrw [derivative_pow, derivative_X_add_C, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\nc : R\n\u22a2 \u2191derivative ((X + \u2191C c) ^ 2) = \u2191C 2 * (X + \u2191C c)\n[PROOFSTEP]\nrw [derivative_sq, derivative_X_add_C, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn k : \u2115\nc : R\n\u22a2 (\u2191derivative)^[k] ((X + \u2191C c) ^ n) = \u2191(\u220f i in range k, (n - i)) * (X + \u2191C c) ^ (n - k)\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nc : R\n\u22a2 (\u2191derivative)^[Nat.zero] ((X + \u2191C c) ^ n) = \u2191(\u220f i in range Nat.zero, (n - i)) * (X + \u2191C c) ^ (n - Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nc : R\nk : \u2115\nIH : (\u2191derivative)^[k] ((X + \u2191C c) ^ n) = \u2191(\u220f i in range k, (n - i)) * (X + \u2191C c) ^ (n - k)\n\u22a2 (\u2191derivative)^[Nat.succ k] ((X + \u2191C c) ^ n) = \u2191(\u220f i in range (Nat.succ k), (n - i)) * (X + \u2191C c) ^ (n - Nat.succ k)\n[PROOFSTEP]\nsimp only [Function.iterate_succ_apply', IH, derivative_mul, zero_mul, derivative_nat_cast, zero_add,\n  Finset.prod_range_succ, C_eq_nat_cast, Nat.sub_sub, \u2190 mul_assoc, derivative_X_add_C_pow, Nat.succ_eq_add_one,\n  Nat.cast_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\n\u22a2 \u2191derivative (comp p q) = \u2191derivative q * comp (\u2191derivative p) q\n[PROOFSTEP]\ninduction p using Polynomial.induction_on'\n[GOAL]\ncase h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\nq p\u271d q\u271d : R[X]\na\u271d\u00b9 : \u2191derivative (comp p\u271d q) = \u2191derivative q * comp (\u2191derivative p\u271d) q\na\u271d : \u2191derivative (comp q\u271d q) = \u2191derivative q * comp (\u2191derivative q\u271d) q\n\u22a2 \u2191derivative (comp (p\u271d + q\u271d) q) = \u2191derivative q * comp (\u2191derivative (p\u271d + q\u271d)) q\n[PROOFSTEP]\nsimp [*, mul_add]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\nq : R[X]\nn\u271d : \u2115\na\u271d : R\n\u22a2 \u2191derivative (comp (\u2191(monomial n\u271d) a\u271d) q) = \u2191derivative q * comp (\u2191derivative (\u2191(monomial n\u271d) a\u271d)) q\n[PROOFSTEP]\nsimp only [derivative_pow, derivative_mul, monomial_comp, derivative_monomial, derivative_C, zero_mul, C_eq_nat_cast,\n  zero_add, RingHom.map_mul]\n[GOAL]\ncase h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\nq : R[X]\nn\u271d : \u2115\na\u271d : R\n\u22a2 \u2191C a\u271d * (\u2191n\u271d * q ^ (n\u271d - 1) * \u2191derivative q) = \u2191derivative q * (\u2191C a\u271d * \u2191n\u271d * q ^ (n\u271d - 1))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nr : R\n\u22a2 \u2191derivative (eval\u2082 C q (\u2191C r)) = eval\u2082 C q (\u2191derivative (\u2191C r)) * \u2191derivative q\n[PROOFSTEP]\nrw [eval\u2082_C, derivative_C, eval\u2082_zero, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q p\u2081 p\u2082 : R[X]\nih\u2081 : \u2191derivative (eval\u2082 C q p\u2081) = eval\u2082 C q (\u2191derivative p\u2081) * \u2191derivative q\nih\u2082 : \u2191derivative (eval\u2082 C q p\u2082) = eval\u2082 C q (\u2191derivative p\u2082) * \u2191derivative q\n\u22a2 \u2191derivative (eval\u2082 C q (p\u2081 + p\u2082)) = eval\u2082 C q (\u2191derivative (p\u2081 + p\u2082)) * \u2191derivative q\n[PROOFSTEP]\nrw [eval\u2082_add, derivative_add, ih\u2081, ih\u2082, derivative_add, eval\u2082_add, add_mul]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nn : \u2115\nr : R\nih : \u2191derivative (eval\u2082 C q (\u2191C r * X ^ n)) = eval\u2082 C q (\u2191derivative (\u2191C r * X ^ n)) * \u2191derivative q\n\u22a2 \u2191derivative (eval\u2082 C q (\u2191C r * X ^ (n + 1))) = eval\u2082 C q (\u2191derivative (\u2191C r * X ^ (n + 1))) * \u2191derivative q\n[PROOFSTEP]\nrw [pow_succ', \u2190 mul_assoc, eval\u2082_mul, eval\u2082_X, derivative_mul, ih, @derivative_mul _ _ _ X, derivative_X, mul_one,\n  eval\u2082_add, @eval\u2082_mul _ _ _ _ X, eval\u2082_X, add_mul, mul_right_comm]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\n[PROOFSTEP]\nrefine' Multiset.induction_on s (by simp) fun i s h => _\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 \u2191derivative (Multiset.prod (Multiset.map f 0)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase 0 i)) * \u2191derivative (f i)) 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\n\u22a2 \u2191derivative (Multiset.prod (Multiset.map f (i ::\u2098 s))) =\n    Multiset.sum\n      (Multiset.map (fun i_1 => Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) i_1)) * \u2191derivative (f i_1))\n        (i ::\u2098 s))\n[PROOFSTEP]\nrw [Multiset.map_cons, Multiset.prod_cons, derivative_mul, Multiset.map_cons _ i s, Multiset.sum_cons,\n  Multiset.erase_cons_head, mul_comm (derivative (f i))]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\n\u22a2 Multiset.prod (Multiset.map f s) * \u2191derivative (f i) + f i * \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.prod (Multiset.map f s) * \u2191derivative (f i) +\n      Multiset.sum\n        (Multiset.map (fun i_1 => Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) i_1)) * \u2191derivative (f i_1))\n          s)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\n\u22a2 f i * \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum\n      (Multiset.map (fun i_1 => Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) i_1)) * \u2191derivative (f i_1)) s)\n[PROOFSTEP]\nrw [h, \u2190 AddMonoidHom.coe_mulLeft, (AddMonoidHom.mulLeft (f i)).map_multiset_sum _, AddMonoidHom.coe_mulLeft]\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\n\u22a2 Multiset.sum\n      (Multiset.map (HMul.hMul (f i))\n        (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)) =\n    Multiset.sum\n      (Multiset.map (fun i_1 => Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) i_1)) * \u2191derivative (f i_1)) s)\n[PROOFSTEP]\nsimp only [Function.comp_apply, Multiset.map_map]\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\n\u22a2 Multiset.sum\n      (Multiset.map (fun x => f i * (Multiset.prod (Multiset.map f (Multiset.erase s x)) * \u2191derivative (f x))) s) =\n    Multiset.sum\n      (Multiset.map (fun x => Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) x)) * \u2191derivative (f x)) s)\n[PROOFSTEP]\nrefine' congr_arg _ (Multiset.map_congr rfl fun j hj => _)\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 f i * (Multiset.prod (Multiset.map f (Multiset.erase s j)) * \u2191derivative (f j)) =\n    Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) j)) * \u2191derivative (f j)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 Multiset.prod_cons, \u2190 Multiset.map_cons]\n[GOAL]\ncase e_a\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 Multiset.prod (Multiset.map f (i ::\u2098 Multiset.erase s j)) * \u2191derivative (f j) =\n    Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) j)) * \u2191derivative (f j)\n[PROOFSTEP]\nby_cases hij : i = j\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\nj : \u03b9\nhj : j \u2208 s\nhij : i = j\n\u22a2 Multiset.prod (Multiset.map f (i ::\u2098 Multiset.erase s j)) * \u2191derivative (f j) =\n    Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) j)) * \u2191derivative (f j)\n[PROOFSTEP]\nsimp [hij, \u2190 Multiset.prod_cons, \u2190 Multiset.map_cons, Multiset.cons_erase hj]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\ns\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\ni : \u03b9\ns : Multiset \u03b9\nh :\n  \u2191derivative (Multiset.prod (Multiset.map f s)) =\n    Multiset.sum (Multiset.map (fun i => Multiset.prod (Multiset.map f (Multiset.erase s i)) * \u2191derivative (f i)) s)\nj : \u03b9\nhj : j \u2208 s\nhij : \u00aci = j\n\u22a2 Multiset.prod (Multiset.map f (i ::\u2098 Multiset.erase s j)) * \u2191derivative (f j) =\n    Multiset.prod (Multiset.map f (Multiset.erase (i ::\u2098 s) j)) * \u2191derivative (f j)\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Ring R\nc : R\n\u22a2 \u2191derivative (X - \u2191C c) = 1\n[PROOFSTEP]\nrw [derivative_sub, derivative_X, derivative_C, sub_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Ring R\nk : \u2115\nf g : R[X]\n\u22a2 (\u2191derivative)^[k] (f - g) = (\u2191derivative)^[k] f - (\u2191derivative)^[k] g\n[PROOFSTEP]\ninduction' k with k ih generalizing f g\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Ring R\nf\u271d g\u271d f g : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (f - g) = (\u2191derivative)^[Nat.zero] f - (\u2191derivative)^[Nat.zero] g\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Ring R\nf\u271d g\u271d : R[X]\nk : \u2115\nih : \u2200 {f g : R[X]}, (\u2191derivative)^[k] (f - g) = (\u2191derivative)^[k] f - (\u2191derivative)^[k] g\nf g : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (f - g) = (\u2191derivative)^[Nat.succ k] f - (\u2191derivative)^[Nat.succ k] g\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\n\u22a2 \u2191derivative \u2191n = 0\n[PROOFSTEP]\nrw [\u2190 C_eq_int_cast n]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\n\u22a2 \u2191derivative (\u2191C \u2191n) = 0\n[PROOFSTEP]\nexact derivative_C\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\nf : R[X]\n\u22a2 \u2191derivative (\u2191n * f) = \u2191n * \u2191derivative f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\nk : \u2115\nf : R[X]\n\u22a2 (\u2191derivative)^[k] (\u2191n * f) = \u2191n * (\u2191derivative)^[k] f\n[PROOFSTEP]\ninduction' k with k ih generalizing f\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\nf\u271d f : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (\u2191n * f) = \u2191n * (\u2191derivative)^[Nat.zero] f\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Ring R\nn : \u2124\nf\u271d : R[X]\nk : \u2115\nih : \u2200 {f : R[X]}, (\u2191derivative)^[k] (\u2191n * f) = \u2191n * (\u2191derivative)^[k] f\nf : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (\u2191n * f) = \u2191n * (\u2191derivative)^[Nat.succ k] f\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np : R[X]\n\u22a2 \u2191derivative (comp p (1 - X)) = -comp (\u2191derivative p) (1 - X)\n[PROOFSTEP]\nsimp [derivative_comp]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np : R[X]\nk : \u2115\n\u22a2 (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X)\n[PROOFSTEP]\ninduction' k with k ih generalizing p\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d p : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (comp p (1 - X)) = (-1) ^ Nat.zero * comp ((\u2191derivative)^[Nat.zero] p) (1 - X)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d : R[X]\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X)\np : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (comp p (1 - X)) = (-1) ^ Nat.succ k * comp ((\u2191derivative)^[Nat.succ k] p) (1 - X)\n[PROOFSTEP]\nsimp [ih (derivative p), iterate_derivative_neg, derivative_comp, pow_succ]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\nS : Multiset R\nr : R\nhr : r \u2208 S\n\u22a2 eval r (\u2191derivative (Multiset.prod (Multiset.map (fun a => X - \u2191C a) S))) =\n    Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase S r))\n[PROOFSTEP]\nnth_rw 1 [\u2190 Multiset.cons_erase hr]\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\nS : Multiset R\nr : R\nhr : r \u2208 S\n\u22a2 eval r (\u2191derivative (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (r ::\u2098 Multiset.erase S r)))) =\n    Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase S r))\n[PROOFSTEP]\nhave := (evalRingHom r).map_multiset_prod (Multiset.map (fun a => X - C a) (S.erase r))\n[GOAL]\nR : Type u\nS\u271d : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\nS : Multiset R\nr : R\nhr : r \u2208 S\nthis :\n  \u2191(evalRingHom r) (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (Multiset.erase S r))) =\n    Multiset.prod (Multiset.map (\u2191(evalRingHom r)) (Multiset.map (fun a => X - \u2191C a) (Multiset.erase S r)))\n\u22a2 eval r (\u2191derivative (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (r ::\u2098 Multiset.erase S r)))) =\n    Multiset.prod (Multiset.map (fun a => r - a) (Multiset.erase S r))\n[PROOFSTEP]\nsimpa using this\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\nc : R\nm : \u2115\n\u22a2 \u2191derivative ((X - \u2191C c) ^ m) = \u2191C \u2191m * (X - \u2191C c) ^ (m - 1)\n[PROOFSTEP]\nrw [derivative_pow, derivative_X_sub_C, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\nc : R\n\u22a2 \u2191derivative ((X - \u2191C c) ^ 2) = \u2191C 2 * (X - \u2191C c)\n[PROOFSTEP]\nrw [derivative_sq, derivative_X_sub_C, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommRing R\nn k : \u2115\nc : R\n\u22a2 (\u2191derivative)^[k] ((X - \u2191C c) ^ n) = \u2191(\u220f i in range k, (n - i)) * (X - \u2191C c) ^ (n - k)\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 C_neg, iterate_derivative_X_add_pow]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Derivative", "llama_tokens": 47078, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473779969194, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.5743687860893159}}
{"text": "[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\n\u22a2 support (deriv f) \u2286 tsupport f\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nx : \ud835\udd5c\n\u22a2 x \u2208 support (deriv f) \u2192 x \u2208 tsupport f\n[PROOFSTEP]\nrw [\u2190 not_imp_not]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nx : \ud835\udd5c\n\u22a2 \u00acx \u2208 tsupport f \u2192 \u00acx \u2208 support (deriv f)\n[PROOFSTEP]\nintro h2x\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nx : \ud835\udd5c\nh2x : \u00acx \u2208 tsupport f\n\u22a2 \u00acx \u2208 support (deriv f)\n[PROOFSTEP]\nrw [not_mem_tsupport_iff_eventuallyEq] at h2x \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nx : \ud835\udd5c\nh2x : f =\u1da0[nhds x] 0\n\u22a2 \u00acx \u2208 support (deriv f)\n[PROOFSTEP]\nexact nmem_support.mpr (h2x.deriv_eq.trans (deriv_const x 0))\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Deriv.Support", "llama_tokens": 555, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388083214155, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.5741578148338145}}
{"text": "[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : DecidableEq l\ninst\u271d : CommRing R\n\u22a2 (J l R)\u1d40 = -J l R\n[PROOFSTEP]\nrw [J, fromBlocks_transpose, \u2190 neg_one_smul R (fromBlocks _ _ _ _ : Matrix (l \u2295 l) (l \u2295 l) R), fromBlocks_smul,\n  Matrix.transpose_zero, Matrix.transpose_one, transpose_neg]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : DecidableEq l\ninst\u271d : CommRing R\n\u22a2 fromBlocks 0 1 (-1\u1d40) 0 = fromBlocks (-1 \u2022 0) (-1 \u2022 -1) (-1 \u2022 1) (-1 \u2022 0)\n[PROOFSTEP]\nsimp [fromBlocks]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 J l R * J l R = -1\n[PROOFSTEP]\nrw [J, fromBlocks_multiply]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 fromBlocks (0 * 0 + -1 * 1) (0 * -1 + -1 * 0) (1 * 0 + 0 * 1) (1 * -1 + 0 * 0) = -1\n[PROOFSTEP]\nsimp only [Matrix.zero_mul, Matrix.neg_mul, zero_add, neg_zero, Matrix.one_mul, add_zero]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 fromBlocks (-1) 0 0 (-1) = -1\n[PROOFSTEP]\nrw [\u2190 neg_zero, \u2190 Matrix.fromBlocks_neg, \u2190 fromBlocks_one]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 (J l R)\u207b\u00b9 = -J l R\n[PROOFSTEP]\nrefine' Matrix.inv_eq_right_inv _\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 J l R * -J l R = 1\n[PROOFSTEP]\nrw [Matrix.mul_neg, J_squared]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 - -1 = 1\n[PROOFSTEP]\nexact neg_neg 1\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 det (J l R) * det (J l R) = 1\n[PROOFSTEP]\nrw [\u2190 det_mul, J_squared, \u2190 one_smul R (-1 : Matrix _ _ R), smul_neg, \u2190 neg_smul, det_smul, Fintype.card_sum, det_one,\n  mul_one]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 (-1) ^ (Fintype.card l + Fintype.card l) = 1\n[PROOFSTEP]\napply Even.neg_one_pow\n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 Even (Fintype.card l + Fintype.card l)\n[PROOFSTEP]\nexact even_add_self _\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\na b : Matrix (l \u2295 l) (l \u2295 l) R\nha : a \u2208 {A | A * J l R * A\u1d40 = J l R}\nhb : b \u2208 {A | A * J l R * A\u1d40 = J l R}\n\u22a2 a * b \u2208 {A | A * J l R * A\u1d40 = J l R}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, transpose_mul] at *\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\na b : Matrix (l \u2295 l) (l \u2295 l) R\nha : a * J l R * a\u1d40 = J l R\nhb : b * J l R * b\u1d40 = J l R\n\u22a2 a * b * J l R * (b\u1d40 * a\u1d40) = J l R\n[PROOFSTEP]\nrw [\u2190 Matrix.mul_assoc, a.mul_assoc, a.mul_assoc, hb]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\na b : Matrix (l \u2295 l) (l \u2295 l) R\nha : a * J l R * a\u1d40 = J l R\nhb : b * J l R * b\u1d40 = J l R\n\u22a2 a * J l R * a\u1d40 = J l R\n[PROOFSTEP]\nexact ha\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype l\n\u22a2 1 \u2208\n    { carrier := {A | A * J l R * A\u1d40 = J l R},\n        mul_mem' :=\n          (_ :\n            \u2200 {a b : Matrix (l \u2295 l) (l \u2295 l) R},\n              a \u2208 {A | A * J l R * A\u1d40 = J l R} \u2192\n                b \u2208 {A | A * J l R * A\u1d40 = J l R} \u2192 a * b \u2208 {A | A * J l R * A\u1d40 = J l R}) }.carrier\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\n\u22a2 A \u2208 symplecticGroup l R \u2194 A * J l R * A\u1d40 = J l R\n[PROOFSTEP]\nsimp [symplecticGroup]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\n\u22a2 J l R \u2208 symplecticGroup l R\n[PROOFSTEP]\nrw [mem_iff, J, fromBlocks_multiply, fromBlocks_transpose, fromBlocks_multiply]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\n\u22a2 fromBlocks ((0 * 0 + -1 * 1) * 0\u1d40 + (0 * -1 + -1 * 0) * (-1)\u1d40) ((0 * 0 + -1 * 1) * 1\u1d40 + (0 * -1 + -1 * 0) * 0\u1d40)\n      ((1 * 0 + 0 * 1) * 0\u1d40 + (1 * -1 + 0 * 0) * (-1)\u1d40) ((1 * 0 + 0 * 1) * 1\u1d40 + (1 * -1 + 0 * 0) * 0\u1d40) =\n    fromBlocks 0 (-1) 1 0\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nh : A \u2208 symplecticGroup l R\n\u22a2 -A \u2208 symplecticGroup l R\n[PROOFSTEP]\nrw [mem_iff] at h \u22a2\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nh : A * J l R * A\u1d40 = J l R\n\u22a2 -A * J l R * (-A)\u1d40 = J l R\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 IsUnit (det A)\n[PROOFSTEP]\nrw [isUnit_iff_exists_inv]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 \u2203 b, det A * b = 1\n[PROOFSTEP]\nuse A.det\n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 det A * det A = 1\n[PROOFSTEP]\nrefine' (isUnit_det_J l R).mul_left_cancel _\n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 det (J l R) * (det A * det A) = det (J l R) * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 det (J l R) * (det A * det A) = det (J l R)\n[PROOFSTEP]\nrw [mem_iff] at hA \n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\n\u22a2 det (J l R) * (det A * det A) = det (J l R)\n[PROOFSTEP]\napply_fun det at hA \n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : det (A * J l R * A\u1d40) = det (J l R)\n\u22a2 det (J l R) * (det A * det A) = det (J l R)\n[PROOFSTEP]\nsimp only [det_mul, det_transpose] at hA \n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : det A * det (J l R) * det A = det (J l R)\n\u22a2 det (J l R) * (det A * det A) = det (J l R)\n[PROOFSTEP]\nrw [mul_comm A.det, mul_assoc] at hA \n[GOAL]\ncase h\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : det (J l R) * (det A * det A) = det (J l R)\n\u22a2 det (J l R) * (det A * det A) = det (J l R)\n[PROOFSTEP]\nexact hA\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 A\u1d40 \u2208 symplecticGroup l R\n[PROOFSTEP]\nrw [mem_iff] at hA \u22a2\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\n\u22a2 A\u1d40 * J l R * A\u1d40\u1d40 = J l R\n[PROOFSTEP]\nrw [transpose_transpose]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\n\u22a2 A\u1d40 * J l R * A = J l R\n[PROOFSTEP]\nhave huA := symplectic_det hA\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\n\u22a2 A\u1d40 * J l R * A = J l R\n[PROOFSTEP]\nhave huAT : IsUnit A\u1d40.det := by\n  rw [Matrix.det_transpose]\n  exact huA\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\n\u22a2 IsUnit (det A\u1d40)\n[PROOFSTEP]\nrw [Matrix.det_transpose]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\n\u22a2 IsUnit (det A)\n[PROOFSTEP]\nexact huA\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 A\u1d40 * J l R * A = J l R\n[PROOFSTEP]\ncalc\n  A\u1d40 * J l R * A = (-A\u1d40) * (J l R)\u207b\u00b9 * A := by\n    rw [J_inv]\n    simp\n  _ = (-A\u1d40) * (A * J l R * A\u1d40)\u207b\u00b9 * A := by rw [hA]\n  _ = -(A\u1d40 * (A\u1d40\u207b\u00b9 * (J l R)\u207b\u00b9)) * A\u207b\u00b9 * A := by simp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]\n  _ = -(J l R)\u207b\u00b9 := by rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]\n  _ = J l R := by simp [J_inv]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 A\u1d40 * J l R * A = -A\u1d40 * (J l R)\u207b\u00b9 * A\n[PROOFSTEP]\nrw [J_inv]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 A\u1d40 * J l R * A = -A\u1d40 * -J l R * A\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 -A\u1d40 * (J l R)\u207b\u00b9 * A = -A\u1d40 * (A * J l R * A\u1d40)\u207b\u00b9 * A\n[PROOFSTEP]\nrw [hA]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 -A\u1d40 * (A * J l R * A\u1d40)\u207b\u00b9 * A = -(A\u1d40 * (A\u1d40\u207b\u00b9 * (J l R)\u207b\u00b9)) * A\u207b\u00b9 * A\n[PROOFSTEP]\nsimp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 -(A\u1d40 * (A\u1d40\u207b\u00b9 * (J l R)\u207b\u00b9)) * A\u207b\u00b9 * A = -(J l R)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A * J l R * A\u1d40 = J l R\nhuA : IsUnit (det A)\nhuAT : IsUnit (det A\u1d40)\n\u22a2 -(J l R)\u207b\u00b9 = J l R\n[PROOFSTEP]\nsimp [J_inv]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A\u1d40 \u2208 symplecticGroup l R\n\u22a2 A \u2208 symplecticGroup l R\n[PROOFSTEP]\nsimpa using transpose_mem hA\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\n\u22a2 A \u2208 symplecticGroup l R \u2194 A\u1d40 * J l R * A = J l R\n[PROOFSTEP]\nrw [\u2190 transpose_mem_iff, mem_iff, transpose_transpose]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 -(J l R * A\u1d40 * J l R * A) = -J l R * (A\u1d40 * J l R * A)\n[PROOFSTEP]\nsimp only [Matrix.mul_assoc, Matrix.neg_mul]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 -J l R * (A\u1d40 * J l R * A) = -J l R * J l R\n[PROOFSTEP]\nrw [mem_iff'] at hA \n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A\u1d40 * J l R * A = J l R\n\u22a2 -J l R * (A\u1d40 * J l R * A) = -J l R * J l R\n[PROOFSTEP]\nrw [hA]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 -J l R * J l R = -1 \u2022 (J l R * J l R)\n[PROOFSTEP]\nsimp only [Matrix.neg_mul, neg_smul, one_smul]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 -1 \u2022 (J l R * J l R) = -1 \u2022 -1\n[PROOFSTEP]\nrw [J_squared]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 -1 \u2022 -1 = 1\n[PROOFSTEP]\nsimp only [neg_smul_neg, one_smul]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA\u271d : Matrix (l \u2295 l) (l \u2295 l) R\nA : { x // x \u2208 symplecticGroup l R }\n\u22a2 \u2191A\u207b\u00b9 = (\u2191A)\u207b\u00b9\n[PROOFSTEP]\nrefine' (coe_inv A).trans (inv_eq_left_inv _).symm\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA\u271d : Matrix (l \u2295 l) (l \u2295 l) R\nA : { x // x \u2208 symplecticGroup l R }\n\u22a2 -J l R * (\u2191A)\u1d40 * J l R * \u2191A = 1\n[PROOFSTEP]\nsimp [inv_left_mul_aux, coe_inv]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA\u271d A : Matrix (l \u2295 l) (l \u2295 l) R\nhA : A \u2208 symplecticGroup l R\n\u22a2 -J l R * A\u1d40 * J l R * A = 1\n[PROOFSTEP]\nsimp only [Matrix.neg_mul, inv_left_mul_aux hA]\n[GOAL]\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA\u271d : Matrix (l \u2295 l) (l \u2295 l) R\nsrc\u271d\u00b9 : Inv { x // x \u2208 symplecticGroup l R } := hasInv\nsrc\u271d : Monoid { x // x \u2208 symplecticGroup l R } := Submonoid.toMonoid (symplecticGroup l R)\nA : { x // x \u2208 symplecticGroup l R }\n\u22a2 A\u207b\u00b9 * A = 1\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase a\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA\u271d : Matrix (l \u2295 l) (l \u2295 l) R\nsrc\u271d\u00b9 : Inv { x // x \u2208 symplecticGroup l R } := hasInv\nsrc\u271d : Monoid { x // x \u2208 symplecticGroup l R } := Submonoid.toMonoid (symplecticGroup l R)\nA : { x // x \u2208 symplecticGroup l R }\n\u22a2 \u2191(A\u207b\u00b9 * A) = \u21911\n[PROOFSTEP]\nsimp only [Submonoid.coe_one, Submonoid.coe_mul, Matrix.neg_mul, coe_inv]\n[GOAL]\ncase a\nl : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : DecidableEq l\ninst\u271d\u00b9 : Fintype l\ninst\u271d : CommRing R\nA\u271d : Matrix (l \u2295 l) (l \u2295 l) R\nsrc\u271d\u00b9 : Inv { x // x \u2208 symplecticGroup l R } := hasInv\nsrc\u271d : Monoid { x // x \u2208 symplecticGroup l R } := Submonoid.toMonoid (symplecticGroup l R)\nA : { x // x \u2208 symplecticGroup l R }\n\u22a2 -(J l R * (\u2191A)\u1d40 * J l R * \u2191A) = 1\n[PROOFSTEP]\nexact inv_left_mul_aux A.2\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.SymplecticGroup", "llama_tokens": 7672, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430353105598, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.5729115625916731}}
{"text": "[GOAL]\nx : \u2102\nh : cos x \u2260 0\n\u22a2 HasStrictDerivAt tan (\u21911 / cos x ^ 2) x\n[PROOFSTEP]\nconvert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1\n[GOAL]\ncase h.e'_7\nx : \u2102\nh : cos x \u2260 0\n\u22a2 \u21911 / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2\n[PROOFSTEP]\nrw_mod_cast [\u2190 sin_sq_add_cos_sq x]\n[GOAL]\ncase h.e'_7\nx : \u2102\nh : \u00accos x = 0\n\u22a2 (sin x ^ 2 + cos x ^ 2) / cos x ^ 2 = (cos x * cos x - sin x * -sin x) / cos x ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nx : \u2102\nhx : cos x = 0\n\u22a2 Tendsto (fun x => \u2191abs (tan x)) (\ud835\udcdd[{x}\u1d9c] x) atTop\n[PROOFSTEP]\nsimp only [tan_eq_sin_div_cos, \u2190 norm_eq_abs, norm_div]\n[GOAL]\nx : \u2102\nhx : cos x = 0\n\u22a2 Tendsto (fun x => \u2016sin x\u2016 / \u2016cos x\u2016) (\ud835\udcdd[{x}\u1d9c] x) atTop\n[PROOFSTEP]\nhave A : sin x \u2260 0 := fun h => by simpa [*, sq] using sin_sq_add_cos_sq x\n[GOAL]\nx : \u2102\nhx : cos x = 0\nh : sin x = 0\n\u22a2 False\n[PROOFSTEP]\nsimpa [*, sq] using sin_sq_add_cos_sq x\n[GOAL]\nx : \u2102\nhx : cos x = 0\nA : sin x \u2260 0\n\u22a2 Tendsto (fun x => \u2016sin x\u2016 / \u2016cos x\u2016) (\ud835\udcdd[{x}\u1d9c] x) atTop\n[PROOFSTEP]\nhave B : Tendsto cos (\ud835\udcdd[\u2260] x) (\ud835\udcdd[\u2260] 0) := hx \u25b8 (hasDerivAt_cos x).tendsto_punctured_nhds (neg_ne_zero.2 A)\n[GOAL]\nx : \u2102\nhx : cos x = 0\nA : sin x \u2260 0\nB : Tendsto cos (\ud835\udcdd[{x}\u1d9c] x) (\ud835\udcdd[{0}\u1d9c] 0)\n\u22a2 Tendsto (fun x => \u2016sin x\u2016 / \u2016cos x\u2016) (\ud835\udcdd[{x}\u1d9c] x) atTop\n[PROOFSTEP]\nexact\n  continuous_sin.continuousWithinAt.norm.mul_atTop (norm_pos_iff.2 A)\n    (tendsto_norm_nhdsWithin_zero.comp B).inv_tendsto_zero\n[GOAL]\nx : \u2102\n\u22a2 ContinuousAt tan x \u2194 cos x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8fun hc h\u2080 => _, fun h => (hasDerivAt_tan h).continuousAt\u27e9\n[GOAL]\nx : \u2102\nhc : ContinuousAt tan x\nh\u2080 : cos x = 0\n\u22a2 False\n[PROOFSTEP]\nexact not_tendsto_nhds_of_tendsto_atTop (tendsto_abs_tan_of_cos_eq_zero h\u2080) _ (hc.norm.tendsto.mono_left inf_le_left)\n[GOAL]\nx : \u2102\nh : cos x = 0\n\u22a2 deriv tan x = \u21911 / cos x ^ 2\n[PROOFSTEP]\nhave : \u00acDifferentiableAt \u2102 tan x := mt differentiableAt_tan.1 (Classical.not_not.2 h)\n[GOAL]\nx : \u2102\nh : cos x = 0\nthis : \u00acDifferentiableAt \u2102 tan x\n\u22a2 deriv tan x = \u21911 / cos x ^ 2\n[PROOFSTEP]\nsimp [deriv_zero_of_not_differentiableAt this, h, sq]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv", "llama_tokens": 1063, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324713956856, "lm_q2_score": 0.7025300698514778, "lm_q1q2_score": 0.5727253250748039}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c : \u03b1\n\u22a2 (a - b) / c = a / c - b / c\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, tsub_mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Field.Canonical.Basic", "llama_tokens": 96, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.5724619218054614}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns\u271d t s : Set H\nc : \u211d\nhc : 0 < c\ny : H\nhy : y \u2208 {y | \u2200 (x : H), x \u2208 s \u2192 0 \u2264 inner x y}\nx : H\nhx : x \u2208 s\n\u22a2 0 \u2264 inner x (c \u2022 y)\n[PROOFSTEP]\nrw [real_inner_smul_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns\u271d t s : Set H\nc : \u211d\nhc : 0 < c\ny : H\nhy : y \u2208 {y | \u2200 (x : H), x \u2208 s \u2192 0 \u2264 inner x y}\nx : H\nhx : x \u2208 s\n\u22a2 0 \u2264 c * inner x y\n[PROOFSTEP]\nexact mul_nonneg hc.le (hy x hx)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns\u271d t s : Set H\nu : H\nhu : u \u2208 {y | \u2200 (x : H), x \u2208 s \u2192 0 \u2264 inner x y}\nv : H\nhv : v \u2208 {y | \u2200 (x : H), x \u2208 s \u2192 0 \u2264 inner x y}\nx : H\nhx : x \u2208 s\n\u22a2 0 \u2264 inner x (u + v)\n[PROOFSTEP]\nrw [inner_add_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns\u271d t s : Set H\nu : H\nhu : u \u2208 {y | \u2200 (x : H), x \u2208 s \u2192 0 \u2264 inner x y}\nv : H\nhv : v \u2208 {y | \u2200 (x : H), x \u2208 s \u2192 0 \u2264 inner x y}\nx : H\nhx : x \u2208 s\n\u22a2 0 \u2264 inner x u + inner x v\n[PROOFSTEP]\nexact add_nonneg (hu x hx) (hv x hx)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 innerDualCone univ = 0\n[PROOFSTEP]\nsuffices \u2200 x : H, x \u2208 (univ : Set H).innerDualCone \u2192 x = 0\n  by\n  apply SetLike.coe_injective\n  exact eq_singleton_iff_unique_mem.mpr \u27e8fun x _ => (inner_zero_right _).ge, this\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nthis : \u2200 (x : H), x \u2208 innerDualCone univ \u2192 x = 0\n\u22a2 innerDualCone univ = 0\n[PROOFSTEP]\napply SetLike.coe_injective\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nthis : \u2200 (x : H), x \u2208 innerDualCone univ \u2192 x = 0\n\u22a2 \u2191(innerDualCone univ) = \u21910\n[PROOFSTEP]\nexact eq_singleton_iff_unique_mem.mpr \u27e8fun x _ => (inner_zero_right _).ge, this\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2200 (x : H), x \u2208 innerDualCone univ \u2192 x = 0\n[PROOFSTEP]\nexact fun x hx => by simpa [\u2190 real_inner_self_nonpos] using hx (-x) (mem_univ _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : H\nhx : x \u2208 innerDualCone univ\n\u22a2 x = 0\n[PROOFSTEP]\nsimpa [\u2190 real_inner_self_nonpos] using hx (-x) (mem_univ _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : H\nx\u271d : x \u2208 s\n\u22a2 0 \u2264 inner x 0\n[PROOFSTEP]\nrw [inner_zero_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns\u271d t : Set H\nx : H\ns : Set H\n\u22a2 innerDualCone (insert x s) = innerDualCone {x} \u2293 innerDualCone s\n[PROOFSTEP]\nrw [insert_eq, innerDualCone_union]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u03b9 : Sort u_6\nf : \u03b9 \u2192 Set H\n\u22a2 innerDualCone (\u22c3 (i : \u03b9), f i) = \u2a05 (i : \u03b9), innerDualCone (f i)\n[PROOFSTEP]\nrefine' le_antisymm (le_iInf fun i x hx y hy => hx _ <| mem_iUnion_of_mem _ hy) _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u03b9 : Sort u_6\nf : \u03b9 \u2192 Set H\n\u22a2 \u2a05 (i : \u03b9), innerDualCone (f i) \u2264 innerDualCone (\u22c3 (i : \u03b9), f i)\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u03b9 : Sort u_6\nf : \u03b9 \u2192 Set H\nx : H\nhx : x \u2208 \u2a05 (i : \u03b9), innerDualCone (f i)\ny : H\nhy : y \u2208 \u22c3 (i : \u03b9), f i\n\u22a2 0 \u2264 inner y x\n[PROOFSTEP]\nrw [ConvexCone.mem_iInf] at hx \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u03b9 : Sort u_6\nf : \u03b9 \u2192 Set H\nx : H\nhx : \u2200 (i : \u03b9), x \u2208 innerDualCone (f i)\ny : H\nhy : y \u2208 \u22c3 (i : \u03b9), f i\n\u22a2 0 \u2264 inner y x\n[PROOFSTEP]\nobtain \u27e8j, hj\u27e9 := mem_iUnion.mp hy\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u03b9 : Sort u_6\nf : \u03b9 \u2192 Set H\nx : H\nhx : \u2200 (i : \u03b9), x \u2208 innerDualCone (f i)\ny : H\nhy : y \u2208 \u22c3 (i : \u03b9), f i\nj : \u03b9\nhj : y \u2208 f j\n\u22a2 0 \u2264 inner y x\n[PROOFSTEP]\nexact hx _ _ hj\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nS : Set (Set H)\n\u22a2 innerDualCone (\u22c3\u2080 S) = sInf (innerDualCone '' S)\n[PROOFSTEP]\nsimp_rw [sInf_image, sUnion_eq_biUnion, innerDualCone_iUnion]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2191(innerDualCone s) = \u22c2 (i : \u2191s), \u2191(innerDualCone {\u2191i})\n[PROOFSTEP]\nrw [\u2190 ConvexCone.coe_iInf, \u2190 innerDualCone_iUnion, iUnion_of_singleton_coe]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 IsClosed \u2191(innerDualCone s)\n[PROOFSTEP]\nrw [innerDualCone_eq_iInter_innerDualCone_singleton]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 IsClosed (\u22c2 (i : \u2191s), \u2191(innerDualCone {\u2191i}))\n[PROOFSTEP]\napply isClosed_iInter\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\n\u22a2 \u2200 (i : \u2191s), IsClosed \u2191(innerDualCone {\u2191i})\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\n\u22a2 IsClosed \u2191(innerDualCone {\u2191x})\n[PROOFSTEP]\nhave h : ({\u2191x} : Set H).innerDualCone = (inner x : H \u2192 \u211d) \u207b\u00b9' Set.Ici 0 := by\n  rw [innerDualCone_singleton, ConvexCone.coe_comap, ConvexCone.coe_positive, inner\u209b\u2097_apply_coe]\n    -- the preimage is closed as `inner x` is continuous and `[0, \u221e)` is closed\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\n\u22a2 \u2191(innerDualCone {\u2191x}) = inner \u2191x \u207b\u00b9' Ici 0\n[PROOFSTEP]\nrw [innerDualCone_singleton, ConvexCone.coe_comap, ConvexCone.coe_positive, inner\u209b\u2097_apply_coe]\n  -- the preimage is closed as `inner x` is continuous and `[0, \u221e)` is closed\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\nh : \u2191(innerDualCone {\u2191x}) = inner \u2191x \u207b\u00b9' Ici 0\n\u22a2 IsClosed \u2191(innerDualCone {\u2191x})\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nx : \u2191s\nh : \u2191(innerDualCone {\u2191x}) = inner \u2191x \u207b\u00b9' Ici 0\n\u22a2 IsClosed (inner \u2191x \u207b\u00b9' Ici 0)\n[PROOFSTEP]\nexact isClosed_Ici.preimage (continuous_const.inner continuous_id')\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\n\u22a2 Pointed K\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := ne\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\n\u22a2 Pointed K\n[PROOFSTEP]\nlet f : \u211d \u2192 H :=\n  (\u00b7 \u2022 x)\n    -- f (0, \u221e) is a subset of K\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\n\u22a2 Pointed K\n[PROOFSTEP]\nhave fI : f '' Set.Ioi 0 \u2286 (K : Set H) := by\n  rintro _ \u27e8_, h, rfl\u27e9\n  exact K.smul_mem (Set.mem_Ioi.1 h) hx\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\n\u22a2 f '' Ioi 0 \u2286 \u2191K\n[PROOFSTEP]\nrintro _ \u27e8_, h, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nw\u271d : \u211d\nh : w\u271d \u2208 Ioi 0\n\u22a2 f w\u271d \u2208 \u2191K\n[PROOFSTEP]\nexact K.smul_mem (Set.mem_Ioi.1 h) hx\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\n\u22a2 Pointed K\n[PROOFSTEP]\nhave clf : closure (f '' Set.Ioi 0) \u2286 (K : Set H) := hc.closure_subset_iff.2 fI\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\n\u22a2 Pointed K\n[PROOFSTEP]\nhave fc : ContinuousWithinAt f (Set.Ioi (0 : \u211d)) 0 := (continuous_id.smul continuous_const).continuousWithinAt\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\n\u22a2 Pointed K\n[PROOFSTEP]\nhave mem\u2080 :=\n  fc.mem_closure_image\n    (by rw [closure_Ioi (0 : \u211d), mem_Ici])\n      -- as 0 \u2208 closure f (0, \u221e) and closure f (0, \u221e) \u2286 K, 0 \u2208 K.\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\n\u22a2 0 \u2208 closure (Ioi 0)\n[PROOFSTEP]\nrw [closure_Ioi (0 : \u211d), mem_Ici]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\nmem\u2080 : f 0 \u2208 closure (f '' Ioi 0)\n\u22a2 Pointed K\n[PROOFSTEP]\nhave f\u2080 : f 0 = 0 := zero_smul \u211d x\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : InnerProductSpace \u211d H\ns t : Set H\nK : ConvexCone \u211d H\nhc : IsClosed \u2191K\nx : H\nhx : x \u2208 \u2191K\nf : \u211d \u2192 H := fun x_1 => x_1 \u2022 x\nfI : f '' Ioi 0 \u2286 \u2191K\nclf : closure (f '' Ioi 0) \u2286 \u2191K\nfc : ContinuousWithinAt f (Ioi 0) 0\nmem\u2080 : f 0 \u2208 closure (f '' Ioi 0)\nf\u2080 : f 0 = 0\n\u22a2 Pointed K\n[PROOFSTEP]\nsimpa only [f\u2080, ConvexCone.Pointed, \u2190 SetLike.mem_coe] using mem_of_subset_of_mem clf mem\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0\n[PROOFSTEP]\nobtain \u27e8z, hzK, infi\u27e9 := exists_norm_eq_iInf_of_complete_convex ne hc.isComplete K.convex b\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0\n[PROOFSTEP]\nhave hinner := (norm_eq_iInf_iff_real_inner_le_zero K.convex hzK).1 infi\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0\n[PROOFSTEP]\nuse z - b\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)) \u2227 inner (z - b) b < 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)\n[PROOFSTEP]\nrintro x hxK\n[GOAL]\ncase h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 inner x (z - b)\n[PROOFSTEP]\nspecialize hinner _ (K.add_mem hxK hzK)\n[GOAL]\ncase h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : inner (b - z) (x + z - z) \u2264 0\n\u22a2 0 \u2264 inner x (z - b)\n[PROOFSTEP]\nrwa [add_sub_cancel, real_inner_comm, \u2190 neg_nonneg, neg_eq_neg_one_mul, \u2190 real_inner_smul_right, neg_smul, one_smul,\n  neg_sub] at hinner \n[GOAL]\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 inner (z - b) b < 0\n[PROOFSTEP]\nhave hinner\u2080 :=\n  hinner 0\n    (K.pointed_of_nonempty_of_isClosed ne hc)\n      -- the rest of the proof is a straightforward calculation\n[GOAL]\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : inner (b - z) (0 - z) \u2264 0\n\u22a2 inner (z - b) b < 0\n[PROOFSTEP]\nrw [zero_sub, inner_neg_right, Right.neg_nonpos_iff] at hinner\u2080 \n[GOAL]\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 inner (z - b) b < 0\n[PROOFSTEP]\nhave hbz : b - z \u2260 0 := by\n  rw [sub_ne_zero]\n  contrapose! hzK\n  rwa [\u2190 hzK]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b - z \u2260 0\n[PROOFSTEP]\nrw [sub_ne_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b \u2260 z\n[PROOFSTEP]\ncontrapose! hzK\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhzK : b = z\n\u22a2 \u00acz \u2208 \u2191K\n[PROOFSTEP]\nrwa [\u2190 hzK]\n[GOAL]\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (z - b) b < 0\n[PROOFSTEP]\nrw [\u2190 neg_zero, lt_neg, \u2190 neg_one_mul, \u2190 real_inner_smul_left, smul_sub, neg_smul, one_smul, neg_smul, neg_sub_neg,\n  one_smul]\n[GOAL]\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 0 < inner (b - z) b\n[PROOFSTEP]\ncalc\n  0 < \u27eab - z, b - z\u27eb_\u211d := lt_of_not_le ((Iff.not real_inner_self_nonpos).2 hbz)\n  _ = \u27eab - z, b - z\u27eb_\u211d + 0 := (add_zero _).symm\n  _ \u2264 \u27eab - z, b - z\u27eb_\u211d + \u27eab - z, z\u27eb_\u211d := (add_le_add rfl.ge hinner\u2080)\n  _ = \u27eab - z, b - z + z\u27eb_\u211d := (inner_add_right _ _ _).symm\n  _ = \u27eab - z, b\u27eb_\u211d := by rw [sub_add_cancel]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (b - z) (b - z + z) = inner (b - z) b\n[PROOFSTEP]\nrw [sub_add_cancel]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\n\u22a2 innerDualCone \u2191(innerDualCone \u2191K) = K\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 innerDualCone \u2191(innerDualCone \u2191K) \u2194 x \u2208 K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 innerDualCone \u2191(innerDualCone \u2191K) \u2192 x \u2208 K\n[PROOFSTEP]\nrw [mem_innerDualCone, \u2190 SetLike.mem_coe]\n[GOAL]\ncase h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\n\u22a2 (\u2200 (x_1 : H), x_1 \u2208 \u2191(innerDualCone \u2191K) \u2192 0 \u2264 inner x_1 x) \u2192 x \u2208 \u2191K\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\n\u22a2 \u00acx \u2208 \u2191K \u2192 \u2203 x_1, x_1 \u2208 \u2191(innerDualCone \u2191K) \u2227 inner x_1 x < 0\n[PROOFSTEP]\nexact K.hyperplane_separation_of_nonempty_of_isClosed_of_nmem ne hc\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\n\u22a2 x \u2208 K \u2192 x \u2208 innerDualCone \u2191(innerDualCone \u2191K)\n[PROOFSTEP]\nrintro hxK y h\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\nhxK : x \u2208 K\ny : H\nh : y \u2208 \u2191(innerDualCone \u2191K)\n\u22a2 0 \u2264 inner y x\n[PROOFSTEP]\nspecialize h x hxK\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nx : H\nhxK : x \u2208 K\ny : H\nh : 0 \u2264 inner x y\n\u22a2 0 \u2264 inner y x\n[PROOFSTEP]\nrwa [real_inner_comm]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Cone.Dual", "llama_tokens": 11741, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8740772384450967, "lm_q2_score": 0.6548947425132314, "lm_q1q2_score": 0.572428588008178}}
{"text": "[GOAL]\na b : \u2124\n\u22a2 a = b \u2194 \u2191a = \u2191b\n[PROOFSTEP]\nsimp only [Int.cast_inj]\n[GOAL]\na b : \u2124\n\u22a2 a \u2260 b \u2194 \u2191a \u2260 \u2191b\n[PROOFSTEP]\nsimp only [ne_eq, Int.cast_inj]\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Qify", "llama_tokens": 92, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.793105951184112, "lm_q2_score": 0.7217432182679956, "lm_q1q2_score": 0.5724188416351208}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (i j : \u2115), \u2203 k, I ^ k \u2022 \u22a4 \u2264 I ^ i \u2022 \u22a4 \u2293 I ^ j \u2022 \u22a4\n[PROOFSTEP]\nsuffices \u2200 i j : \u2115, \u2203 k, I ^ k \u2264 I ^ i \u2227 I ^ k \u2264 I ^ j by\n  simpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nthis : \u2200 (i j : \u2115), \u2203 k, I ^ k \u2264 I ^ i \u2227 I ^ k \u2264 I ^ j\n\u22a2 \u2200 (i j : \u2115), \u2203 k, I ^ k \u2022 \u22a4 \u2264 I ^ i \u2022 \u22a4 \u2293 I ^ j \u2022 \u22a4\n[PROOFSTEP]\nsimpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (i j : \u2115), \u2203 k, I ^ k \u2264 I ^ i \u2227 I ^ k \u2264 I ^ j\n[PROOFSTEP]\nintro i j\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\ni j : \u2115\n\u22a2 \u2203 k, I ^ k \u2264 I ^ i \u2227 I ^ k \u2264 I ^ j\n[PROOFSTEP]\nexact \u27e8max i j, pow_le_pow (le_max_left i j), pow_le_pow (le_max_right i j)\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (a : R) (i : \u2115), \u2203 j, a \u2022 I ^ j \u2022 \u22a4 \u2264 I ^ i \u2022 \u22a4\n[PROOFSTEP]\nsuffices \u2200 (a : R) (i : \u2115), \u2203 j : \u2115, a \u2022 I ^ j \u2264 I ^ i by\n  simpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nthis : \u2200 (a : R) (i : \u2115), \u2203 j, a \u2022 I ^ j \u2264 I ^ i\n\u22a2 \u2200 (a : R) (i : \u2115), \u2203 j, a \u2022 I ^ j \u2022 \u22a4 \u2264 I ^ i \u2022 \u22a4\n[PROOFSTEP]\nsimpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (a : R) (i : \u2115), \u2203 j, a \u2022 I ^ j \u2264 I ^ i\n[PROOFSTEP]\nintro r n\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nr : R\nn : \u2115\n\u22a2 \u2203 j, r \u2022 I ^ j \u2264 I ^ n\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nr : R\nn : \u2115\n\u22a2 r \u2022 I ^ n \u2264 I ^ n\n[PROOFSTEP]\nrintro a \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase h.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nr : R\nn : \u2115\nx : R\nhx : x \u2208 \u2191(I ^ n)\n\u22a2 \u2191(DistribMulAction.toLinearMap R R r) x \u2208 I ^ n\n[PROOFSTEP]\nexact (I ^ n).smul_mem r hx\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (i : \u2115), \u2203 j, \u2191(I ^ j \u2022 \u22a4) * \u2191(I ^ j \u2022 \u22a4) \u2286 \u2191(I ^ i \u2022 \u22a4)\n[PROOFSTEP]\nsuffices \u2200 i : \u2115, \u2203 j : \u2115, (I ^ j : Set R) * (I ^ j : Set R) \u2286 (I ^ i : Set R) by\n  simpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nthis : \u2200 (i : \u2115), \u2203 j, \u2191(I ^ j) * \u2191(I ^ j) \u2286 \u2191(I ^ i)\n\u22a2 \u2200 (i : \u2115), \u2203 j, \u2191(I ^ j \u2022 \u22a4) * \u2191(I ^ j \u2022 \u22a4) \u2286 \u2191(I ^ i \u2022 \u22a4)\n[PROOFSTEP]\nsimpa only [smul_top_eq_map, Algebra.id.map_eq_id, map_id] using this\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (i : \u2115), \u2203 j, \u2191(I ^ j) * \u2191(I ^ j) \u2286 \u2191(I ^ i)\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nn : \u2115\n\u22a2 \u2203 j, \u2191(I ^ j) * \u2191(I ^ j) \u2286 \u2191(I ^ n)\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nn : \u2115\n\u22a2 \u2191(I ^ n) * \u2191(I ^ n) \u2286 \u2191(I ^ n)\n[PROOFSTEP]\nrintro a \u27e8x, b, _hx, hb, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nn : \u2115\nx b : R\n_hx : x \u2208 \u2191(I ^ n)\nhb : b \u2208 \u2191(I ^ n)\n\u22a2 (fun x x_1 => x * x_1) x b \u2208 \u2191(I ^ n)\n[PROOFSTEP]\nexact (I ^ n).smul_mem x hb\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\n\u22a2 \u2200 (t : Set R), t \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(I ^ i) \u2286 t\n[PROOFSTEP]\nintro U\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\n\u22a2 U \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(I ^ i) \u2286 U\n[PROOFSTEP]\nrw [I.ringFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\n\u22a2 (\u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 U) \u2194 \u2203 i, True \u2227 \u2191(I ^ i) \u2286 U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\n\u22a2 (\u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 U) \u2192 \u2203 i, True \u2227 \u2191(I ^ i) \u2286 U\n[PROOFSTEP]\nrintro \u27e8-, \u27e8i, rfl\u27e9, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\ni : \u2115\nh : id \u2191((fun i => toAddSubgroup (I ^ i \u2022 \u22a4)) i) \u2286 U\n\u22a2 \u2203 i, True \u2227 \u2191(I ^ i) \u2286 U\n[PROOFSTEP]\nreplace h : \u2191(I ^ i) \u2286 U := by simpa using h\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\ni : \u2115\nh : id \u2191((fun i => toAddSubgroup (I ^ i \u2022 \u22a4)) i) \u2286 U\n\u22a2 \u2191(I ^ i) \u2286 U\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mp.intro.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\ni : \u2115\nh : \u2191(I ^ i) \u2286 U\n\u22a2 \u2203 i, True \u2227 \u2191(I ^ i) \u2286 U\n[PROOFSTEP]\nexact \u27e8i, trivial, h\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\n\u22a2 (\u2203 i, True \u2227 \u2191(I ^ i) \u2286 U) \u2192 \u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 U\n[PROOFSTEP]\nrintro \u27e8i, -, h\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\ni : \u2115\nh : \u2191(I ^ i) \u2286 U\n\u22a2 \u2203 i, i \u2208 RingFilterBasis.toAddGroupFilterBasis \u2227 id i \u2286 U\n[PROOFSTEP]\nexact \u27e8(I ^ i : Ideal R), \u27e8i, by simp\u27e9, h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nU : Set R\ni : \u2115\nh : \u2191(I ^ i) \u2286 U\n\u22a2 \u2191(I ^ i) = \u2191((fun i => toAddSubgroup (I ^ i \u2022 \u22a4)) i)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)\n[PROOFSTEP]\nletI := I.adicTopology\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\nthis : TopologicalSpace R := adicTopology I\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)\n[PROOFSTEP]\nhave := I.hasBasis_nhds_zero_adic.map fun y => x + y\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\nthis\u271d : TopologicalSpace R := adicTopology I\nthis : HasBasis (Filter.map (fun y => x + y) (\ud835\udcdd 0)) (fun _n => True) fun i => (fun y => x + y) '' \u2191(I ^ i)\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)\n[PROOFSTEP]\nrwa [map_add_left_nhds_zero x] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\ni : \u2115\n\u22a2 \u2191(I ^ i \u2022 \u22a4) = \u2191((fun i => toAddSubgroup (I ^ i \u2022 \u22a4)) i)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\ni : \u2115\na : R\na_in : a \u2208 \u2191(I ^ i \u2022 \u22a4)\n\u22a2 a \u2208 (fun x => x \u2022 m) \u207b\u00b9' \u2191(I ^ i \u2022 \u22a4)\n[PROOFSTEP]\nreplace a_in : a \u2208 I ^ i := by simpa [(I ^ i).mul_top] using a_in\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\ni : \u2115\na : R\na_in : a \u2208 \u2191(I ^ i \u2022 \u22a4)\n\u22a2 a \u2208 I ^ i\n[PROOFSTEP]\nsimpa [(I ^ i).mul_top] using a_in\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nm : M\ni : \u2115\na : R\na_in : a \u2208 I ^ i\n\u22a2 a \u2208 (fun x => x \u2022 m) \u207b\u00b9' \u2191(I ^ i \u2022 \u22a4)\n[PROOFSTEP]\nexact smul_mem_smul a_in mem_top\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nn : \u2115\n\u22a2 OpenAddSubgroup R\n[PROOFSTEP]\nletI := I.adicTopology\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nn : \u2115\nthis : TopologicalSpace R := adicTopology I\n\u22a2 OpenAddSubgroup R\n[PROOFSTEP]\nrefine \u27e8(I ^ n).toAddSubgroup, ?_\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nn : \u2115\nthis : TopologicalSpace R := adicTopology I\n\u22a2 IsOpen (toAddSubgroup (I ^ n)).toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nconvert (I.adic_basis.toRing_subgroups_basis.openAddSubgroup n).isOpen\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nn : \u2115\nthis : TopologicalSpace R := adicTopology I\n\u22a2 (toAddSubgroup (I ^ n)).toAddSubmonoid.toAddSubsemigroup.carrier =\n    \u2191(RingSubgroupsBasis.openAddSubgroup (_ : RingSubgroupsBasis fun i => toAddSubgroup (I ^ i \u2022 \u22a4)) n)\n[PROOFSTEP]\nchange (I ^ n : Set R) = (I ^ n \u2022 (\u22a4 : Ideal R) : Set R)\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nI : Ideal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nn : \u2115\nthis : TopologicalSpace R := adicTopology I\n\u22a2 \u2191(I ^ n) = \u2191(I ^ n \u2022 \u22a4)\n[PROOFSTEP]\nsimp [smul_top_eq_map, Algebra.id.map_eq_id, map_id, restrictScalars_self]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 IsAdic J \u2194 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 IsAdic J \u2192 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : IsAdic J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nchange _ = _ at H \n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nrw [H]\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nletI := J.adicTopology\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase mp.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\nn : \u2115\n\u22a2 IsOpen \u2191(J ^ n)\n[PROOFSTEP]\nexact (J.openAddSubgroup n).isOpen'\n[GOAL]\ncase mp.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase mp.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\ns : Set R\nhs : s \u2208 \ud835\udcdd 0\n\u22a2 \u2203 n, \u2191(J ^ n) \u2286 s\n[PROOFSTEP]\nsimpa using J.hasBasis_nhds_zero_adic.mem_iff.mp hs\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 ((\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s) \u2192 IsAdic J\n[PROOFSTEP]\nrintro \u27e8H\u2081, H\u2082\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 IsAdic J\n[PROOFSTEP]\napply TopologicalAddGroup.ext\n[GOAL]\ncase mpr.intro.tg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 TopologicalAddGroup R\n[PROOFSTEP]\napply @TopologicalRing.to_topologicalAddGroup\n[GOAL]\ncase mpr.intro.tg'\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 TopologicalAddGroup R\n[PROOFSTEP]\napply (RingSubgroupsBasis.toRingFilterBasis _).toAddGroupFilterBasis.isTopologicalAddGroup\n[GOAL]\ncase mpr.intro.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 \ud835\udcdd 0 = \ud835\udcdd 0\n[PROOFSTEP]\next s\n[GOAL]\ncase mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 s \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nletI := Ideal.adic_basis J\n[GOAL]\ncase mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 s \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrw [J.hasBasis_nhds_zero_adic.mem_iff]\n[GOAL]\ncase mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.h.a.mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2192 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr.intro.h.a.mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 (\u2203 i, True \u2227 \u2191(J ^ i) \u2286 s) \u2192 s \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr.intro.h.a.mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : s \u2208 \ud835\udcdd 0\n\u22a2 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n[PROOFSTEP]\nrcases H\u2082 s H with \u27e8n, h\u27e9\n[GOAL]\ncase mpr.intro.h.a.mp.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : s \u2208 \ud835\udcdd 0\nn : \u2115\nh : \u2191(J ^ n) \u2286 s\n\u22a2 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n[PROOFSTEP]\nexact \u27e8n, trivial, h\u27e9\n[GOAL]\ncase mpr.intro.h.a.mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n\u22a2 s \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrcases H with \u27e8n, -, hn\u27e9\n[GOAL]\ncase mpr.intro.h.a.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nn : \u2115\nhn : \u2191(J ^ n) \u2286 s\n\u22a2 s \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrw [mem_nhds_iff]\n[GOAL]\ncase mpr.intro.h.a.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nn : \u2115\nhn : \u2191(J ^ n) \u2286 s\n\u22a2 \u2203 t, t \u2286 s \u2227 IsOpen t \u2227 0 \u2208 t\n[PROOFSTEP]\nrefine' \u27e8_, hn, H\u2081 n, (J ^ n).zero_mem\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : IsAdic J\nn : \u2115\nhn : 0 < n\n\u22a2 IsAdic (J ^ n)\n[PROOFSTEP]\nrw [isAdic_iff] at h \u22a2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\n\u22a2 (\u2200 (n_1 : \u2115), IsOpen \u2191((J ^ n) ^ n_1)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n_1, \u2191((J ^ n) ^ n_1) \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\n\u22a2 \u2200 (n_1 : \u2115), IsOpen \u2191((J ^ n) ^ n_1)\n[PROOFSTEP]\nintro m\n[GOAL]\ncase left\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\nm : \u2115\n\u22a2 IsOpen \u2191((J ^ n) ^ m)\n[PROOFSTEP]\nrw [\u2190 pow_mul]\n[GOAL]\ncase left\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\nm : \u2115\n\u22a2 IsOpen \u2191(J ^ (n * m))\n[PROOFSTEP]\napply h.left\n[GOAL]\ncase right\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\n\u22a2 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n_1, \u2191((J ^ n) ^ n_1) \u2286 s\n[PROOFSTEP]\nintro V hV\n[GOAL]\ncase right\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\n\u22a2 \u2203 n_1, \u2191((J ^ n) ^ n_1) \u2286 V\n[PROOFSTEP]\ncases' h.right V hV with m hm\n[GOAL]\ncase right.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\n\u22a2 \u2203 n_1, \u2191((J ^ n) ^ n_1) \u2286 V\n[PROOFSTEP]\nuse m\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\n\u22a2 \u2191((J ^ n) ^ m) \u2286 V\n[PROOFSTEP]\nrefine' Set.Subset.trans _ hm\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nn : \u2115\nhn : 0 < n\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\n\u22a2 \u2191((J ^ n) ^ m) \u2286 \u2191(J ^ m)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase h.zero\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\nhn : 0 < Nat.zero\n\u22a2 \u2191((J ^ Nat.zero) ^ m) \u2286 \u2191(J ^ m)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h.zero.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\nhn : 0 < Nat.zero\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_succ_le_zero 0 hn\n[GOAL]\ncase h.succ\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\nn\u271d : \u2115\nhn : 0 < Nat.succ n\u271d\n\u22a2 \u2191((J ^ Nat.succ n\u271d) ^ m) \u2286 \u2191(J ^ m)\n[PROOFSTEP]\nrw [\u2190 pow_mul, Nat.succ_mul]\n[GOAL]\ncase h.succ\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\nn\u271d : \u2115\nhn : 0 < Nat.succ n\u271d\n\u22a2 \u2191(J ^ (n\u271d * m + m)) \u2286 \u2191(J ^ m)\n[PROOFSTEP]\napply Ideal.pow_le_pow\n[GOAL]\ncase h.succ.h\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nh : (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\nV : Set R\nhV : V \u2208 \ud835\udcdd 0\nm : \u2115\nhm : \u2191(J ^ m) \u2286 V\nn\u271d : \u2115\nhn : 0 < Nat.succ n\u271d\n\u22a2 m \u2264 n\u271d * m + m\n[PROOFSTEP]\napply Nat.le_add_left\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\n\u22a2 IsAdic \u22a5 \u2194 DiscreteTopology A\n[PROOFSTEP]\nrw [isAdic_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\n\u22a2 ((\u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)) \u2227 \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s) \u2194 DiscreteTopology A\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\n\u22a2 ((\u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)) \u2227 \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s) \u2192 DiscreteTopology A\n[PROOFSTEP]\nrintro \u27e8h, _h'\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\nh : \u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)\n_h' : \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s\n\u22a2 DiscreteTopology A\n[PROOFSTEP]\nrw [discreteTopology_iff_open_singleton_zero]\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\nh : \u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)\n_h' : \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s\n\u22a2 IsOpen {0}\n[PROOFSTEP]\nsimpa using h 1\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\n\u22a2 DiscreteTopology A \u2192 (\u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)) \u2227 \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s\n[PROOFSTEP]\nintros\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\na\u271d : DiscreteTopology A\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)) \u2227 \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.left\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\na\u271d : DiscreteTopology A\n\u22a2 \u2200 (n : \u2115), IsOpen \u2191(\u22a5 ^ n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.right\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\na\u271d : DiscreteTopology A\n\u22a2 \u2200 (s : Set A), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(\u22a5 ^ n) \u2286 s\n[PROOFSTEP]\nintro U U_nhds\n[GOAL]\ncase mpr.right\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\na\u271d : DiscreteTopology A\nU : Set A\nU_nhds : U \u2208 \ud835\udcdd 0\n\u22a2 \u2203 n, \u2191(\u22a5 ^ n) \u2286 U\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : TopologicalSpace R\ninst\u271d\u00b3 : TopologicalRing R\nA : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : TopologicalRing A\na\u271d : DiscreteTopology A\nU : Set A\nU_nhds : U \u2208 \ud835\udcdd 0\n\u22a2 \u2191(\u22a5 ^ 1) \u2286 U\n[PROOFSTEP]\nsimp [mem_of_mem_nhds U_nhds]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : WithIdeal R\n\u22a2 NonarchimedeanRing R\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : WithIdeal R\n\u22a2 TopologicalRing (UniformSpace.Completion R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : WithIdeal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 TopologicalAddGroup M\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : WithIdeal R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 ContinuousSMul R M\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology", "llama_tokens": 12644, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.695958331339634, "lm_q1q2_score": 0.5722093753236411}}
{"text": "[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1 \u03b2 : Type u\u2080\ne : \u03b1 \u2243 \u03b2\nx : f \u03b1\n\u22a2 map e.symm (map e x) = x\n[PROOFSTEP]\nconvert (congr_fun (EquivFunctor.map_trans' e e.symm) x).symm\n[GOAL]\ncase h.e'_3\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1 \u03b2 : Type u\u2080\ne : \u03b1 \u2243 \u03b2\nx : f \u03b1\n\u22a2 x = map (e.trans e.symm) x\n[PROOFSTEP]\nsimp\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1 \u03b2 : Type u\u2080\ne : \u03b1 \u2243 \u03b2\ny : f \u03b2\n\u22a2 map e (map e.symm y) = y\n[PROOFSTEP]\nconvert (congr_fun (EquivFunctor.map_trans' e.symm e) y).symm\n[GOAL]\ncase h.e'_3\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1 \u03b2 : Type u\u2080\ne : \u03b1 \u2243 \u03b2\ny : f \u03b2\n\u22a2 y = map (e.symm.trans e) y\n[PROOFSTEP]\nsimp\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1\u271d \u03b2 : Type u\u2080\ne : \u03b1\u271d \u2243 \u03b2\n\u03b1 : Type u\u2080\n\u22a2 mapEquiv f (Equiv.refl \u03b1) = Equiv.refl (f \u03b1)\n[PROOFSTEP]\nsimp [EquivFunctor.mapEquiv]\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1\u271d \u03b2 : Type u\u2080\ne : \u03b1\u271d \u2243 \u03b2\n\u03b1 : Type u\u2080\n\u22a2 { toFun := id, invFun := id, left_inv := (_ : LeftInverse id id), right_inv := (_ : Function.RightInverse id id) } =\n    Equiv.refl (f \u03b1)\n[PROOFSTEP]\nrfl\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d : EquivFunctor f\n\u03b1 \u03b2 : Type u\u2080\ne : \u03b1 \u2243 \u03b2\n\u03b3 : Type u\u2080\nab : \u03b1 \u2243 \u03b2\nbc : \u03b2 \u2243 \u03b3\nx : f \u03b1\n\u22a2 \u2191((mapEquiv f ab).trans (mapEquiv f bc)) x = \u2191(mapEquiv f (ab.trans bc)) x\n[PROOFSTEP]\nsimp [mapEquiv, map_trans']\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\n\u03b1 : Type u\u2080\n\u22a2 (fun {\u03b1 \u03b2} e => Functor.map \u2191e) (Equiv.refl \u03b1) = id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\n\u03b1 : Type u\u2080\nx\u271d : f \u03b1\n\u22a2 (fun {\u03b1 \u03b2} e => Functor.map \u2191e) (Equiv.refl \u03b1) x\u271d = id x\u271d\n[PROOFSTEP]\napply LawfulFunctor.id_map\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\n\u03b1 \u03b2 \u03b3 : Type u\u2080\nk : \u03b1 \u2243 \u03b2\nh : \u03b2 \u2243 \u03b3\n\u22a2 (fun {\u03b1 \u03b2} e => Functor.map \u2191e) (k.trans h) = (fun {\u03b1 \u03b2} e => Functor.map \u2191e) h \u2218 (fun {\u03b1 \u03b2} e => Functor.map \u2191e) k\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\n\u03b1 \u03b2 \u03b3 : Type u\u2080\nk : \u03b1 \u2243 \u03b2\nh : \u03b2 \u2243 \u03b3\nx : f \u03b1\n\u22a2 (fun {\u03b1 \u03b2} e => Functor.map \u2191e) (k.trans h) x =\n    ((fun {\u03b1 \u03b2} e => Functor.map \u2191e) h \u2218 (fun {\u03b1 \u03b2} e => Functor.map \u2191e) k) x\n[PROOFSTEP]\napply LawfulFunctor.comp_map k h x\n[GOAL]\nf : Type u\u2080 \u2192 Type u\u2081\ninst\u271d\u00b9 : Applicative f\ninst\u271d : LawfulApplicative f\n\u03b1 \u03b2 : Type u\u2080\nh : \u2200 (\u03b3 : Type u\u2080), Injective pure\ne\u2081 e\u2082 : \u03b1 \u2243 \u03b2\nH : mapEquiv f e\u2081 = mapEquiv f e\u2082\nx : \u03b1\n\u22a2 pure (\u2191e\u2081 x) = pure (\u2191e\u2082 x)\n[PROOFSTEP]\nsimpa [EquivFunctor.map] using Equiv.congr_fun H (pure x)\n", "meta": {"mathlib_filename": "Mathlib.Control.EquivFunctor", "llama_tokens": 1311, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.782662489091802, "lm_q2_score": 0.7310585727705126, "lm_q1q2_score": 0.5721721222364696}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SMul M (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SMul N (\u03b1 i)\na\u271d : M\ni : \u03b9\nb\u271d : \u03b1 i\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : SMul M N\ninst\u271d : \u2200 (i : \u03b9), IsScalarTower M N (\u03b1 i)\na : M\nb : N\nx : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 (a \u2022 b) \u2022 x = a \u2022 b \u2022 x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SMul M (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SMul N (\u03b1 i)\na\u271d : M\ni : \u03b9\nb\u271d : \u03b1 i\nx : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : SMul M N\ninst\u271d : \u2200 (i : \u03b9), IsScalarTower M N (\u03b1 i)\na : M\nb : N\nfst\u271d : \u03b9\nsnd\u271d : \u03b1 fst\u271d\n\u22a2 (a \u2022 b) \u2022 { fst := fst\u271d, snd := snd\u271d } = a \u2022 b \u2022 { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrw [smul_mk, smul_mk, smul_mk, smul_assoc]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SMul M (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SMul N (\u03b1 i)\na\u271d : M\ni : \u03b9\nb\u271d : \u03b1 i\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d : \u2200 (i : \u03b9), SMulCommClass M N (\u03b1 i)\na : M\nb : N\nx : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 a \u2022 b \u2022 x = b \u2022 a \u2022 x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SMul M (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SMul N (\u03b1 i)\na\u271d : M\ni : \u03b9\nb\u271d : \u03b1 i\nx : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d : \u2200 (i : \u03b9), SMulCommClass M N (\u03b1 i)\na : M\nb : N\nfst\u271d : \u03b9\nsnd\u271d : \u03b1 fst\u271d\n\u22a2 a \u2022 b \u2022 { fst := fst\u271d, snd := snd\u271d } = b \u2022 a \u2022 { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrw [smul_mk, smul_mk, smul_mk, smul_mk, smul_comm]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SMul M (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SMul N (\u03b1 i)\na\u271d : M\ni : \u03b9\nb : \u03b1 i\nx\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SMul M\u1d50\u1d52\u1d56 (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), IsCentralScalar M (\u03b1 i)\na : M\nx : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 MulOpposite.op a \u2022 x = a \u2022 x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SMul M (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SMul N (\u03b1 i)\na\u271d : M\ni : \u03b9\nb : \u03b1 i\nx : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SMul M\u1d50\u1d52\u1d56 (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), IsCentralScalar M (\u03b1 i)\na : M\nfst\u271d : \u03b9\nsnd\u271d : \u03b1 fst\u271d\n\u22a2 MulOpposite.op a \u2022 { fst := fst\u271d, snd := snd\u271d } = a \u2022 { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrw [smul_mk, smul_mk, op_smul_eq_smul]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\nm : Monoid M\ninst\u271d : (i : \u03b9) \u2192 MulAction M (\u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 1 \u2022 x = x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\nm : Monoid M\ninst\u271d : (i : \u03b9) \u2192 MulAction M (\u03b1 i)\nfst\u271d : \u03b9\nsnd\u271d : \u03b1 fst\u271d\n\u22a2 1 \u2022 { fst := fst\u271d, snd := snd\u271d } = { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrw [smul_mk, one_smul]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\nm : Monoid M\ninst\u271d : (i : \u03b9) \u2192 MulAction M (\u03b1 i)\na b : M\nx : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 (a * b) \u2022 x = a \u2022 b \u2022 x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b1 : \u03b9 \u2192 Type u_4\nm : Monoid M\ninst\u271d : (i : \u03b9) \u2192 MulAction M (\u03b1 i)\na b : M\nfst\u271d : \u03b9\nsnd\u271d : \u03b1 fst\u271d\n\u22a2 (a * b) \u2022 { fst := fst\u271d, snd := snd\u271d } = a \u2022 b \u2022 { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrw [smul_mk, smul_mk, smul_mk, mul_smul]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.GroupAction.Sigma", "llama_tokens": 1883, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267694452331, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.572050744135763}}
{"text": "[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x \u2208 antidiagonal n \u2194 x.fst + x.snd = n\n[PROOFSTEP]\nrw [antidiagonal, mem_map]\n[GOAL]\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 (\u2203 a, a \u2208 range (n + 1) \u2227 (a, n - a) = x) \u2194 x.fst + x.snd = n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 (\u2203 a, a \u2208 range (n + 1) \u2227 (a, n - a) = x) \u2192 x.fst + x.snd = n\n[PROOFSTEP]\nrintro \u27e8i, hi, rfl\u27e9\n[GOAL]\ncase mp.intro.intro\nn i : \u2115\nhi : i \u2208 range (n + 1)\n\u22a2 (i, n - i).fst + (i, n - i).snd = n\n[PROOFSTEP]\nrw [mem_range, lt_succ_iff] at hi \n[GOAL]\ncase mp.intro.intro\nn i : \u2115\nhi : i \u2264 n\n\u22a2 (i, n - i).fst + (i, n - i).snd = n\n[PROOFSTEP]\nexact add_tsub_cancel_of_le hi\n[GOAL]\ncase mpr\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.fst + x.snd = n \u2192 \u2203 a, a \u2208 range (n + 1) \u2227 (a, n - a) = x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nx : \u2115 \u00d7 \u2115\n\u22a2 \u2203 a, a \u2208 range (x.fst + x.snd + 1) \u2227 (a, x.fst + x.snd - a) = x\n[PROOFSTEP]\nrefine' \u27e8x.fst, _, _\u27e9\n[GOAL]\ncase mpr.refine'_1\nx : \u2115 \u00d7 \u2115\n\u22a2 x.fst \u2208 range (x.fst + x.snd + 1)\n[PROOFSTEP]\nrw [mem_range, add_assoc, lt_add_iff_pos_right]\n[GOAL]\ncase mpr.refine'_1\nx : \u2115 \u00d7 \u2115\n\u22a2 0 < x.snd + 1\n[PROOFSTEP]\nexact zero_lt_succ _\n[GOAL]\ncase mpr.refine'_2\nx : \u2115 \u00d7 \u2115\n\u22a2 (x.fst, x.fst + x.snd - x.fst) = x\n[PROOFSTEP]\nexact Prod.ext rfl (by simp only [add_tsub_cancel_left])\n[GOAL]\nx : \u2115 \u00d7 \u2115\n\u22a2 (x.fst, x.fst + x.snd - x.fst).snd = x.snd\n[PROOFSTEP]\nsimp only [add_tsub_cancel_left]\n[GOAL]\nn : \u2115\n\u22a2 length (antidiagonal n) = n + 1\n[PROOFSTEP]\nrw [antidiagonal, length_map, length_range]\n[GOAL]\nn : \u2115\n\u22a2 antidiagonal (n + 1) = (0, n + 1) :: map (Prod.map succ id) (antidiagonal n)\n[PROOFSTEP]\nsimp only [antidiagonal, range_succ_eq_map, map_cons, true_and_iff, Nat.add_succ_sub_one, add_zero, id.def,\n  eq_self_iff_true, tsub_zero, map_map, Prod.map_mk]\n[GOAL]\nn : \u2115\n\u22a2 (0, n + 1) :: (succ 0, n) :: map ((fun i => (i, n + 1 - i)) \u2218 succ \u2218 succ) (range n) =\n    (0, n + 1) :: (succ 0, n) :: map (Prod.map succ id \u2218 (fun i => (i, n - i)) \u2218 succ) (range n)\n[PROOFSTEP]\napply congr rfl (congr rfl _)\n[GOAL]\nn : \u2115\n\u22a2 map ((fun i => (i, n + 1 - i)) \u2218 succ \u2218 succ) (range n) =\n    map (Prod.map succ id \u2218 (fun i => (i, n - i)) \u2218 succ) (range n)\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nn n\u271d : \u2115\na\u271d : \u2115 \u00d7 \u2115\n\u22a2 a\u271d \u2208 get? (map ((fun i => (i, n + 1 - i)) \u2218 succ \u2218 succ) (range n)) n\u271d \u2194\n    a\u271d \u2208 get? (map (Prod.map succ id \u2218 (fun i => (i, n - i)) \u2218 succ) (range n)) n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 antidiagonal (n + 1) = map (Prod.map id succ) (antidiagonal n) ++ [(n + 1, 0)]\n[PROOFSTEP]\nsimp only [antidiagonal, range_succ, add_tsub_cancel_left, map_append, append_assoc, tsub_self, singleton_append,\n  map_map, map]\n[GOAL]\nn : \u2115\n\u22a2 map (fun i => (i, n + 1 - i)) (range n) ++ [(n, 1), (n + 1, 0)] =\n    map (Prod.map id succ \u2218 fun i => (i, n - i)) (range n) ++ [Prod.map id succ (n, 0), (n + 1, 0)]\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nn : \u2115\n\u22a2 map (fun i => (i, n + 1 - i)) (range n) = map (Prod.map id succ \u2218 fun i => (i, n - i)) (range n)\n[PROOFSTEP]\napply map_congr\n[GOAL]\ncase e_a.a\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 range n \u2192 (x, n + 1 - x) = (Prod.map id succ \u2218 fun i => (i, n - i)) x\n[PROOFSTEP]\nsimp (config := { contextual := true }) [le_of_lt, Nat.succ_eq_add_one, Nat.sub_add_comm]\n[GOAL]\nn : \u2115\n\u22a2 antidiagonal (n + 2) = (0, n + 2) :: map (Prod.map succ succ) (antidiagonal n) ++ [(n + 2, 0)]\n[PROOFSTEP]\nrw [antidiagonal_succ']\n[GOAL]\nn : \u2115\n\u22a2 map (Prod.map id succ) (antidiagonal (n + 1)) ++ [(n + 1 + 1, 0)] =\n    (0, n + 2) :: map (Prod.map succ succ) (antidiagonal n) ++ [(n + 2, 0)]\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 map (Prod.map id succ \u2218 Prod.map succ id) (antidiagonal n) = map (Prod.map succ succ) (antidiagonal n)\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nn n\u271d : \u2115\na\u271d : \u2115 \u00d7 \u2115\n\u22a2 a\u271d \u2208 get? (map (Prod.map id succ \u2218 Prod.map succ id) (antidiagonal n)) n\u271d \u2194\n    a\u271d \u2208 get? (map (Prod.map succ succ) (antidiagonal n)) n\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 map Prod.swap (antidiagonal n) = reverse (antidiagonal n)\n[PROOFSTEP]\nrw [antidiagonal, map_map, \u2190 List.map_reverse, range_eq_range', reverse_range', \u2190 range_eq_range', map_map]\n[GOAL]\nn : \u2115\n\u22a2 map (Prod.swap \u2218 fun i => (i, n - i)) (range (n + 1)) =\n    map ((fun i => (i, n - i)) \u2218 fun x => 0 + (n + 1) - 1 - x) (range (n + 1))\n[PROOFSTEP]\napply map_congr\n[GOAL]\ncase a\nn : \u2115\n\u22a2 \u2200 (x : \u2115),\n    x \u2208 range (n + 1) \u2192 (Prod.swap \u2218 fun i => (i, n - i)) x = ((fun i => (i, n - i)) \u2218 fun x => 0 + (n + 1) - 1 - x) x\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Nat.sub_sub_self, lt_succ_iff]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.NatAntidiagonal", "llama_tokens": 2316, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.7122321903471563, "lm_q1q2_score": 0.5720462322087169}}
{"text": "[GOAL]\nG : Type u_1\nS : Type u_2\ninst\u271d : Semigroup S\na\u271d b c : S\nh : Commute b c\na : S\n\u22a2 a * b * c = a * c * b\n[PROOFSTEP]\nsimp only [mul_assoc, h.eq]\n[GOAL]\nG : Type u_1\nS : Type u_2\ninst\u271d : Semigroup S\na b c\u271d : S\nh : Commute a b\nc : S\n\u22a2 a * (b * c) = b * (a * c)\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, h.eq]\n[GOAL]\nG : Type u_1\nS : Type u_2\ninst\u271d : Semigroup S\na\u271d b c : S\nhbc : Commute b c\na d : S\n\u22a2 a * b * (c * d) = a * c * (b * d)\n[PROOFSTEP]\nsimp only [hbc.left_comm, mul_assoc]\n[GOAL]\nG : Type u_1\nM : Type u_2\ninst\u271d : Monoid M\na\u271d b\u271d : M\nu\u271d u\u2081 u\u2082 u : M\u02e3\na b : M\nhu : a * b = \u2191u\nhc : Commute a b\n\u22a2 a * (b * \u2191u\u207b\u00b9) = 1\n[PROOFSTEP]\nrw [\u2190 mul_assoc, hu, u.mul_inv]\n[GOAL]\nG : Type u_1\nM : Type u_2\ninst\u271d : Monoid M\na\u271d b\u271d : M\nu\u271d u\u2081 u\u2082 u : M\u02e3\na b : M\nhu : a * b = \u2191u\nhc : Commute a b\n\u22a2 b * \u2191u\u207b\u00b9 * a = 1\n[PROOFSTEP]\nhave : Commute a u := hu \u25b8 (Commute.refl _).mul_right hc\n[GOAL]\nG : Type u_1\nM : Type u_2\ninst\u271d : Monoid M\na\u271d b\u271d : M\nu\u271d u\u2081 u\u2082 u : M\u02e3\na b : M\nhu : a * b = \u2191u\nhc : Commute a b\nthis : Commute a \u2191u\n\u22a2 b * \u2191u\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrw [\u2190 this.units_inv_right.right_comm, \u2190 hc.eq, hu, u.mul_inv]\n[GOAL]\nG : Type u_1\ninst\u271d : DivisionMonoid G\na b c d : G\nhab : Commute a b\n\u22a2 (a * b)\u207b\u00b9 = a\u207b\u00b9 * b\u207b\u00b9\n[PROOFSTEP]\nrw [hab.eq, mul_inv_rev]\n[GOAL]\nG : Type u_1\ninst\u271d : DivisionMonoid G\na b c d : G\nhab : Commute a b\n\u22a2 (a * b)\u207b\u00b9 = a\u207b\u00b9 * b\u207b\u00b9\n[PROOFSTEP]\nrw [hab.eq, mul_inv_rev]\n[GOAL]\nG : Type u_1\ninst\u271d : DivisionMonoid G\na b c d : G\nhbd : Commute b d\nhbc : Commute b\u207b\u00b9 c\n\u22a2 a / b * (c / d) = a * c / (b * d)\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, mul_inv_rev, hbd.inv_inv.symm.eq, hbc.mul_mul_mul_comm]\n[GOAL]\nG : Type u_1\ninst\u271d : DivisionMonoid G\na b c d : G\nhbc : Commute b c\nhbd : Commute b\u207b\u00b9 d\nhcd : Commute c\u207b\u00b9 d\n\u22a2 a / b / (c / d) = a / c / (b / d)\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, mul_inv_rev, inv_inv, hbd.symm.eq, hcd.symm.eq, hbc.inv_inv.mul_mul_mul_comm]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\na b : G\nh : Commute a b\n\u22a2 a\u207b\u00b9 * b * a = b\n[PROOFSTEP]\nrw [h.inv_left.eq, inv_mul_cancel_right]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\na b : G\nh : Commute a b\n\u22a2 a\u207b\u00b9 * (b * a) = b\n[PROOFSTEP]\nrw [\u2190 mul_assoc, h.inv_mul_cancel]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\na b : G\nh : Commute a b\n\u22a2 a * b * a\u207b\u00b9 = b\n[PROOFSTEP]\nrw [h.eq, mul_inv_cancel_right]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\na b : G\nh : Commute a b\n\u22a2 a * (b * a\u207b\u00b9) = b\n[PROOFSTEP]\nrw [\u2190 mul_assoc, h.mul_inv_cancel]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.Commute", "llama_tokens": 1311, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019595, "lm_q2_score": 0.7718434978390747, "lm_q1q2_score": 0.5712987774545981}}
{"text": "[GOAL]\nx y z\u271d : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\nz : ZFSet\nhz : z \u2208 x \u2229 y\nw : ZFSet\nhw : w \u2208 z\n\u22a2 w \u2208 x \u2229 y\n[PROOFSTEP]\nrw [mem_inter] at hz \u22a2\n[GOAL]\nx y z\u271d : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\nz : ZFSet\nhz : z \u2208 x \u2227 z \u2208 y\nw : ZFSet\nhw : w \u2208 z\n\u22a2 w \u2208 x \u2227 w \u2208 y\n[PROOFSTEP]\nexact \u27e8hx.mem_trans hw hz.1, hy.mem_trans hw hz.2\u27e9\n[GOAL]\nx y\u271d z\u271d : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y \u2208 \u22c3\u2080 x\nz : ZFSet\nhz : z \u2208 y\n\u22a2 z \u2208 \u22c3\u2080 x\n[PROOFSTEP]\nrcases mem_sUnion.1 hy with \u27e8w, hw, hw'\u27e9\n[GOAL]\ncase intro.intro\nx y\u271d z\u271d : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y \u2208 \u22c3\u2080 x\nz : ZFSet\nhz : z \u2208 y\nw : ZFSet\nhw : w \u2208 x\nhw' : y \u2208 w\n\u22a2 z \u2208 \u22c3\u2080 x\n[PROOFSTEP]\nexact mem_sUnion_of_mem hz (h.mem_trans hw' hw)\n[GOAL]\nx y\u271d z\u271d : ZFSet\nH : \u2200 (y : ZFSet), y \u2208 x \u2192 IsTransitive y\ny : ZFSet\nhy : y \u2208 \u22c3\u2080 x\nz : ZFSet\nhz : z \u2208 y\n\u22a2 z \u2208 \u22c3\u2080 x\n[PROOFSTEP]\nrcases mem_sUnion.1 hy with \u27e8w, hw, hw'\u27e9\n[GOAL]\ncase intro.intro\nx y\u271d z\u271d : ZFSet\nH : \u2200 (y : ZFSet), y \u2208 x \u2192 IsTransitive y\ny : ZFSet\nhy : y \u2208 \u22c3\u2080 x\nz : ZFSet\nhz : z \u2208 y\nw : ZFSet\nhw : w \u2208 x\nhw' : y \u2208 w\n\u22a2 z \u2208 \u22c3\u2080 x\n[PROOFSTEP]\nexact mem_sUnion_of_mem ((H w hw).mem_trans hz hw') hw\n[GOAL]\nx y z : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\n\u22a2 IsTransitive (x \u222a y)\n[PROOFSTEP]\nrw [\u2190 sUnion_pair]\n[GOAL]\nx y z : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\n\u22a2 IsTransitive (\u22c3\u2080 {x, y})\n[PROOFSTEP]\napply IsTransitive.sUnion' fun z => _\n[GOAL]\nx y z : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\n\u22a2 \u2200 (z : ZFSet), z \u2208 {x, y} \u2192 IsTransitive z\n[PROOFSTEP]\nintro\n[GOAL]\nx y z : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\nz\u271d : ZFSet\n\u22a2 z\u271d \u2208 {x, y} \u2192 IsTransitive z\u271d\n[PROOFSTEP]\nrw [mem_pair]\n[GOAL]\nx y z : ZFSet\nhx : IsTransitive x\nhy : IsTransitive y\nz\u271d : ZFSet\n\u22a2 z\u271d = x \u2228 z\u271d = y \u2192 IsTransitive z\u271d\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\ny z : ZFSet\nhy : IsTransitive y\nz\u271d : ZFSet\nhx : IsTransitive z\u271d\n\u22a2 IsTransitive z\u271d\ncase inr x z : ZFSet hx : IsTransitive x z\u271d : ZFSet hy : IsTransitive z\u271d \u22a2 IsTransitive z\u271d\n[PROOFSTEP]\nassumption'\n[GOAL]\nx y\u271d z\u271d : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y \u2208 powerset x\nz : ZFSet\nhz : z \u2208 y\n\u22a2 z \u2208 powerset x\n[PROOFSTEP]\nrw [mem_powerset] at hy \u22a2\n[GOAL]\nx y\u271d z\u271d : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y \u2286 x\nz : ZFSet\nhz : z \u2208 y\n\u22a2 z \u2286 x\n[PROOFSTEP]\nexact h.subset_of_mem (hy hz)\n[GOAL]\nx y\u271d z : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y \u2208 \u22c3\u2080 x\n\u22a2 y \u2208 x\n[PROOFSTEP]\nrcases mem_sUnion.1 hy with \u27e8z, hz, hz'\u27e9\n[GOAL]\ncase intro.intro\nx y\u271d z\u271d : ZFSet\nh : IsTransitive x\ny : ZFSet\nhy : y \u2208 \u22c3\u2080 x\nz : ZFSet\nhz : z \u2208 x\nhz' : y \u2208 z\n\u22a2 y \u2208 x\n[PROOFSTEP]\nexact h.mem_trans hz' hz\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.ZFC.Ordinal", "llama_tokens": 1454, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.7401743505760728, "lm_q1q2_score": 0.5712987597594015}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Group \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 : \u03b1\n\u22a2 max a 1 / max a\u207b\u00b9 1 = a\n[PROOFSTEP]\nrcases le_total a 1 with (h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Group \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 : \u03b1\nh : a \u2264 1\n\u22a2 max a 1 / max a\u207b\u00b9 1 = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Group \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 : \u03b1\nh : 1 \u2264 a\n\u22a2 max a 1 / max a\u207b\u00b9 1 = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommGroup \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 min (a / c) (b / c) = min a b / c\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using min_mul_mul_right a b c\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommGroup \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 max (a / c) (b / c) = max a b / c\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using max_mul_mul_right a b c\u207b\u00b9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommGroup \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 min (a / b) (a / c) = a / max b c\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommGroup \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 max (a / b) (a / c) = a / min b c\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 max a b - max c d \u2264 max (a - c) (b - d)\n[PROOFSTEP]\nsimp only [sub_le_iff_le_add, max_le_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 a \u2264 max (a - c) (b - d) + max c d \u2227 b \u2264 max (a - c) (b - d) + max c d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 a \u2264 max (a - c) (b - d) + max c d\ncase right \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 a\u271d b\u271d c\u271d a b c d : \u03b1 \u22a2 b \u2264 max (a - c) (b - d) + max c d\n[PROOFSTEP]\ncalc\n  a = a - c + c := (sub_add_cancel a c).symm\n  _ \u2264 max (a - c) (b - d) + max c d := add_le_add (le_max_left _ _) (le_max_left _ _)\n[GOAL]\ncase right\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 b \u2264 max (a - c) (b - d) + max c d\n[PROOFSTEP]\ncalc\n  b = b - d + d := (sub_add_cancel b d).symm\n  _ \u2264 max (a - c) (b - d) + max c d := add_le_add (le_max_right _ _) (le_max_right _ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 |max a b - max c d| \u2264 max |a - c| |b - d|\n[PROOFSTEP]\nrefine' abs_sub_le_iff.2 \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 max a b - max c d \u2264 max |a - c| |b - d|\n[PROOFSTEP]\nexact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 max c d - max a b \u2264 max |a - c| |b - d|\n[PROOFSTEP]\nrw [abs_sub_comm a c, abs_sub_comm b d]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 max c d - max a b \u2264 max |c - a| |d - b|\n[PROOFSTEP]\nexact (max_sub_max_le_max _ _ _ _).trans (max_le_max (le_abs_self _) (le_abs_self _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c d : \u03b1\n\u22a2 |min a b - min c d| \u2264 max |a - c| |b - d|\n[PROOFSTEP]\nsimpa only [max_neg_neg, neg_sub_neg, abs_sub_comm] using abs_max_sub_max_le_max (-a) (-b) (-c) (-d)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 |max a c - max b c| \u2264 |a - b|\n[PROOFSTEP]\nsimpa only [sub_self, abs_zero, max_eq_left (abs_nonneg (a - b))] using abs_max_sub_max_le_max a c b c\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Group.MinMax", "llama_tokens": 1894, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056322076481139, "lm_q2_score": 0.7090191460821871, "lm_q1q2_score": 0.571208659922973}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 Ico 0 n = range n\n[PROOFSTEP]\nrw [Ico, tsub_zero, range_eq_range']\n[GOAL]\nn m : \u2115\n\u22a2 List.length (Ico n m) = m - n\n[PROOFSTEP]\ndsimp [Ico]\n[GOAL]\nn m : \u2115\n\u22a2 List.length (range' n (m - n)) = m - n\n[PROOFSTEP]\nsimp only [length_range']\n[GOAL]\nn m : \u2115\n\u22a2 Pairwise (fun x x_1 => x < x_1) (Ico n m)\n[PROOFSTEP]\ndsimp [Ico]\n[GOAL]\nn m : \u2115\n\u22a2 Pairwise (fun x x_1 => x < x_1) (range' n (m - n))\n[PROOFSTEP]\nsimp only [pairwise_lt_range']\n[GOAL]\nn m : \u2115\n\u22a2 Nodup (Ico n m)\n[PROOFSTEP]\ndsimp [Ico]\n[GOAL]\nn m : \u2115\n\u22a2 Nodup (range' n (m - n))\n[PROOFSTEP]\nsimp only [nodup_range']\n[GOAL]\nn m l : \u2115\n\u22a2 l \u2208 Ico n m \u2194 n \u2264 l \u2227 l < m\n[PROOFSTEP]\nsuffices n \u2264 l \u2227 l < n + (m - n) \u2194 n \u2264 l \u2227 l < m by simp [Ico, this]\n[GOAL]\nn m l : \u2115\nthis : n \u2264 l \u2227 l < n + (m - n) \u2194 n \u2264 l \u2227 l < m\n\u22a2 l \u2208 Ico n m \u2194 n \u2264 l \u2227 l < m\n[PROOFSTEP]\nsimp [Ico, this]\n[GOAL]\nn m l : \u2115\n\u22a2 n \u2264 l \u2227 l < n + (m - n) \u2194 n \u2264 l \u2227 l < m\n[PROOFSTEP]\ncases' le_total n m with hnm hmn\n[GOAL]\ncase inl\nn m l : \u2115\nhnm : n \u2264 m\n\u22a2 n \u2264 l \u2227 l < n + (m - n) \u2194 n \u2264 l \u2227 l < m\n[PROOFSTEP]\nrw [add_tsub_cancel_of_le hnm]\n[GOAL]\ncase inr\nn m l : \u2115\nhmn : m \u2264 n\n\u22a2 n \u2264 l \u2227 l < n + (m - n) \u2194 n \u2264 l \u2227 l < m\n[PROOFSTEP]\nrw [tsub_eq_zero_iff_le.mpr hmn, add_zero]\n[GOAL]\ncase inr\nn m l : \u2115\nhmn : m \u2264 n\n\u22a2 n \u2264 l \u2227 l < n \u2194 n \u2264 l \u2227 l < m\n[PROOFSTEP]\nexact and_congr_right fun hnl => Iff.intro (fun hln => (not_le_of_gt hln hnl).elim) fun hlm => lt_of_lt_of_le hlm hmn\n[GOAL]\nn m : \u2115\nh : m \u2264 n\n\u22a2 Ico n m = []\n[PROOFSTEP]\nsimp [Ico, tsub_eq_zero_iff_le.mpr h]\n[GOAL]\nn m k : \u2115\n\u22a2 map ((fun x x_1 => x + x_1) k) (Ico n m) = Ico (n + k) (m + k)\n[PROOFSTEP]\nrw [Ico, Ico, map_add_range', add_tsub_add_eq_tsub_right m k, add_comm n k]\n[GOAL]\nn m k : \u2115\nh\u2081 : k \u2264 n\n\u22a2 map (fun x => x - k) (Ico n m) = Ico (n - k) (m - k)\n[PROOFSTEP]\nrw [Ico, Ico, tsub_tsub_tsub_cancel_right h\u2081, map_sub_range' _ _ _ h\u2081]\n[GOAL]\nn m : \u2115\nh : Ico n m = []\n\u22a2 m - n = 0\n[PROOFSTEP]\nrw [\u2190 length, h, List.length]\n[GOAL]\nn m l : \u2115\nhnm : n \u2264 m\nhml : m \u2264 l\n\u22a2 Ico n m ++ Ico m l = Ico n l\n[PROOFSTEP]\ndsimp only [Ico]\n[GOAL]\nn m l : \u2115\nhnm : n \u2264 m\nhml : m \u2264 l\n\u22a2 range' n (m - n) ++ range' m (l - m) = range' n (l - n)\n[PROOFSTEP]\nconvert range'_append n (m - n) (l - m) 1 using 2\n[GOAL]\ncase h.e'_2.h.e'_6\nn m l : \u2115\nhnm : n \u2264 m\nhml : m \u2264 l\n\u22a2 range' m (l - m) = range' (n + 1 * (m - n)) (l - m)\n[PROOFSTEP]\nrw [one_mul, add_tsub_cancel_of_le hnm]\n[GOAL]\ncase h.e'_3.h.e'_2\nn m l : \u2115\nhnm : n \u2264 m\nhml : m \u2264 l\n\u22a2 l - n = l - m + (m - n)\n[PROOFSTEP]\nrw [tsub_add_tsub_cancel hml hnm]\n[GOAL]\nn m l : \u2115\n\u22a2 Ico n m \u2229 Ico m l = []\n[PROOFSTEP]\napply eq_nil_iff_forall_not_mem.2\n[GOAL]\nn m l : \u2115\n\u22a2 \u2200 (a : \u2115), \u00aca \u2208 Ico n m \u2229 Ico m l\n[PROOFSTEP]\nintro a\n[GOAL]\nn m l a : \u2115\n\u22a2 \u00aca \u2208 Ico n m \u2229 Ico m l\n[PROOFSTEP]\nsimp only [and_imp, not_and, not_lt, List.mem_inter_iff, List.Ico.mem]\n[GOAL]\nn m l a : \u2115\n\u22a2 n \u2264 a \u2192 a < m \u2192 m \u2264 a \u2192 l \u2264 a\n[PROOFSTEP]\nintro _ h\u2082 h\u2083\n[GOAL]\nn m l a : \u2115\na\u271d : n \u2264 a\nh\u2082 : a < m\nh\u2083 : m \u2264 a\n\u22a2 l \u2264 a\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nn m l a : \u2115\na\u271d : n \u2264 a\nh\u2082 : a < m\nh\u2083 : m \u2264 a\n\u22a2 False\n[PROOFSTEP]\nexact not_lt_of_ge h\u2083 h\u2082\n[GOAL]\nn : \u2115\n\u22a2 Ico n (n + 1) = [n]\n[PROOFSTEP]\ndsimp [Ico]\n[GOAL]\nn : \u2115\n\u22a2 range' n (n + 1 - n) = [n]\n[PROOFSTEP]\nsimp [range', add_tsub_cancel_left]\n[GOAL]\nn m : \u2115\nh : n \u2264 m\n\u22a2 Ico n (m + 1) = Ico n m ++ [m]\n[PROOFSTEP]\nrwa [\u2190 succ_singleton, append_consecutive]\n[GOAL]\ncase hml\nn m : \u2115\nh : n \u2264 m\n\u22a2 m \u2264 m + 1\n[PROOFSTEP]\nexact Nat.le_succ _\n[GOAL]\nn m : \u2115\nh : n < m\n\u22a2 Ico n m = n :: Ico (n + 1) m\n[PROOFSTEP]\nrw [\u2190 append_consecutive (Nat.le_succ n) h, succ_singleton]\n[GOAL]\nn m : \u2115\nh : n < m\n\u22a2 [n] ++ Ico (succ n) m = n :: Ico (n + 1) m\n[PROOFSTEP]\nrfl\n[GOAL]\nm : \u2115\nh : 0 < m\n\u22a2 Ico (m - 1) m = [m - 1]\n[PROOFSTEP]\ndsimp [Ico]\n[GOAL]\nm : \u2115\nh : 0 < m\n\u22a2 range' (m - 1) (m - (m - 1)) = [m - 1]\n[PROOFSTEP]\nrw [tsub_tsub_cancel_of_le (succ_le_of_lt h)]\n[GOAL]\nm : \u2115\nh : 0 < m\n\u22a2 range' (m - 1) (succ 0) = [m - 1]\n[PROOFSTEP]\nsimp [\u2190 Nat.one_eq_succ_zero]\n[GOAL]\nn m : \u2115\n\u22a2 Chain' (fun a b => b = succ a) (Ico n m)\n[PROOFSTEP]\nby_cases n < m\n[GOAL]\nn m : \u2115\n\u22a2 Chain' (fun a b => b = succ a) (Ico n m)\n[PROOFSTEP]\nby_cases n < m\n[GOAL]\ncase pos\nn m : \u2115\nh : n < m\n\u22a2 Chain' (fun a b => b = succ a) (Ico n m)\n[PROOFSTEP]\nrw [eq_cons h]\n[GOAL]\ncase pos\nn m : \u2115\nh : n < m\n\u22a2 Chain' (fun a b => b = succ a) (n :: Ico (n + 1) m)\n[PROOFSTEP]\nexact chain_succ_range' _ _ 1\n[GOAL]\ncase neg\nn m : \u2115\nh : \u00acn < m\n\u22a2 Chain' (fun a b => b = succ a) (Ico n m)\n[PROOFSTEP]\nrw [eq_nil_of_le (le_of_not_gt h)]\n[GOAL]\ncase neg\nn m : \u2115\nh : \u00acn < m\n\u22a2 Chain' (fun a b => b = succ a) []\n[PROOFSTEP]\ntrivial\n[GOAL]\nn m : \u2115\n\u22a2 \u00acm \u2208 Ico n m\n[PROOFSTEP]\nsimp\n[GOAL]\nn m l : \u2115\nhml : m \u2264 l\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 decide (k < l) = true\n[PROOFSTEP]\nsimp only [(lt_of_lt_of_le (mem.1 hk).2 hml), decide_True]\n[GOAL]\nn m l : \u2115\nhln : l \u2264 n\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 \u00acdecide (k < l) = true\n[PROOFSTEP]\nsimp only [decide_eq_true_eq, not_lt]\n[GOAL]\nn m l : \u2115\nhln : l \u2264 n\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 l \u2264 k\n[PROOFSTEP]\napply le_trans hln\n[GOAL]\nn m l : \u2115\nhln : l \u2264 n\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 n \u2264 k\n[PROOFSTEP]\nexact (mem.1 hk).1\n[GOAL]\nn m l : \u2115\nhlm : l \u2264 m\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n l\n[PROOFSTEP]\ncases' le_total n l with hnl hln\n[GOAL]\ncase inl\nn m l : \u2115\nhlm : l \u2264 m\nhnl : n \u2264 l\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n l\n[PROOFSTEP]\nrw [\u2190 append_consecutive hnl hlm, filter_append, filter_lt_of_top_le (le_refl l), filter_lt_of_le_bot (le_refl l),\n  append_nil]\n[GOAL]\ncase inr\nn m l : \u2115\nhlm : l \u2264 m\nhln : l \u2264 n\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n l\n[PROOFSTEP]\nrw [eq_nil_of_le hln, filter_lt_of_le_bot hln]\n[GOAL]\nn m l : \u2115\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)\n[PROOFSTEP]\ncases' le_total m l with hml hlm\n[GOAL]\ncase inl\nn m l : \u2115\nhml : m \u2264 l\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)\n[PROOFSTEP]\nrw [min_eq_left hml, filter_lt_of_top_le hml]\n[GOAL]\ncase inr\nn m l : \u2115\nhlm : l \u2264 m\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n (min m l)\n[PROOFSTEP]\nrw [min_eq_right hlm, filter_lt_of_ge hlm]\n[GOAL]\nn m l : \u2115\nhln : l \u2264 n\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 decide (l \u2264 k) = true\n[PROOFSTEP]\nrw [decide_eq_true_eq]\n[GOAL]\nn m l : \u2115\nhln : l \u2264 n\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 l \u2264 k\n[PROOFSTEP]\nexact le_trans hln (mem.1 hk).1\n[GOAL]\nn m l : \u2115\nhml : m \u2264 l\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 \u00acdecide (l \u2264 k) = true\n[PROOFSTEP]\nrw [decide_eq_true_eq]\n[GOAL]\nn m l : \u2115\nhml : m \u2264 l\nk : \u2115\nhk : k \u2208 Ico n m\n\u22a2 \u00acl \u2264 k\n[PROOFSTEP]\nexact not_le_of_gt (lt_of_lt_of_le (mem.1 hk).2 hml)\n[GOAL]\nn m l : \u2115\nhnl : n \u2264 l\n\u22a2 filter (fun x => decide (l \u2264 x)) (Ico n m) = Ico l m\n[PROOFSTEP]\ncases' le_total l m with hlm hml\n[GOAL]\ncase inl\nn m l : \u2115\nhnl : n \u2264 l\nhlm : l \u2264 m\n\u22a2 filter (fun x => decide (l \u2264 x)) (Ico n m) = Ico l m\n[PROOFSTEP]\nrw [\u2190 append_consecutive hnl hlm, filter_append, filter_le_of_top_le (le_refl l), filter_le_of_le_bot (le_refl l),\n  nil_append]\n[GOAL]\ncase inr\nn m l : \u2115\nhnl : n \u2264 l\nhml : m \u2264 l\n\u22a2 filter (fun x => decide (l \u2264 x)) (Ico n m) = Ico l m\n[PROOFSTEP]\nrw [eq_nil_of_le hml, filter_le_of_top_le hml]\n[GOAL]\nn m l : \u2115\n\u22a2 filter (fun x => decide (l \u2264 x)) (Ico n m) = Ico (max n l) m\n[PROOFSTEP]\ncases' le_total n l with hnl hln\n[GOAL]\ncase inl\nn m l : \u2115\nhnl : n \u2264 l\n\u22a2 filter (fun x => decide (l \u2264 x)) (Ico n m) = Ico (max n l) m\n[PROOFSTEP]\nrw [max_eq_right hnl, filter_le_of_le hnl]\n[GOAL]\ncase inr\nn m l : \u2115\nhln : l \u2264 n\n\u22a2 filter (fun x => decide (l \u2264 x)) (Ico n m) = Ico (max n l) m\n[PROOFSTEP]\nrw [max_eq_left hln, filter_le_of_le_bot hln]\n[GOAL]\nn m : \u2115\nhnm : n < m\n\u22a2 filter (fun x => decide (x < n + 1)) (Ico n m) = [n]\n[PROOFSTEP]\nhave r : min m (n + 1) = n + 1 := (@inf_eq_right _ _ m (n + 1)).mpr hnm\n[GOAL]\nn m : \u2115\nhnm : n < m\nr : min m (n + 1) = n + 1\n\u22a2 filter (fun x => decide (x < n + 1)) (Ico n m) = [n]\n[PROOFSTEP]\nsimp [filter_lt n m (n + 1), r]\n[GOAL]\nn m : \u2115\nhnm : n < m\n\u22a2 filter (fun x => decide (x \u2264 n)) (Ico n m) = [n]\n[PROOFSTEP]\nrw [\u2190 filter_lt_of_succ_bot hnm]\n[GOAL]\nn m : \u2115\nhnm : n < m\n\u22a2 filter (fun x => decide (x \u2264 n)) (Ico n m) = filter (fun x => decide (x < n + 1)) (Ico n m)\n[PROOFSTEP]\nexact\n  filter_congr' fun _ _ => by\n    rw [decide_eq_true_eq, decide_eq_true_eq]\n    exact lt_succ_iff.symm\n[GOAL]\nn m : \u2115\nhnm : n < m\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 Ico n m\n\u22a2 decide (x\u271d\u00b9 \u2264 n) = true \u2194 decide (x\u271d\u00b9 < n + 1) = true\n[PROOFSTEP]\nrw [decide_eq_true_eq, decide_eq_true_eq]\n[GOAL]\nn m : \u2115\nhnm : n < m\nx\u271d\u00b9 : \u2115\nx\u271d : x\u271d\u00b9 \u2208 Ico n m\n\u22a2 x\u271d\u00b9 \u2264 n \u2194 x\u271d\u00b9 < n + 1\n[PROOFSTEP]\nexact lt_succ_iff.symm\n[GOAL]\nn a b : \u2115\n\u22a2 n < a \u2228 b \u2264 n \u2228 n \u2208 Ico a b\n[PROOFSTEP]\nby_cases h\u2081 : n < a\n[GOAL]\ncase pos\nn a b : \u2115\nh\u2081 : n < a\n\u22a2 n < a \u2228 b \u2264 n \u2228 n \u2208 Ico a b\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.h\nn a b : \u2115\nh\u2081 : n < a\n\u22a2 n < a\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\ncase neg\nn a b : \u2115\nh\u2081 : \u00acn < a\n\u22a2 n < a \u2228 b \u2264 n \u2228 n \u2208 Ico a b\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\nn a b : \u2115\nh\u2081 : \u00acn < a\n\u22a2 b \u2264 n \u2228 n \u2208 Ico a b\n[PROOFSTEP]\nby_cases h\u2082 : n \u2208 Ico a b\n[GOAL]\ncase pos\nn a b : \u2115\nh\u2081 : \u00acn < a\nh\u2082 : n \u2208 Ico a b\n\u22a2 b \u2264 n \u2228 n \u2208 Ico a b\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.h\nn a b : \u2115\nh\u2081 : \u00acn < a\nh\u2082 : n \u2208 Ico a b\n\u22a2 n \u2208 Ico a b\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase neg\nn a b : \u2115\nh\u2081 : \u00acn < a\nh\u2082 : \u00acn \u2208 Ico a b\n\u22a2 b \u2264 n \u2228 n \u2208 Ico a b\n[PROOFSTEP]\nleft\n[GOAL]\ncase neg.h\nn a b : \u2115\nh\u2081 : \u00acn < a\nh\u2082 : \u00acn \u2208 Ico a b\n\u22a2 b \u2264 n\n[PROOFSTEP]\nsimp only [Ico.mem, not_and, not_lt] at *\n[GOAL]\ncase neg.h\nn a b : \u2115\nh\u2081 : a \u2264 n\nh\u2082 : a \u2264 n \u2192 b \u2264 n\n\u22a2 b \u2264 n\n[PROOFSTEP]\nexact h\u2082 h\u2081\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Intervals", "llama_tokens": 5245, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933183101078, "lm_q2_score": 0.6959583313396339, "lm_q1q2_score": 0.5706115856876179}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx y : \u03b9 \u2192 \u03b1\nh : \u2200 (i : \u03b9), x i < y i\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (i : \u03b9), x i + \u03b5 < y i\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx y : \u03b9 \u2192 \u03b1\nh : \u2200 (i : \u03b9), x i < y i\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (i : \u03b9), x i + \u03b5 < y i\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase intro.inl\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx y : \u03b9 \u2192 \u03b1\nh : \u2200 (i : \u03b9), x i < y i\nval\u271d : Fintype \u03b9\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (i : \u03b9), x i + \u03b5 < y i\n[PROOFSTEP]\nexact \u27e81, zero_lt_one, isEmptyElim\u27e9\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx y : \u03b9 \u2192 \u03b1\nh : \u2200 (i : \u03b9), x i < y i\nval\u271d : Fintype \u03b9\nh\u271d : Nonempty \u03b9\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (i : \u03b9), x i + \u03b5 < y i\n[PROOFSTEP]\nchoose \u03b5 h\u03b5 hx\u03b5 using fun i => exists_pos_add_of_lt' (h i)\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx y : \u03b9 \u2192 \u03b1\nh : \u2200 (i : \u03b9), x i < y i\nval\u271d : Fintype \u03b9\nh\u271d : Nonempty \u03b9\n\u03b5 : \u03b9 \u2192 \u03b1\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nhx\u03b5 : \u2200 (i : \u03b9), x i + \u03b5 i = y i\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (i : \u03b9), x i + \u03b5 < y i\n[PROOFSTEP]\nobtain rfl : x + \u03b5 = y := funext hx\u03b5\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx : \u03b9 \u2192 \u03b1\nval\u271d : Fintype \u03b9\nh\u271d : Nonempty \u03b9\n\u03b5 : \u03b9 \u2192 \u03b1\nh\u03b5 : \u2200 (i : \u03b9), 0 < \u03b5 i\nh : \u2200 (i : \u03b9), x i < (x + \u03b5) i\nhx\u03b5 : \u2200 (i : \u03b9), x i + \u03b5 i = (x + \u03b5) i\n\u22a2 \u2203 \u03b5_1, 0 < \u03b5_1 \u2227 \u2200 (i : \u03b9), x i + \u03b5_1 < (x + \u03b5) i\n[PROOFSTEP]\nhave h\u03b5 : 0 < Finset.univ.inf' Finset.univ_nonempty \u03b5 := (Finset.lt_inf'_iff _).2 fun i _ => h\u03b5 _\n[GOAL]\ncase intro.inr\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : LinearOrderedSemifield \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : Finite \u03b9\nx : \u03b9 \u2192 \u03b1\nval\u271d : Fintype \u03b9\nh\u271d : Nonempty \u03b9\n\u03b5 : \u03b9 \u2192 \u03b1\nh\u03b5\u271d : \u2200 (i : \u03b9), 0 < \u03b5 i\nh : \u2200 (i : \u03b9), x i < (x + \u03b5) i\nhx\u03b5 : \u2200 (i : \u03b9), x i + \u03b5 i = (x + \u03b5) i\nh\u03b5 : 0 < Finset.inf' Finset.univ (_ : Finset.Nonempty Finset.univ) \u03b5\n\u22a2 \u2203 \u03b5_1, 0 < \u03b5_1 \u2227 \u2200 (i : \u03b9), x i + \u03b5_1 < (x + \u03b5) i\n[PROOFSTEP]\nexact \u27e8_, half_pos h\u03b5, fun i => add_lt_add_left ((half_lt_self h\u03b5).trans_le <| Finset.inf'_le _ <| Finset.mem_univ _) _\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Field.Pi", "llama_tokens": 1369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303236047049, "lm_q2_score": 0.7217432122827968, "lm_q1q2_score": 0.5705598951654186}}
{"text": "[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\nR : Type u_4\ninst\u271d : AddGroup R\na b : R\nh : a - b = 0\n\u22a2 a + -b = b + -b\n[PROOFSTEP]\nrw [\u2190 sub_eq_add_neg, h, add_neg_self]\n  -- Porting note:\n  -- This theorem was introduced during ad-hoc porting\n  -- and hopefully can be removed again after `Mathlib.Algebra.Ring.Basic` is fully ported.\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nrcases exists_pair_ne M\u2080 with \u27e8x, y, hx\u27e9\n[GOAL]\ncase intro.intro\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\nx y : M\u2080\nhx : x \u2260 y\n\u22a2 False\n[PROOFSTEP]\napply hx\n[GOAL]\ncase intro.intro\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\nx y : M\u2080\nhx : x \u2260 y\n\u22a2 x = y\n[PROOFSTEP]\ncalc\n  x = 1 * x := by rw [one_mul]\n  _ = 0 := by rw [h, zero_mul]\n  _ = 1 * y := by rw [h, zero_mul]\n  _ = y := by rw [one_mul]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\nx y : M\u2080\nhx : x \u2260 y\n\u22a2 x = 1 * x\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\nx y : M\u2080\nhx : x \u2260 y\n\u22a2 1 * x = 0\n[PROOFSTEP]\nrw [h, zero_mul]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\nx y : M\u2080\nhx : x \u2260 y\n\u22a2 0 = 1 * y\n[PROOFSTEP]\nrw [h, zero_mul]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroOneClass M\u2080\ninst\u271d : Nontrivial M\u2080\na b : M\u2080\nh : 1 = 0\nx y : M\u2080\nhx : x \u2260 y\n\u22a2 1 * y = y\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b3 : MulZeroOneClass M\u2080\ninst\u271d\u00b2 : Nontrivial M\u2080\na b : M\u2080\ninst\u271d\u00b9 : Zero M\u2080'\ninst\u271d : One M\u2080'\nf : M\u2080' \u2192 M\u2080\nzero : f 0 = 0\none : f 1 = 1\n\u22a2 \u00acf 0 = f 1\n[PROOFSTEP]\nrw [zero, one]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b3 : MulZeroOneClass M\u2080\ninst\u271d\u00b2 : Nontrivial M\u2080\na b : M\u2080\ninst\u271d\u00b9 : Zero M\u2080'\ninst\u271d : One M\u2080'\nf : M\u2080' \u2192 M\u2080\nzero : f 0 = 0\none : f 1 = 1\n\u22a2 \u00ac0 = 1\n[PROOFSTEP]\nexact zero_ne_one\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroClass M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na b : M\u2080\n\u22a2 0 = a * b \u2194 a = 0 \u2228 b = 0\n[PROOFSTEP]\nrw [eq_comm, mul_eq_zero]\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroClass M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na b : M\u2080\n\u22a2 a * a = 0 \u2194 a = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nG\u2080 : Type u\nM\u2080 : Type u_1\nM\u2080' : Type u_2\nG\u2080' : Type u_3\ninst\u271d\u00b9 : MulZeroClass M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na b : M\u2080\n\u22a2 0 = a * a \u2194 a = 0\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupWithZero.Defs", "llama_tokens": 1646, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527631, "lm_q2_score": 0.7057850154599562, "lm_q1q2_score": 0.5703217308100207}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\n\u22a2 q = 0 \u2228 IsPrimePow q\n[PROOFSTEP]\napply or_iff_not_imp_left.2\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\n\u22a2 \u00acq = 0 \u2192 IsPrimePow q\n[PROOFSTEP]\nintro q_pos\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nlet K := LocalRing.ResidueField R\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhaveI RM_char := ringChar.charP K\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nlet r := ringChar K\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nlet n := q.factorization r\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\n\u22a2 IsPrimePow q\n[PROOFSTEP]\ncases' CharP.char_is_prime_or_zero K r with r_prime r_zero\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nlet a := q / r ^ n\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave q_eq_a_mul_rn : q = r ^ n * a := by rw [Nat.mul_div_cancel' (Nat.ord_proj_dvd q r)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\n\u22a2 q = r ^ n * a\n[PROOFSTEP]\nrw [Nat.mul_div_cancel' (Nat.ord_proj_dvd q r)]\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave r_ne_dvd_a := Nat.not_dvd_ord_compl r_prime q_pos\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave rn_dvd_q : r ^ n \u2223 q := \u27e8a, q_eq_a_mul_rn\u27e9\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = r ^ n * a\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nrw [mul_comm] at q_eq_a_mul_rn \n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave a_unit : IsUnit (a : R) := by\n  by_contra g\n  rw [\u2190 mem_nonunits_iff] at g \n  rw [\u2190 LocalRing.mem_maximalIdeal] at g \n  have a_cast_zero := Ideal.Quotient.eq_zero_iff_mem.2 g\n  rw [map_natCast] at a_cast_zero \n  have r_dvd_a := (ringChar.spec K a).1 a_cast_zero\n  exact absurd r_dvd_a r_ne_dvd_a\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\n\u22a2 IsUnit \u2191a\n[PROOFSTEP]\nby_contra g\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\ng : \u00acIsUnit \u2191a\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 mem_nonunits_iff] at g \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\ng : \u2191a \u2208 nonunits R\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 LocalRing.mem_maximalIdeal] at g \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\ng\u271d : \u2191a \u2208 nonunits R\ng : \u2191a \u2208 LocalRing.maximalIdeal R\n\u22a2 False\n[PROOFSTEP]\nhave a_cast_zero := Ideal.Quotient.eq_zero_iff_mem.2 g\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\ng\u271d : \u2191a \u2208 nonunits R\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191(Ideal.Quotient.mk (LocalRing.maximalIdeal R)) \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [map_natCast] at a_cast_zero \n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\ng\u271d : \u2191a \u2208 nonunits R\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nhave r_dvd_a := (ringChar.spec K a).1 a_cast_zero\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\ng\u271d : \u2191a \u2208 nonunits R\ng : \u2191a \u2208 LocalRing.maximalIdeal R\na_cast_zero : \u2191a = 0\nr_dvd_a : ringChar K \u2223 a\n\u22a2 False\n[PROOFSTEP]\nexact absurd r_dvd_a r_ne_dvd_a\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\n\u22a2 IsPrimePow q\n[PROOFSTEP]\ncases' a_unit.exists_left_inv with a_inv h_inv_mul_a\n[GOAL]\ncase inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave rn_cast_zero : \u2191(r ^ n) = (0 : R) :=\n  by\n  rw [\u2190 @mul_one R _ (r ^ n), mul_comm, \u2190 Classical.choose_spec a_unit.exists_left_inv, mul_assoc, \u2190 Nat.cast_mul, \u2190\n    q_eq_a_mul_rn, CharP.cast_eq_zero R q]\n  simp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 \u2191(r ^ n) = 0\n[PROOFSTEP]\nrw [\u2190 @mul_one R _ (r ^ n), mul_comm, \u2190 Classical.choose_spec a_unit.exists_left_inv, mul_assoc, \u2190 Nat.cast_mul, \u2190\n  q_eq_a_mul_rn, CharP.cast_eq_zero R q]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\n\u22a2 Classical.choose (_ : \u2203 b, b * \u2191a = 1) * 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave q_eq_rn := Nat.dvd_antisymm ((CharP.cast_eq_zero_iff R q (r ^ n)).mp rn_cast_zero) rn_dvd_q\n[GOAL]\ncase inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave n_pos : n \u2260 0 := fun n_zero =>\n  absurd (by simpa [n_zero] using q_eq_rn)\n    (CharP.char_ne_one R q)\n      -- Definition of prime power: `\u2203 r n, Prime r \u2227 0 < n \u2227 r ^ n = q`.\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\nn_zero : n = 0\n\u22a2 q = 1\n[PROOFSTEP]\nsimpa [n_zero] using q_eq_rn\n[GOAL]\ncase inl.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_prime : Nat.Prime r\na : \u2115 := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : \u00acr \u2223 q / r ^ \u2191(Nat.factorization q) r\nrn_dvd_q : r ^ n \u2223 q\na_unit : IsUnit \u2191a\na_inv : R\nh_inv_mul_a : a_inv * \u2191a = 1\nrn_cast_zero : \u2191(r ^ n) = 0\nq_eq_rn : q = r ^ n\nn_pos : n \u2260 0\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nexact \u27e8r, \u27e8n, \u27e8r_prime.prime, \u27e8pos_iff_ne_zero.mpr n_pos, q_eq_rn.symm\u27e9\u27e9\u27e9\u27e9\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_zero : r = 0\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhaveI K_char_p_0 := ringChar.of_eq r_zero\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhaveI K_char_zero : CharZero K := CharP.charP_to_charZero K\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhaveI R_char_zero :=\n  RingHom.charZero\n    (LocalRing.residue R)\n      -- Finally, `r = 0` would lead to a contradiction:\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\nR_char_zero : CharZero R\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nhave q_zero := CharP.eq R char_R_q (CharP.ofCharZero R)\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nq : \u2115\nchar_R_q : CharP R q\nq_pos : \u00acq = 0\nK : Type u_1 := LocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : \u2115 := ringChar K\nn : (fun x => \u2115) r := \u2191(Nat.factorization q) r\nr_zero : r = 0\nK_char_p_0 : CharP K 0\nK_char_zero : CharZero K\nR_char_zero : CharZero R\nq_zero : q = 0\n\u22a2 IsPrimePow q\n[PROOFSTEP]\nexact absurd q_zero q_pos\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.LocalRing", "llama_tokens": 7824, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835452961425, "lm_q2_score": 0.6825737214979746, "lm_q1q2_score": 0.5702791127631096}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b3\na b c : \u03b1\nh : a \u2264 b\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u2191(Icc a b)\nhs : IsOpen (projIcc a b h \u207b\u00b9' s)\n\u22a2 Subtype.val \u207b\u00b9' (projIcc a b h \u207b\u00b9' s) = s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b3\na b c : \u03b1\nh : a \u2264 b\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u2191(Icc a b)\nhs : IsOpen (projIcc a b h \u207b\u00b9' s)\nx\u271d : { x // x \u2208 Icc a b }\n\u22a2 x\u271d \u2208 Subtype.val \u207b\u00b9' (projIcc a b h \u207b\u00b9' s) \u2194 x\u271d \u2208 s\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.ProjIcc", "llama_tokens": 349, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835330070839, "lm_q2_score": 0.6825737279551494, "lm_q1q2_score": 0.5702791097697844}}
{"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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : V\n\u22a2 affineSegment R x y = segment R x y\n[PROOFSTEP]\nrw [segment_eq_image_lineMap, affineSegment]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 affineSegment R x y = affineSegment R y x\n[PROOFSTEP]\nrefine' Set.ext fun z => _\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\n\u22a2 z \u2208 affineSegment R x y \u2194 z \u2208 affineSegment R 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\n\u22a2 z \u2208 affineSegment R x y \u2192 z \u2208 affineSegment R y x\n[PROOFSTEP]\nrintro \u27e8t, ht, hxy\u27e9\n[GOAL]\ncase mp.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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nt : R\nht : t \u2208 Set.Icc 0 1\nhxy : \u2191(lineMap x y) t = z\n\u22a2 z \u2208 affineSegment R y x\n[PROOFSTEP]\nrefine' \u27e81 - t, _, _\u27e9\n[GOAL]\ncase mp.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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nt : R\nht : t \u2208 Set.Icc 0 1\nhxy : \u2191(lineMap x y) t = z\n\u22a2 1 - t \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrwa [Set.sub_mem_Icc_iff_right, sub_self, sub_zero]\n[GOAL]\ncase mp.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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nt : R\nht : t \u2208 Set.Icc 0 1\nhxy : \u2191(lineMap x y) t = z\n\u22a2 \u2191(lineMap y x) (1 - t) = z\n[PROOFSTEP]\nrwa [lineMap_apply_one_sub]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\n\u22a2 z \u2208 affineSegment R y x \u2192 z \u2208 affineSegment R x y\n[PROOFSTEP]\nrintro \u27e8t, ht, hxy\u27e9\n[GOAL]\ncase mpr.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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nt : R\nht : t \u2208 Set.Icc 0 1\nhxy : \u2191(lineMap y x) t = z\n\u22a2 z \u2208 affineSegment R x y\n[PROOFSTEP]\nrefine' \u27e81 - t, _, _\u27e9\n[GOAL]\ncase mpr.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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nt : R\nht : t \u2208 Set.Icc 0 1\nhxy : \u2191(lineMap y x) t = z\n\u22a2 1 - t \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrwa [Set.sub_mem_Icc_iff_right, sub_self, sub_zero]\n[GOAL]\ncase mpr.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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nt : R\nht : t \u2208 Set.Icc 0 1\nhxy : \u2191(lineMap y x) t = z\n\u22a2 \u2191(lineMap x y) (1 - t) = z\n[PROOFSTEP]\nrwa [lineMap_apply_one_sub]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 affineSegment R x x = {x}\n[PROOFSTEP]\nrw [affineSegment]\n  -- Note: when adding \"simp made no progress\" in lean4#2336,\n    -- had to change `lineMap_same` to `lineMap_same _`. Not sure why?\n    -- porting note: added `_ _` and `Function.const`\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 \u2191(lineMap x x) '' Set.Icc 0 1 = {x}\n[PROOFSTEP]\nsimp_rw [lineMap_same _, AffineMap.coe_const _ _, Function.const, (Set.nonempty_Icc.mpr zero_le_one).image_const]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nf : P \u2192\u1d43[R] P'\nx y : P\n\u22a2 \u2191f '' affineSegment R x y = affineSegment R (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nrw [affineSegment, affineSegment, Set.image_image, \u2190 comp_lineMap]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nf : P \u2192\u1d43[R] P'\nx y : P\n\u22a2 (fun x_1 => \u2191f (\u2191(lineMap x y) x_1)) '' Set.Icc 0 1 = \u2191(comp f (lineMap x y)) '' Set.Icc 0 1\n[PROOFSTEP]\nrfl\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nv : V\n\u22a2 v +\u1d65 z \u2208 affineSegment R (v +\u1d65 x) (v +\u1d65 y) \u2194 z \u2208 affineSegment R x y\n[PROOFSTEP]\nrw [\u2190 affineSegment_const_vadd_image, (AddAction.injective v).mem_set_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : V\np : P\n\u22a2 z +\u1d65 p \u2208 affineSegment R (x +\u1d65 p) (y +\u1d65 p) \u2194 z \u2208 affineSegment R x y\n[PROOFSTEP]\nrw [\u2190 affineSegment_vadd_const_image, (vadd_right_injective p).mem_set_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 p -\u1d65 z \u2208 affineSegment R (p -\u1d65 x) (p -\u1d65 y) \u2194 z \u2208 affineSegment R x y\n[PROOFSTEP]\nrw [\u2190 affineSegment_const_vsub_image, (vsub_right_injective p).mem_set_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 z -\u1d65 p \u2208 affineSegment R (x -\u1d65 p) (y -\u1d65 p) \u2194 z \u2208 affineSegment R x y\n[PROOFSTEP]\nrw [\u2190 affineSegment_vsub_const_image, (vsub_left_injective p).mem_set_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nh : Wbtw R x y z\nf : P \u2192\u1d43[R] P'\n\u22a2 Wbtw R (\u2191f x) (\u2191f y) (\u2191f z)\n[PROOFSTEP]\nrw [Wbtw, \u2190 affineSegment_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nh : Wbtw R x y z\nf : P \u2192\u1d43[R] P'\n\u22a2 \u2191f y \u2208 \u2191f '' affineSegment R x z\n[PROOFSTEP]\nexact Set.mem_image_of_mem _ 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2192\u1d43[R] P'\nhf : Injective \u2191f\n\u22a2 Wbtw R (\u2191f x) (\u2191f y) (\u2191f z) \u2194 Wbtw R x y z\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2192\u1d43[R] P'\nhf : Injective \u2191f\nh : Wbtw R (\u2191f x) (\u2191f y) (\u2191f z)\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrwa [Wbtw, \u2190 affineSegment_image, hf.mem_set_image] 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2192\u1d43[R] P'\nhf : Injective \u2191f\n\u22a2 Sbtw R (\u2191f x) (\u2191f y) (\u2191f z) \u2194 Sbtw R x y z\n[PROOFSTEP]\nsimp_rw [Sbtw, hf.wbtw_map_iff, hf.ne_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2243\u1d43[R] P'\n\u22a2 Wbtw R (\u2191f x) (\u2191f y) (\u2191f z) \u2194 Wbtw R x y z\n[PROOFSTEP]\nrefine' Function.Injective.wbtw_map_iff (_ : Function.Injective f.toAffineMap)\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2243\u1d43[R] P'\n\u22a2 Function.Injective \u2191\u2191f\n[PROOFSTEP]\nexact f.injective\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2243\u1d43[R] P'\n\u22a2 Sbtw R (\u2191f x) (\u2191f y) (\u2191f z) \u2194 Sbtw R x y z\n[PROOFSTEP]\nrefine' Function.Injective.sbtw_map_iff (_ : Function.Injective f.toAffineMap)\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nf : P \u2243\u1d43[R] P'\n\u22a2 Function.Injective \u2191\u2191f\n[PROOFSTEP]\nexact f.injective\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nv : V\n\u22a2 Sbtw R (v +\u1d65 x) (v +\u1d65 y) (v +\u1d65 z) \u2194 Sbtw R x y z\n[PROOFSTEP]\nrw [Sbtw, Sbtw, wbtw_const_vadd_iff, (AddAction.injective v).ne_iff, (AddAction.injective v).ne_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : V\np : P\n\u22a2 Sbtw R (x +\u1d65 p) (y +\u1d65 p) (z +\u1d65 p) \u2194 Sbtw R x y z\n[PROOFSTEP]\nrw [Sbtw, Sbtw, wbtw_vadd_const_iff, (vadd_right_injective p).ne_iff, (vadd_right_injective p).ne_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 Sbtw R (p -\u1d65 x) (p -\u1d65 y) (p -\u1d65 z) \u2194 Sbtw R x y z\n[PROOFSTEP]\nrw [Sbtw, Sbtw, wbtw_const_vsub_iff, (vsub_right_injective p).ne_iff, (vsub_right_injective p).ne_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 Sbtw R (x -\u1d65 p) (y -\u1d65 p) (z -\u1d65 p) \u2194 Sbtw R x y z\n[PROOFSTEP]\nrw [Sbtw, Sbtw, wbtw_vsub_const_iff, (vsub_left_injective p).ne_iff, (vsub_left_injective p).ne_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nh : Sbtw R x y z\n\u22a2 y \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1\n[PROOFSTEP]\nrcases h with \u27e8\u27e8t, ht, rfl\u27e9, hyx, hyz\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nt : R\nht : t \u2208 Set.Icc 0 1\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\n\u22a2 \u2191(lineMap x z) t \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1\n[PROOFSTEP]\nrcases Set.eq_endpoints_or_mem_Ioo_of_mem_Icc ht with (rfl | rfl | ho)\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nht : 0 \u2208 Set.Icc 0 1\nhyx : \u2191(lineMap x z) 0 \u2260 x\nhyz : \u2191(lineMap x z) 0 \u2260 z\n\u22a2 \u2191(lineMap x z) 0 \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase intro.intro.intro.intro.inl.h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nht : 0 \u2208 Set.Icc 0 1\nhyx : \u2191(lineMap x z) 0 \u2260 x\nhyz : \u2191(lineMap x z) 0 \u2260 z\n\u22a2 False\n[PROOFSTEP]\nexact hyx (lineMap_apply_zero _ _)\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nht : 1 \u2208 Set.Icc 0 1\nhyx : \u2191(lineMap x z) 1 \u2260 x\nhyz : \u2191(lineMap x z) 1 \u2260 z\n\u22a2 \u2191(lineMap x z) 1 \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase intro.intro.intro.intro.inr.inl.h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nht : 1 \u2208 Set.Icc 0 1\nhyx : \u2191(lineMap x z) 1 \u2260 x\nhyz : \u2191(lineMap x z) 1 \u2260 z\n\u22a2 False\n[PROOFSTEP]\nexact hyz (lineMap_apply_one _ _)\n[GOAL]\ncase intro.intro.intro.intro.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nt : R\nht : t \u2208 Set.Icc 0 1\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\nho : t \u2208 Set.Ioo 0 1\n\u22a2 \u2191(lineMap x z) t \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1\n[PROOFSTEP]\nexact \u27e8t, ho, 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nh : Wbtw R x y z\n\u22a2 y \u2208 affineSpan R {x, z}\n[PROOFSTEP]\nrcases h with \u27e8r, \u27e8-, rfl\u27e9\u27e9\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 z : P\nr : R\n\u22a2 \u2191(lineMap x z) r \u2208 affineSpan R {x, z}\n[PROOFSTEP]\nexact lineMap_mem_affineSpan_pair _ _ _\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\n\u22a2 Wbtw R x y z \u2194 Wbtw R z y x\n[PROOFSTEP]\nrw [Wbtw, Wbtw, affineSegment_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\n\u22a2 Sbtw R x y z \u2194 Sbtw R z y x\n[PROOFSTEP]\nrw [Sbtw, Sbtw, wbtw_comm, \u2190 and_assoc, \u2190 and_assoc, and_right_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\n\u22a2 Wbtw R x y x \u2194 y = x\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nh : Wbtw R x y x\n\u22a2 y = x\n[PROOFSTEP]\nhave \u27e8_, _, h\u2082\u27e9 := h\n[GOAL]\ncase 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nh : Wbtw R x y x\nw\u271d : R\nleft\u271d : w\u271d \u2208 Set.Icc 0 1\nh\u2082 : \u2191(lineMap x x) w\u271d = y\n\u22a2 y = x\n[PROOFSTEP]\nrw [h\u2082.symm, lineMap_same_apply]\n[GOAL]\ncase 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nh : y = x\n\u22a2 Wbtw R x y x\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nh : y = x\n\u22a2 Wbtw R x x x\n[PROOFSTEP]\nexact wbtw_self_left R x 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\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nh : Wbtw R x y z\nhne : y \u2260 x\n\u22a2 x \u2260 z\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nhne : y \u2260 x\nh : Wbtw R x y x\n\u22a2 False\n[PROOFSTEP]\nrw [wbtw_self_iff] 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nhne : y \u2260 x\nh : y = x\n\u22a2 False\n[PROOFSTEP]\nexact hne 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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\nh : Wbtw R x y z\nhne : y \u2260 z\n\u22a2 x \u2260 z\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nh : Wbtw R x y x\nhne : y \u2260 x\n\u22a2 False\n[PROOFSTEP]\nrw [wbtw_self_iff] 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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : P\nh : y = x\nhne : y \u2260 x\n\u22a2 False\n[PROOFSTEP]\nexact hne 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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Sbtw R x y z \u2194 y \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1 \u2227 x \u2260 z\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h.mem_image_Ioo, h.left_ne_right\u27e9, fun 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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\nh : y \u2208 \u2191(lineMap x z) '' Set.Ioo 0 1 \u2227 x \u2260 z\n\u22a2 Sbtw R x y z\n[PROOFSTEP]\nrcases h with \u27e8\u27e8t, ht, rfl\u27e9, hxz\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nhxz : x \u2260 z\nt : R\nht : t \u2208 Set.Ioo 0 1\n\u22a2 Sbtw R x (\u2191(lineMap x z) t) z\n[PROOFSTEP]\nrefine' \u27e8\u27e8t, Set.mem_Icc_of_Ioo ht, rfl\u27e9, _\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nhxz : x \u2260 z\nt : R\nht : t \u2208 Set.Ioo 0 1\n\u22a2 \u2191(lineMap x z) t \u2260 x \u2227 \u2191(lineMap x z) t \u2260 z\n[PROOFSTEP]\nrw [lineMap_apply, \u2190 @vsub_ne_zero V, \u2190 @vsub_ne_zero V _ _ _ _ z, vadd_vsub_assoc, vsub_self, vadd_vsub_assoc, \u2190\n  neg_vsub_eq_vsub_rev z x, \u2190 @neg_one_smul R, \u2190 add_smul, \u2190 sub_eq_add_neg]\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nhxz : x \u2260 z\nt : R\nht : t \u2208 Set.Ioo 0 1\n\u22a2 t \u2022 (z -\u1d65 x) + 0 \u2260 0 \u2227 (t - 1) \u2022 (z -\u1d65 x) \u2260 0\n[PROOFSTEP]\nsimp [smul_ne_zero, sub_eq_zero, ht.1.ne.symm, ht.2.ne, hxz.symm]\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R x y z \u2227 Wbtw R y x z \u2194 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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R x y z \u2227 Wbtw R y x z \u2192 x = y\n[PROOFSTEP]\nrintro \u27e8hxyz, hyxz\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\nhxyz : Wbtw R x y z\nhyxz : Wbtw R y x z\n\u22a2 x = y\n[PROOFSTEP]\nrcases hxyz with \u27e8ty, hty, rfl\u27e9\n[GOAL]\ncase mp.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\nhyxz : Wbtw R (\u2191(lineMap x z) ty) x z\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrcases hyxz with \u27e8tx, htx, hx\u27e9\n[GOAL]\ncase mp.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nhx : \u2191(lineMap (\u2191(lineMap x z) ty) z) tx = x\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrw [lineMap_apply, lineMap_apply, \u2190 add_vadd] at hx \n[GOAL]\ncase mp.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nhx : tx \u2022 (z -\u1d65 (ty \u2022 (z -\u1d65 x) +\u1d65 x)) + ty \u2022 (z -\u1d65 x) +\u1d65 x = x\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrw [\u2190 @vsub_eq_zero_iff_eq V, vadd_vsub, vsub_vadd_eq_vsub_sub, smul_sub, smul_smul, \u2190 sub_smul, \u2190 add_smul,\n  smul_eq_zero] at hx \n[GOAL]\ncase mp.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nhx : tx - tx * ty + ty = 0 \u2228 z -\u1d65 x = 0\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrcases hx with (h | h)\n[GOAL]\ncase mp.intro.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : tx - tx * ty + ty = 0\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nnth_rw 1 [\u2190 mul_one tx] at h \n[GOAL]\ncase mp.intro.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : tx * 1 - tx * ty + ty = 0\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrw [\u2190 mul_sub, add_eq_zero_iff_neg_eq] at h \n[GOAL]\ncase mp.intro.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : -(tx * (1 - ty)) = ty\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nhave h' : ty = 0 := by\n  refine' le_antisymm _ hty.1\n  rw [\u2190 h, Left.neg_nonpos_iff]\n  exact mul_nonneg htx.1 (sub_nonneg.2 hty.2)\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : -(tx * (1 - ty)) = ty\n\u22a2 ty = 0\n[PROOFSTEP]\nrefine' le_antisymm _ hty.1\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : -(tx * (1 - ty)) = ty\n\u22a2 ty \u2264 0\n[PROOFSTEP]\nrw [\u2190 h, Left.neg_nonpos_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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : -(tx * (1 - ty)) = ty\n\u22a2 0 \u2264 tx * (1 - ty)\n[PROOFSTEP]\nexact mul_nonneg htx.1 (sub_nonneg.2 hty.2)\n[GOAL]\ncase mp.intro.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : -(tx * (1 - ty)) = ty\nh' : ty = 0\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nsimp [h']\n[GOAL]\ncase mp.intro.intro.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : z -\u1d65 x = 0\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase mp.intro.intro.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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\nty : R\nhty : ty \u2208 Set.Icc 0 1\ntx : R\nhtx : tx \u2208 Set.Icc 0 1\nh : z = x\n\u22a2 x = \u2191(lineMap x z) ty\n[PROOFSTEP]\nrw [h, lineMap_same_apply]\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 x = y \u2192 Wbtw R x y z \u2227 Wbtw R y x z\n[PROOFSTEP]\nrintro rfl\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx z : P\n\u22a2 Wbtw R x x z \u2227 Wbtw R x x z\n[PROOFSTEP]\nexact \u27e8wbtw_self_left _ _ _, wbtw_self_left _ _ _\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R x y z \u2227 Wbtw R x z y \u2194 y = z\n[PROOFSTEP]\nrw [wbtw_comm, wbtw_comm (z := y), 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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R z y x \u2227 Wbtw R y z x \u2194 z = y\n[PROOFSTEP]\nexact wbtw_swap_left_iff 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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R x y z \u2227 Wbtw R z x y \u2194 x = y\n[PROOFSTEP]\nrw [wbtw_comm, wbtw_swap_right_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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\nh : Wbtw R x y z\n\u22a2 Wbtw R y x z \u2194 x = y\n[PROOFSTEP]\nrw [\u2190 wbtw_swap_left_iff R z, and_iff_right 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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\nh : Wbtw R x y z\n\u22a2 Wbtw R x z y \u2194 y = z\n[PROOFSTEP]\nrw [\u2190 wbtw_swap_right_iff R x, and_iff_right 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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\nh : Wbtw R x y z\n\u22a2 Wbtw R z x y \u2194 x = y\n[PROOFSTEP]\nrw [\u2190 wbtw_rotate_iff R x, and_iff_right 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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\n\u22a2 Wbtw R x (\u2191(lineMap x y) r) y \u2194 x = y \u2228 r \u2208 Set.Icc 0 1\n[PROOFSTEP]\nby_cases hxy : x = y\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x = y\n\u22a2 Wbtw R x (\u2191(lineMap x y) r) y \u2194 x = y \u2228 r \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrw [hxy, lineMap_same_apply]\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x = y\n\u22a2 Wbtw R y y y \u2194 y = y \u2228 r \u2208 Set.Icc 0 1\n[PROOFSTEP]\nsimp\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : \u00acx = y\n\u22a2 Wbtw R x (\u2191(lineMap x y) r) y \u2194 x = y \u2228 r \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrw [or_iff_right hxy, Wbtw, affineSegment, (lineMap_injective R hxy).mem_set_image]\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\n\u22a2 Sbtw R x (\u2191(lineMap x y) r) y \u2194 x \u2260 y \u2227 r \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nrw [sbtw_iff_mem_image_Ioo_and_ne, and_comm, and_congr_right]\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\n\u22a2 x \u2260 y \u2192 (\u2191(lineMap x y) r \u2208 \u2191(lineMap x y) '' Set.Ioo 0 1 \u2194 r \u2208 Set.Ioo 0 1)\n[PROOFSTEP]\nintro hxy\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y : P\nr : R\nhxy : x \u2260 y\n\u22a2 \u2191(lineMap x y) r \u2208 \u2191(lineMap x y) '' Set.Ioo 0 1 \u2194 r \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nrw [(lineMap_injective R hxy).mem_set_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 Wbtw R 0 x 1 \u2194 x \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrw [Wbtw, affineSegment, Set.mem_image]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 (\u2203 x_1, x_1 \u2208 Set.Icc 0 1 \u2227 \u2191(lineMap 0 1) x_1 = x) \u2194 x \u2208 Set.Icc 0 1\n[PROOFSTEP]\nsimp_rw [lineMap_apply_ring]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 (\u2203 x_1, x_1 \u2208 Set.Icc 0 1 \u2227 (1 - x_1) * 0 + x_1 * 1 = x) \u2194 x \u2208 Set.Icc 0 1\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 Wbtw R 1 x 0 \u2194 x \u2208 Set.Icc 0 1\n[PROOFSTEP]\nrw [wbtw_comm, wbtw_zero_one_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 Sbtw R 0 x 1 \u2194 x \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nrw [Sbtw, wbtw_zero_one_iff, Set.mem_Icc, Set.mem_Ioo]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 (0 \u2264 x \u2227 x \u2264 1) \u2227 x \u2260 0 \u2227 x \u2260 1 \u2194 0 < x \u2227 x < 1\n[PROOFSTEP]\nexact \u27e8fun h => \u27e8h.1.1.lt_of_ne (Ne.symm h.2.1), h.1.2.lt_of_ne h.2.2\u27e9, fun h => \u27e8\u27e8h.1.le, h.2.le\u27e9, h.1.ne', h.2.ne\u27e9\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module 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 : R\n\u22a2 Sbtw R 1 x 0 \u2194 x \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nrw [sbtw_comm, sbtw_zero_one_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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nw x y z : P\nh\u2081 : Wbtw R w y z\nh\u2082 : Wbtw R w x y\n\u22a2 Wbtw R w x z\n[PROOFSTEP]\nrcases h\u2081 with \u27e8t\u2081, ht\u2081, rfl\u27e9\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nw x z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nh\u2082 : Wbtw R w x (\u2191(lineMap w z) t\u2081)\n\u22a2 Wbtw R w x z\n[PROOFSTEP]\nrcases h\u2082 with \u27e8t\u2082, ht\u2082, rfl\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 Wbtw R w (\u2191(lineMap w (\u2191(lineMap w z) t\u2081)) t\u2082) z\n[PROOFSTEP]\nrefine' \u27e8t\u2082 * t\u2081, \u27e8mul_nonneg ht\u2082.1 ht\u2081.1, mul_le_one ht\u2082.2 ht\u2081.1 ht\u2081.2\u27e9, _\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 \u2191(lineMap w z) (t\u2082 * t\u2081) = \u2191(lineMap w (\u2191(lineMap w z) t\u2081)) t\u2082\n[PROOFSTEP]\nrw [lineMap_apply, lineMap_apply, lineMap_vsub_left, smul_smul]\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nw x y z : P\nh\u2081 : Wbtw R w x z\nh\u2082 : Wbtw R x y z\n\u22a2 Wbtw R w y z\n[PROOFSTEP]\nrw [wbtw_comm] at *\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 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Wbtw R z y x\n\u22a2 Wbtw R z y w\n[PROOFSTEP]\nexact h\u2081.trans_left h\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R w y z\nh\u2082 : Sbtw R w x y\n\u22a2 Sbtw R w x z\n[PROOFSTEP]\nrefine' \u27e8h\u2081.trans_left h\u2082.wbtw, h\u2082.ne_left, _\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R w y z\nh\u2082 : Sbtw R w x y\n\u22a2 x \u2260 z\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y : P\nh\u2082 : Sbtw R w x y\nh\u2081 : Wbtw R w y x\n\u22a2 False\n[PROOFSTEP]\nexact h\u2082.right_ne ((wbtw_swap_right_iff R w).1 \u27e8h\u2081, h\u2082.wbtw\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R w x z\nh\u2082 : Sbtw R x y z\n\u22a2 Sbtw R w y z\n[PROOFSTEP]\nrw [wbtw_comm] at *\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Sbtw R x y z\n\u22a2 Sbtw R w y z\n[PROOFSTEP]\nrw [sbtw_comm] at *\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 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Sbtw R z y x\n\u22a2 Sbtw R z y w\n[PROOFSTEP]\nexact h\u2081.trans_sbtw_left h\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R w y z\nh\u2082 : Wbtw R w x y\nh : y \u2260 z\n\u22a2 x \u2260 z\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y : P\nh\u2082 : Wbtw R w x y\nh\u2081 : Wbtw R w y x\nh : y \u2260 x\n\u22a2 False\n[PROOFSTEP]\nexact h (h\u2081.swap_right_iff.1 h\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R w x z\nh\u2082 : Wbtw R x y z\nh : w \u2260 x\n\u22a2 w \u2260 y\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\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x z : P\nh\u2081 : Wbtw R w x z\nh : w \u2260 x\nh\u2082 : Wbtw R x w z\n\u22a2 False\n[PROOFSTEP]\nexact h (h\u2081.swap_left_iff.1 h\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nh :\n  \u2191(Finset.affineCombination R s p) w \u2208\n    affineSpan R {\u2191(Finset.affineCombination R s p) w\u2081, \u2191(Finset.affineCombination R s p) w\u2082}\ni : \u03b9\nhis : i \u2208 s\nhs : Sbtw R (w\u2081 i) (w i) (w\u2082 i)\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081) (\u2191(Finset.affineCombination R s p) w)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\nrw [affineCombination_mem_affineSpan_pair ha hw hw\u2081 hw\u2082] 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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\nh : \u2203 r, \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\ni : \u03b9\nhis : i \u2208 s\nhs : Sbtw R (w\u2081 i) (w i) (w\u2082 i)\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081) (\u2191(Finset.affineCombination R s p) w)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\nrcases h with \u27e8r, hr\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nhs : Sbtw R (w\u2081 i) (w i) (w\u2082 i)\nr : R\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081) (\u2191(Finset.affineCombination R s p) w)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\nrw [hr i his, sbtw_mul_sub_add_iff] at hs \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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = r * (w\u2082 i - w\u2081 i) + w\u2081 i\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081) (\u2191(Finset.affineCombination R s p) w)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\nchange \u2200 i \u2208 s, w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i at hr \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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081) (\u2191(Finset.affineCombination R s p) w)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\nrw [s.affineCombination_congr hr fun _ _ => rfl]\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081)\n    (\u2191(Finset.affineCombination R s fun x => p x) fun i => (r \u2022 (w\u2082 - w\u2081) + w\u2081) i)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\ndsimp only\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R s p) w\u2081)\n    (\u2191(Finset.affineCombination R s fun x => p x) fun i => (r \u2022 (w\u2082 - w\u2081) + w\u2081) i)\n    (\u2191(Finset.affineCombination R s p) w\u2082)\n[PROOFSTEP]\nrw [\u2190 s.weightedVSub_vadd_affineCombination, s.weightedVSub_const_smul, \u2190 s.affineCombination_vsub, \u2190 lineMap_apply,\n  sbtw_lineMap_iff, and_iff_left hs.2, \u2190 @vsub_ne_zero V, s.affineCombination_vsub]\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\n\u22a2 \u2191(Finset.weightedVSub s p) (w\u2081 - w\u2082) \u2260 0\n[PROOFSTEP]\nintro hz\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\nhz : \u2191(Finset.weightedVSub s p) (w\u2081 - w\u2082) = 0\n\u22a2 False\n[PROOFSTEP]\nhave hw\u2081w\u2082 : (\u2211 i in s, (w\u2081 - w\u2082) i) = 0 := by simp_rw [Pi.sub_apply, Finset.sum_sub_distrib, hw\u2081, hw\u2082, sub_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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\nhz : \u2191(Finset.weightedVSub s p) (w\u2081 - w\u2082) = 0\n\u22a2 \u2211 i in s, (w\u2081 - w\u2082) i = 0\n[PROOFSTEP]\nsimp_rw [Pi.sub_apply, Finset.sum_sub_distrib, hw\u2081, hw\u2082, sub_self]\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\nhz : \u2191(Finset.weightedVSub s p) (w\u2081 - w\u2082) = 0\nhw\u2081w\u2082 : \u2211 i in s, (w\u2081 - w\u2082) i = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' hs.1 _\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\nhz : \u2191(Finset.weightedVSub s p) (w\u2081 - w\u2082) = 0\nhw\u2081w\u2082 : \u2211 i in s, (w\u2081 - w\u2082) i = 0\n\u22a2 w\u2081 i = w\u2082 i\n[PROOFSTEP]\nhave ha' := ha s (w\u2081 - w\u2082) hw\u2081w\u2082 hz i his\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\u2078 : OrderedRing R\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module R V\ninst\u271d\u2075 : AddTorsor V P\ninst\u271d\u2074 : AddCommGroup V'\ninst\u271d\u00b3 : Module R V'\ninst\u271d\u00b2 : AddTorsor V' P'\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : NoZeroSMulDivisors R V\n\u03b9 : Type u_6\np : \u03b9 \u2192 P\nha : AffineIndependent R p\nw w\u2081 w\u2082 : \u03b9 \u2192 R\ns : Finset \u03b9\nhw : \u2211 i in s, w i = 1\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\ni : \u03b9\nhis : i \u2208 s\nr : R\nhs : w\u2081 i \u2260 w\u2082 i \u2227 r \u2208 Set.Ioo 0 1\nhr : \u2200 (i : \u03b9), i \u2208 s \u2192 w i = (r \u2022 (w\u2082 - w\u2081) + w\u2081) i\nhz : \u2191(Finset.weightedVSub s p) (w\u2081 - w\u2082) = 0\nhw\u2081w\u2082 : \u2211 i in s, (w\u2081 - w\u2082) i = 0\nha' : (w\u2081 - w\u2082) i = 0\n\u22a2 w\u2081 i = w\u2082 i\n[PROOFSTEP]\nrwa [Pi.sub_apply, sub_eq_zero] at ha' \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 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 SameRay R (y -\u1d65 x) (z -\u1d65 y)\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\n\u22a2 SameRay R (\u2191(lineMap x z) t -\u1d65 x) (z -\u1d65 \u2191(lineMap x z) t)\n[PROOFSTEP]\nsimp_rw [lineMap_apply]\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\n\u22a2 SameRay R (t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (t \u2022 (z -\u1d65 x) +\u1d65 x))\n[PROOFSTEP]\nrcases ht0.lt_or_eq with (ht0' | 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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nht0' : 0 < t\n\u22a2 SameRay R (t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (t \u2022 (z -\u1d65 x) +\u1d65 x))\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nht0 : 0 \u2264 0\nht1 : 0 \u2264 1\n\u22a2 SameRay R (0 \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (0 \u2022 (z -\u1d65 x) +\u1d65 x))\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nht0 : 0 \u2264 0\nht1 : 0 \u2264 1\n\u22a2 SameRay R (0 \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (0 \u2022 (z -\u1d65 x) +\u1d65 x))\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nht0' : 0 < t\n\u22a2 SameRay R (t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (t \u2022 (z -\u1d65 x) +\u1d65 x))\n[PROOFSTEP]\nrcases ht1.lt_or_eq with (ht1' | 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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nht0' : 0 < t\nht1' : t < 1\n\u22a2 SameRay R (t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (t \u2022 (z -\u1d65 x) +\u1d65 x))\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nht0 : 0 \u2264 1\nht1 : 1 \u2264 1\nht0' : 0 < 1\n\u22a2 SameRay R (1 \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (1 \u2022 (z -\u1d65 x) +\u1d65 x))\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nht0 : 0 \u2264 1\nht1 : 1 \u2264 1\nht0' : 0 < 1\n\u22a2 SameRay R (1 \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (1 \u2022 (z -\u1d65 x) +\u1d65 x))\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nht0' : 0 < t\nht1' : t < 1\n\u22a2 SameRay R (t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) (z -\u1d65 (t \u2022 (z -\u1d65 x) +\u1d65 x))\n[PROOFSTEP]\nrefine' Or.inr (Or.inr \u27e81 - t, t, sub_pos.2 ht1', ht0', _\u27e9)\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nht0' : 0 < t\nht1' : t < 1\n\u22a2 (1 - t) \u2022 (t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 x) = t \u2022 (z -\u1d65 (t \u2022 (z -\u1d65 x) +\u1d65 x))\n[PROOFSTEP]\nsimp [vsub_vadd_eq_vsub_sub, smul_sub, smul_smul, \u2190 sub_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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nht0' : 0 < t\nht1' : t < 1\n\u22a2 ((1 - t) * t) \u2022 (z -\u1d65 x) = (t - t * t) \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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 SameRay R (y -\u1d65 x) (z -\u1d65 x)\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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nht0 : 0 \u2264 t\nright\u271d : t \u2264 1\n\u22a2 SameRay R (\u2191(lineMap x z) t -\u1d65 x) (z -\u1d65 x)\n[PROOFSTEP]\nsimpa [lineMap_apply] using SameRay.sameRay_nonneg_smul_left (z -\u1d65 x) 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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 SameRay R (z -\u1d65 x) (z -\u1d65 y)\n[PROOFSTEP]\nrcases h with \u27e8t, \u27e8_, 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\u00b3 : StrictOrderedCommRing R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\nleft\u271d : 0 \u2264 t\nht1 : t \u2264 1\n\u22a2 SameRay R (z -\u1d65 x) (z -\u1d65 \u2191(lineMap x z) t)\n[PROOFSTEP]\nsimpa [lineMap_apply, vsub_vadd_eq_vsub_sub, sub_smul] using\n  SameRay.sameRay_nonneg_smul_right (z -\u1d65 x) (sub_nonneg.2 ht1)\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nletI : DecidableRel ((\u00b7 < \u00b7) : R \u2192 R \u2192 Prop) := LinearOrderedRing.decidableLT\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave h\u2081\u2083 : i\u2081 \u2260 i\u2083 := by\n  rintro rfl\n  simp at h\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\n\u22a2 i\u2081 \u2260 i\u2083\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2081)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2081)\n\u22a2 False\n[PROOFSTEP]\nsimp at h\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave h\u2082\u2083 : i\u2082 \u2260 i\u2083 := by\n  rintro rfl\n  simp at h\u2081 \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\n\u22a2 i\u2082 \u2260 i\u2083\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2082)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2082)\nh\u2081\u2083 : i\u2081 \u2260 i\u2082\n\u22a2 False\n[PROOFSTEP]\nsimp at h\u2081 \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave h3 : \u2200 i : Fin 3, i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083 :=\n  by\n  clear h\u2081 h\u2082 h\u2081' h\u2082'\n  intro i\n  fin_cases i <;> fin_cases i\u2081 <;> fin_cases i\u2082 <;> fin_cases i\u2083 <;> simp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\n\u22a2 \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\n[PROOFSTEP]\nclear h\u2081 h\u2082 h\u2081' h\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\n\u22a2 \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\ni : Fin 3\n\u22a2 i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = i\u2081 \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = i\u2082 \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2081\n[GOAL]\ncase tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2081 \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2082 \u2228 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2081\n[GOAL]\ncase tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2081 \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2082 \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2081\n[GOAL]\ncase head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2082\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = i\u2082 \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2082\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = i\u2082 \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2082\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = i\u2082 \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2082\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2082 \u2228 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2082\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2082 \u2228 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2082\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2082 \u2228 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2082\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2082 \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2082\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2082 \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2082\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2082 \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228 { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = i\u2083\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 0, isLt := (_ : 0 < 3) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) } = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 0, isLt := (_ : 0 < 3) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } =\n        { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave hu : (Finset.univ : Finset (Fin 3)) = { i\u2081, i\u2082, i\u2083 } :=\n  by\n  clear h\u2081 h\u2082 h\u2081' h\u2082'\n  fin_cases i\u2081 <;> fin_cases i\u2082 <;> fin_cases i\u2083 <;> simp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\n\u22a2 Finset.univ = {i\u2081, i\u2082, i\u2083}\n[PROOFSTEP]\nclear h\u2081 h\u2082 h\u2081' h\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\n\u22a2 Finset.univ = {i\u2081, i\u2082, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2081\n[GOAL]\ncase head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2082\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = i\u2082 \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 0, isLt := (_ : 0 < 3) }, i\u2082, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2082\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = i\u2082 \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, i\u2082, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2082\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = i\u2082 \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, i\u2082, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2082\n[GOAL]\ncase head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 : \u2200 (i : Fin 3), i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 0, isLt := (_ : 0 < 3) }, { val := 0, isLt := (_ : 0 < 3) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 : \u2200 (i : Fin 3), i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 0, isLt := (_ : 0 < 3) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 0, isLt := (_ : 0 < 3) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 : \u2200 (i : Fin 3), i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 0, isLt := (_ : 0 < 3) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = i\u2083\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = i\u2083\n\u22a2 Finset.univ = {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 i\u2083\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = i\u2083\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2083 : Fin 3\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 i\u2083\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = i\u2083\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, i\u2083}\n[PROOFSTEP]\nfin_cases i\u2083\n[GOAL]\ncase head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 0, isLt := (_ : 0 < 3) }, { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 h\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n        i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n        i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 0, isLt := (_ : 0 < 3) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n        i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 0, isLt := (_ : 0 < 3) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 h\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n        i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n        i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n        i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 1, isLt := (_ : (fun a => a < 3) 1) }, { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 0, isLt := (_ : 0 < 3) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 0, isLt := (_ : 0 < 3) } \u2228 i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228 i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh\u2081\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2082\u2083 : { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2260 { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 1, isLt := (_ : (fun a => a < 3) 1) } \u2228\n        i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 0, isLt := (_ : 0 < 3) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228 i = { val := 0, isLt := (_ : 0 < 3) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 0, isLt := (_ : 0 < 3) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh\u2081\u2083 h\u2082\u2083 : { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 3) 1) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n        i = { val := 1, isLt := (_ : (fun a => a < 3) 1) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }, { val := 1, isLt := (_ : (fun a => a < 3) 1) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\ncase tail.tail.head.tail.tail.head.tail.tail.head\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\np\u2081 p\u2082 p : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2082 h\u2081\u2083 h\u2082\u2083 :\n  { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2260\n    { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\nh3 :\n  \u2200 (i : Fin 3),\n    i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n      i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) } \u2228\n        i = { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }\n\u22a2 Finset.univ =\n    {{ val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) },\n      { val := 2, isLt := (_ : (fun a => (fun a => a < 3) a) 2) }}\n[PROOFSTEP]\nsimp at h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 \u22a2\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave hp : p \u2208 affineSpan R (Set.range t.points) :=\n  by\n  have hle : line[R, t.points i\u2081, p\u2081] \u2264 affineSpan R (Set.range t.points) :=\n    by\n    refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _\n    have hle : line[R, t.points i\u2082, t.points i\u2083] \u2264 affineSpan R (Set.range t.points) :=\n      by\n      refine' affineSpan_mono R _\n      simp [Set.insert_subset_iff]\n    rw [AffineSubspace.le_def'] at hle \n    exact hle _ h\u2081.wbtw.mem_affineSpan\n  rw [AffineSubspace.le_def'] at hle \n  exact hle _ h\u2081'\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\n\u22a2 p \u2208 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nhave hle : line[R, t.points i\u2081, p\u2081] \u2264 affineSpan R (Set.range t.points) :=\n  by\n  refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _\n  have hle : line[R, t.points i\u2082, t.points i\u2083] \u2264 affineSpan R (Set.range t.points) :=\n    by\n    refine' affineSpan_mono R _\n    simp [Set.insert_subset_iff]\n  rw [AffineSubspace.le_def'] at hle \n  exact hle _ h\u2081.wbtw.mem_affineSpan\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\n\u22a2 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081} \u2264 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nrefine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\n\u22a2 p\u2081 \u2208 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nhave hle : line[R, t.points i\u2082, t.points i\u2083] \u2264 affineSpan R (Set.range t.points) :=\n  by\n  refine' affineSpan_mono R _\n  simp [Set.insert_subset_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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\n\u22a2 affineSpan R {Affine.Simplex.points t i\u2082, Affine.Simplex.points t i\u2083} \u2264 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nrefine' affineSpan_mono R _\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\n\u22a2 {Affine.Simplex.points t i\u2082, Affine.Simplex.points t i\u2083} \u2286 Set.range t.points\n[PROOFSTEP]\nsimp [Set.insert_subset_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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhle : affineSpan R {Affine.Simplex.points t i\u2082, Affine.Simplex.points t i\u2083} \u2264 affineSpan R (Set.range t.points)\n\u22a2 p\u2081 \u2208 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nrw [AffineSubspace.le_def'] at hle \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhle :\n  \u2200 (p : P),\n    p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, Affine.Simplex.points t i\u2083} \u2192 p \u2208 affineSpan R (Set.range t.points)\n\u22a2 p\u2081 \u2208 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nexact hle _ h\u2081.wbtw.mem_affineSpan\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhle : affineSpan R {Affine.Simplex.points t i\u2081, p\u2081} \u2264 affineSpan R (Set.range t.points)\n\u22a2 p \u2208 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nrw [AffineSubspace.le_def'] at hle \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhle : \u2200 (p : P), p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081} \u2192 p \u2208 affineSpan R (Set.range t.points)\n\u22a2 p \u2208 affineSpan R (Set.range t.points)\n[PROOFSTEP]\nexact hle _ h\u2081'\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhp : p \u2208 affineSpan R (Set.range t.points)\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave h\u2081i := h\u2081.mem_image_Ioo\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhp : p \u2208 affineSpan R (Set.range t.points)\nh\u2081i : p\u2081 \u2208 \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) '' Set.Ioo 0 1\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nhave h\u2082i := h\u2082.mem_image_Ioo\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhp : p \u2208 affineSpan R (Set.range t.points)\nh\u2081i : p\u2081 \u2208 \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) '' Set.Ioo 0 1\nh\u2082i : p\u2082 \u2208 \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) '' Set.Ioo 0 1\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nrw [Set.mem_image] at h\u2081i h\u2082i \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2081 p\u2082 p : P\nh\u2081 : Sbtw R (Affine.Simplex.points t i\u2082) p\u2081 (Affine.Simplex.points t i\u2083)\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2081' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, p\u2081}\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhp : p \u2208 affineSpan R (Set.range t.points)\nh\u2081i : \u2203 x, x \u2208 Set.Ioo 0 1 \u2227 \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) x = p\u2081\nh\u2082i : \u2203 x, x \u2208 Set.Ioo 0 1 \u2227 \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) x = p\u2082\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p p\u2081\n[PROOFSTEP]\nrcases h\u2081i with \u27e8r\u2081, \u27e8hr\u20810, hr\u20811\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np\u2082 p : P\nh\u2082 : Sbtw R (Affine.Simplex.points t i\u2081) p\u2082 (Affine.Simplex.points t i\u2083)\nh\u2082' : p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, p\u2082}\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhp : p \u2208 affineSpan R (Set.range t.points)\nh\u2082i : \u2203 x, x \u2208 Set.Ioo 0 1 \u2227 \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) x = p\u2082\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nh\u2081' :\n  p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081}\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n[PROOFSTEP]\nrcases h\u2082i with \u27e8r\u2082, \u27e8hr\u20820, hr\u20821\u27e9, 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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\np : P\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nhp : p \u2208 affineSpan R (Set.range t.points)\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nh\u2081' :\n  p \u2208 affineSpan R {Affine.Simplex.points t i\u2081, \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081}\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nh\u2082' :\n  p \u2208 affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) p (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n[PROOFSTEP]\nrcases eq_affineCombination_of_mem_affineSpan_of_fintype hp with \u27e8w, hw, 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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2081, \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(Finset.affineCombination R Finset.univ t.points) w)\n    (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n[PROOFSTEP]\nhave h\u2081s :=\n  sign_eq_of_affineCombination_mem_affineSpan_single_lineMap t.Independent hw (Finset.mem_univ _) (Finset.mem_univ _)\n    (Finset.mem_univ _) h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 hr\u20810 hr\u20811 h\u2081'\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2081, \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(Finset.affineCombination R Finset.univ t.points) w)\n    (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n[PROOFSTEP]\nhave h\u2082s :=\n  sign_eq_of_affineCombination_mem_affineSpan_single_lineMap t.Independent hw (Finset.mem_univ _) (Finset.mem_univ _)\n    (Finset.mem_univ _) h\u2081\u2082.symm h\u2082\u2083 h\u2081\u2083 hr\u20820 hr\u20821 h\u2082'\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2081, \u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\n\u22a2 Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(Finset.affineCombination R Finset.univ t.points) w)\n    (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n[PROOFSTEP]\nrw [\u2190 Finset.univ.affineCombination_affineCombinationSingleWeights R t.points (Finset.mem_univ i\u2081), \u2190\n  Finset.univ.affineCombination_affineCombinationLineMapWeights t.points (Finset.mem_univ _) (Finset.mem_univ _)] at h\u2081'\n  \u22a2\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\n\u22a2 Sbtw R (\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081))\n    (\u2191(Finset.affineCombination R Finset.univ t.points) w)\n    (\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081))\n[PROOFSTEP]\nrefine'\n  Sbtw.affineCombination_of_mem_affineSpan_pair t.Independent hw\n    (Finset.univ.sum_affineCombinationSingleWeights R (Finset.mem_univ _))\n    (Finset.univ.sum_affineCombinationLineMapWeights (Finset.mem_univ _) (Finset.mem_univ _) _) h\u2081' (Finset.mem_univ i\u2081)\n    _\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\n\u22a2 Sbtw R (Finset.affineCombinationSingleWeights R i\u2081 i\u2081) (w i\u2081) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081 i\u2081)\n[PROOFSTEP]\nrw [Finset.affineCombinationSingleWeights_apply_self, Finset.affineCombinationLineMapWeights_apply_of_ne h\u2081\u2082 h\u2081\u2083,\n  sbtw_one_zero_iff]\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\n\u22a2 w i\u2081 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nhave hs : \u2200 i : Fin 3, SignType.sign (w i) = SignType.sign (w i\u2083) :=\n  by\n  intro i\n  rcases h3 i with (rfl | rfl | rfl)\n  \u00b7 exact h\u2082s\n  \u00b7 exact h\u2081s\n  \u00b7 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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\n\u22a2 \u2200 (i : Fin 3), \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\ni : Fin 3\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nrcases h3 i with (rfl | rfl | rfl)\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2082 i\u2083 : Fin 3\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\ni : Fin 3\nh\u2081\u2082 : i \u2260 i\u2082\nh\u2081\u2083 : i \u2260 i\u2083\nh3 : \u2200 (i_1 : Fin 3), i_1 = i \u2228 i_1 = i\u2082 \u2228 i_1 = i\u2083\nhu : Finset.univ = {i, i\u2082, i\u2083}\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i) (\u2191(lineMap (Affine.Simplex.points t i) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2082s : \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nexact h\u2082s\n[GOAL]\ncase inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2083 : Fin 3\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\ni : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\nh\u2082\u2083 : i \u2260 i\u2083\nh3 : \u2200 (i_1 : Fin 3), i_1 = i\u2081 \u2228 i_1 = i \u2228 i_1 = i\u2083\nhu : Finset.univ = {i\u2081, i, i\u2083}\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i) (\u2191(lineMap (Affine.Simplex.points t i) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n[PROOFSTEP]\nexact h\u2081s\n[GOAL]\ncase inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\ni : Fin 3\nh\u2081\u2083 : i\u2081 \u2260 i\nh\u2082\u2083 : i\u2082 \u2260 i\nh3 : \u2200 (i_1 : Fin 3), i_1 = i\u2081 \u2228 i_1 = i\u2082 \u2228 i_1 = i\nhu : Finset.univ = {i\u2081, i\u2082, i}\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i)) r\u2081)\n    (Affine.Simplex.points t i)\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i)) r\u2082)\n    (Affine.Simplex.points t i)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i)\n\u22a2 \u2191SignType.sign (w i) = \u2191SignType.sign (w i)\n[PROOFSTEP]\nrfl\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n\u22a2 w i\u2081 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nhave hss : SignType.sign (\u2211 i, w i) = 1 := by simp [hw]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\n\u22a2 \u2191SignType.sign (\u2211 i : Fin (2 + 1), w i) = 1\n[PROOFSTEP]\nsimp [hw]\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (\u2211 i : Fin (2 + 1), w i) = 1\n\u22a2 w i\u2081 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nhave hs' := sign_sum Finset.univ_nonempty (SignType.sign (w i\u2083)) fun i _ => hs i\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (\u2211 i : Fin (2 + 1), w i) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\n\u22a2 w i\u2081 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nrw [hs'] at hss \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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), \u2191SignType.sign (w i) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\n\u22a2 w i\u2081 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nsimp_rw [hss, sign_eq_one_iff] at hs \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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\n\u22a2 w i\u2081 \u2208 Set.Ioo 0 1\n[PROOFSTEP]\nrefine' \u27e8hs i\u2081, _\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i : Fin (2 + 1), w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\n\u22a2 w i\u2081 < 1\n[PROOFSTEP]\nrw [hu] at hw \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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i in {i\u2081, i\u2082, i\u2083}, w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\n\u22a2 w i\u2081 < 1\n[PROOFSTEP]\nrw [Finset.sum_insert, Finset.sum_insert, Finset.sum_singleton] at hw \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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : w i\u2081 + (w i\u2082 + w i\u2083) = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\n\u22a2 w i\u2081 < 1\n[PROOFSTEP]\nby_contra hle\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : w i\u2081 + (w i\u2082 + w i\u2083) = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\nhle : \u00acw i\u2081 < 1\n\u22a2 False\n[PROOFSTEP]\nrw [not_lt] at hle \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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : w i\u2081 + (w i\u2082 + w i\u2083) = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\nhle : 1 \u2264 w i\u2081\n\u22a2 False\n[PROOFSTEP]\nexact (hle.trans_lt (lt_add_of_pos_right _ (Left.add_pos (hs i\u2082) (hs i\u2083)))).ne' hw\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : w i\u2081 + \u2211 x in {i\u2082, i\u2083}, w x = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\n\u22a2 \u00aci\u2082 \u2208 {i\u2083}\n[PROOFSTEP]\nsimpa using h\u2082\u2083\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\u2074 : LinearOrderedRing R\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module R V\ninst\u271d\u00b9 : AddTorsor V P\ninst\u271d : NoZeroSMulDivisors R V\nt : Affine.Triangle R P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nthis : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh3 : \u2200 (i : Fin 3), i = i\u2081 \u2228 i = i\u2082 \u2228 i = i\u2083\nhu : Finset.univ = {i\u2081, i\u2082, i\u2083}\nr\u2081 : R\nhr\u20810 : 0 < r\u2081\nhr\u20811 : r\u2081 < 1\nh\u2081 :\n  Sbtw R (Affine.Simplex.points t i\u2082) (\u2191(lineMap (Affine.Simplex.points t i\u2082) (Affine.Simplex.points t i\u2083)) r\u2081)\n    (Affine.Simplex.points t i\u2083)\nr\u2082 : R\nhr\u20820 : 0 < r\u2082\nhr\u20821 : r\u2082 < 1\nh\u2082 :\n  Sbtw R (Affine.Simplex.points t i\u2081) (\u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082)\n    (Affine.Simplex.points t i\u2083)\nw : Fin (2 + 1) \u2192 R\nhw : \u2211 i in {i\u2081, i\u2082, i\u2083}, w i = 1\nhp : \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208 affineSpan R (Set.range t.points)\nh\u2081' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R\n      {\u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i\u2081),\n        \u2191(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i\u2082 i\u2083 r\u2081)}\nh\u2082' :\n  \u2191(Finset.affineCombination R Finset.univ t.points) w \u2208\n    affineSpan R {Affine.Simplex.points t i\u2082, \u2191(lineMap (Affine.Simplex.points t i\u2081) (Affine.Simplex.points t i\u2083)) r\u2082}\nh\u2081s : \u2191SignType.sign (w i\u2082) = \u2191SignType.sign (w i\u2083)\nh\u2082s : \u2191SignType.sign (w i\u2081) = \u2191SignType.sign (w i\u2083)\nhss : \u2191SignType.sign (w i\u2083) = 1\nhs' : \u2191SignType.sign (\u2211 i : Fin 3, w i) = \u2191SignType.sign (w i\u2083)\nhs : \u2200 (i : Fin 3), 0 < w i\n\u22a2 \u00aci\u2081 \u2208 {i\u2082, i\u2083}\n[PROOFSTEP]\nsimpa [not_or] using \u27e8h\u2081\u2082, h\u2081\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 Wbtw R x y z \u2194 x = y \u2228 z \u2208 \u2191(lineMap x y) '' Set.Ici 1\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 x = y \u2228 z \u2208 \u2191(lineMap x y) '' Set.Ici 1\n[PROOFSTEP]\nrcases h with \u27e8r, \u27e8hr0, hr1\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\n\u22a2 x = \u2191(lineMap x z) r \u2228 z \u2208 \u2191(lineMap x (\u2191(lineMap x z) r)) '' Set.Ici 1\n[PROOFSTEP]\nrcases hr0.lt_or_eq with (hr0' | rfl)\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhr0' : 0 < r\n\u22a2 x = \u2191(lineMap x z) r \u2228 z \u2208 \u2191(lineMap x (\u2191(lineMap x z) r)) '' Set.Ici 1\n[PROOFSTEP]\nrw [Set.mem_image]\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhr0' : 0 < r\n\u22a2 x = \u2191(lineMap x z) r \u2228 \u2203 x_1, x_1 \u2208 Set.Ici 1 \u2227 \u2191(lineMap x (\u2191(lineMap x z) r)) x_1 = z\n[PROOFSTEP]\nrefine' Or.inr \u27e8r\u207b\u00b9, one_le_inv hr0' hr1, _\u27e9\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhr0' : 0 < r\n\u22a2 \u2191(lineMap x (\u2191(lineMap x z) r)) r\u207b\u00b9 = z\n[PROOFSTEP]\nsimp only [lineMap_apply, smul_smul, vadd_vsub]\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhr0' : 0 < r\n\u22a2 (r\u207b\u00b9 * r) \u2022 (z -\u1d65 x) +\u1d65 x = z\n[PROOFSTEP]\nrw [inv_mul_cancel hr0'.ne', one_smul, vsub_vadd]\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nhr0 : 0 \u2264 0\nhr1 : 0 \u2264 1\n\u22a2 x = \u2191(lineMap x z) 0 \u2228 z \u2208 \u2191(lineMap x (\u2191(lineMap x z) 0)) '' Set.Ici 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : x = y \u2228 z \u2208 \u2191(lineMap x y) '' Set.Ici 1\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrcases h with (rfl | \u27e8r, \u27e8hr, rfl\u27e9\u27e9)\n[GOAL]\ncase refine'_2.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\n\u22a2 Wbtw R x x z\n[PROOFSTEP]\nexact wbtw_self_left _ _ _\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : r \u2208 Set.Ici 1\n\u22a2 Wbtw R x y (\u2191(lineMap x y) r)\n[PROOFSTEP]\nrw [Set.mem_Ici] at hr \n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : 1 \u2264 r\n\u22a2 Wbtw R x y (\u2191(lineMap x y) r)\n[PROOFSTEP]\nrefine' \u27e8r\u207b\u00b9, \u27e8inv_nonneg.2 (zero_le_one.trans hr), inv_le_one hr\u27e9, _\u27e9\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : 1 \u2264 r\n\u22a2 \u2191(lineMap x (\u2191(lineMap x y) r)) r\u207b\u00b9 = y\n[PROOFSTEP]\nsimp only [lineMap_apply, smul_smul, vadd_vsub]\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : 1 \u2264 r\n\u22a2 (r\u207b\u00b9 * r) \u2022 (y -\u1d65 x) +\u1d65 x = y\n[PROOFSTEP]\nrw [inv_mul_cancel (one_pos.trans_le hr).ne', 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\nhne : x \u2260 y\n\u22a2 z \u2208 affineSpan R {x, y}\n[PROOFSTEP]\nrcases h.right_mem_image_Ici_of_left_ne hne with \u27e8r, \u27e8-, rfl\u27e9\u27e9\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nh : Wbtw R x y (\u2191(lineMap x y) r)\n\u22a2 \u2191(lineMap x y) r \u2208 affineSpan R {x, y}\n[PROOFSTEP]\nexact lineMap_mem_affineSpan_pair _ _ _\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 Sbtw R x y z \u2194 x \u2260 y \u2227 z \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h.left_ne, _\u27e9, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Sbtw R x y z\n\u22a2 z \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n[PROOFSTEP]\nobtain \u27e8r, \u27e8hr, rfl\u27e9\u27e9 := h.wbtw.right_mem_image_Ici_of_left_ne h.left_ne\n[GOAL]\ncase refine'_1.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : r \u2208 Set.Ici 1\nh : Sbtw R x y (\u2191(lineMap x y) r)\n\u22a2 \u2191(lineMap x y) r \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n[PROOFSTEP]\nrw [Set.mem_Ici] at hr \n[GOAL]\ncase refine'_1.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : 1 \u2264 r\nh : Sbtw R x y (\u2191(lineMap x y) r)\n\u22a2 \u2191(lineMap x y) r \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n[PROOFSTEP]\nrcases hr.lt_or_eq with (hrlt | rfl)\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nr : R\nhr : 1 \u2264 r\nh : Sbtw R x y (\u2191(lineMap x y) r)\nhrlt : 1 < r\n\u22a2 \u2191(lineMap x y) r \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n[PROOFSTEP]\nexact Set.mem_image_of_mem _ hrlt\n[GOAL]\ncase refine'_1.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhr : 1 \u2264 1\nh : Sbtw R x y (\u2191(lineMap x y) 1)\n\u22a2 \u2191(lineMap x y) 1 \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase refine'_1.intro.intro.inr.h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhr : 1 \u2264 1\nh : Sbtw R x y (\u2191(lineMap x y) 1)\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : x \u2260 y \u2227 z \u2208 \u2191(lineMap x y) '' Set.Ioi 1\n\u22a2 Sbtw R x y z\n[PROOFSTEP]\nrcases h with \u27e8hne, r, hr, rfl\u27e9\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nhr : r \u2208 Set.Ioi 1\n\u22a2 Sbtw R x y (\u2191(lineMap x y) r)\n[PROOFSTEP]\nrw [Set.mem_Ioi] at hr \n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nhr : 1 < r\n\u22a2 Sbtw R x y (\u2191(lineMap x y) r)\n[PROOFSTEP]\nrefine'\n  \u27e8wbtw_iff_left_eq_or_right_mem_image_Ici.2\n      (Or.inr (Set.mem_image_of_mem _ (Set.mem_of_mem_of_subset hr Set.Ioi_subset_Ici_self))),\n    hne.symm, _\u27e9\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nhr : 1 < r\n\u22a2 y \u2260 \u2191(lineMap x y) r\n[PROOFSTEP]\nrw [lineMap_apply, \u2190 @vsub_ne_zero V, vsub_vadd_eq_vsub_sub]\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nhr : 1 < r\n\u22a2 y -\u1d65 x - r \u2022 (y -\u1d65 x) \u2260 0\n[PROOFSTEP]\nnth_rw 1 [\u2190 one_smul R (y -\u1d65 x)]\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nhr : 1 < r\n\u22a2 1 \u2022 (y -\u1d65 x) - r \u2022 (y -\u1d65 x) \u2260 0\n[PROOFSTEP]\nrw [\u2190 sub_smul, smul_ne_zero_iff, vsub_ne_zero, sub_ne_zero]\n[GOAL]\ncase refine'_2.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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nhr : 1 < r\n\u22a2 1 \u2260 r \u2227 y \u2260 x\n[PROOFSTEP]\nexact \u27e8hr.ne, hne.symm\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 Wbtw R x y z \u2194 z = y \u2228 x \u2208 \u2191(lineMap z y) '' Set.Ici 1\n[PROOFSTEP]\nrw [wbtw_comm, wbtw_iff_left_eq_or_right_mem_image_Ici]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 Sbtw R x y z \u2194 z \u2260 y \u2227 x \u2208 \u2191(lineMap z y) '' Set.Ioi 1\n[PROOFSTEP]\nrw [sbtw_comm, sbtw_iff_left_ne_and_right_mem_image_Ioi]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrefine' \u27e8r\u2081 / r\u2082, \u27e8div_nonneg hr\u2081 (hr\u2081.trans hr\u2082), div_le_one_of_le hr\u2082 (hr\u2081.trans hr\u2082)\u27e9, _\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x\n[PROOFSTEP]\nby_cases h : r\u2081 = 0\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\nh : r\u2081 = 0\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x\n[PROOFSTEP]\nsimp [h]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\nh : \u00acr\u2081 = 0\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x\n[PROOFSTEP]\nsimp [lineMap_apply, smul_smul, ((hr\u2081.lt_of_ne' h).trans_le hr\u2082).ne.symm]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrcases le_total r\u2081 r\u2082 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\nh : r\u2081 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr\u2081 h)\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : 0 \u2264 r\u2082\nh : r\u2082 \u2264 r\u2081\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inr (wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x v hr\u2082 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : r\u2082 \u2264 r\u2081\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nconvert wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x (-v) (Left.nonneg_neg_iff.2 hr\u2081) (neg_le_neg_iff.2 hr\u2082) using 1\n[GOAL]\ncase h.e'_9\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : r\u2082 \u2264 r\u2081\n\u22a2 r\u2081 \u2022 v +\u1d65 x = -r\u2081 \u2022 -v +\u1d65 x\n[PROOFSTEP]\nrw [neg_smul_neg]\n[GOAL]\ncase h.e'_10\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : r\u2082 \u2264 r\u2081\n\u22a2 r\u2082 \u2022 v +\u1d65 x = -r\u2082 \u2022 -v +\u1d65 x\n[PROOFSTEP]\nrw [neg_smul_neg]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : r\u2082 \u2264 0\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrcases le_total r\u2081 r\u2082 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : r\u2082 \u2264 0\nh : r\u2081 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_le x v hr\u2082 h)\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : r\u2082 \u2264 0\nh : r\u2082 \u2264 r\u2081\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x) \u2228 Wbtw R x (r\u2082 \u2022 v +\u1d65 x) (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inl (wbtw_smul_vadd_smul_vadd_of_nonpos_of_le x v hr\u2081 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : 0 \u2264 r\u2082\n\u22a2 Wbtw R (r\u2081 \u2022 v +\u1d65 x) x (r\u2082 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nconvert\n  wbtw_smul_vadd_smul_vadd_of_nonneg_of_le (r\u2081 \u2022 v +\u1d65 x) v (Left.nonneg_neg_iff.2 hr\u2081)\n    (neg_le_sub_iff_le_add.2 ((le_add_iff_nonneg_left r\u2081).2 hr\u2082)) using\n  1\n[GOAL]\ncase h.e'_9\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : 0 \u2264 r\u2082\n\u22a2 x = -r\u2081 \u2022 v +\u1d65 (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nsimp [sub_smul, \u2190 add_vadd]\n[GOAL]\ncase h.e'_10\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : r\u2081 \u2264 0\nhr\u2082 : 0 \u2264 r\u2082\n\u22a2 r\u2082 \u2022 v +\u1d65 x = (r\u2082 - r\u2081) \u2022 v +\u1d65 (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nsimp [sub_smul, \u2190 add_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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2082 \u2264 0\n\u22a2 Wbtw R (r\u2081 \u2022 v +\u1d65 x) x (r\u2082 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrw [wbtw_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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2082 \u2264 0\n\u22a2 Wbtw R (r\u2082 \u2022 v +\u1d65 x) x (r\u2081 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg x v hr\u2082 hr\u2081\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw x y z : P\nh\u2081 : Wbtw R w y z\nh\u2082 : Wbtw R w x y\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrcases h\u2081 with \u27e8t\u2081, ht\u2081, rfl\u27e9\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw x z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nh\u2082 : Wbtw R w x (\u2191(lineMap w z) t\u2081)\n\u22a2 Wbtw R x (\u2191(lineMap w z) t\u2081) z\n[PROOFSTEP]\nrcases h\u2082 with \u27e8t\u2082, ht\u2082, rfl\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 Wbtw R (\u2191(lineMap w (\u2191(lineMap w z) t\u2081)) t\u2082) (\u2191(lineMap w z) t\u2081) z\n[PROOFSTEP]\nrefine'\n  \u27e8(t\u2081 - t\u2082 * t\u2081) / (1 - t\u2082 * t\u2081),\n    \u27e8div_nonneg (sub_nonneg.2 (mul_le_of_le_one_left ht\u2081.1 ht\u2082.2)) (sub_nonneg.2 (mul_le_one ht\u2082.2 ht\u2081.1 ht\u2081.2)),\n      div_le_one_of_le (sub_le_sub_right ht\u2081.2 _) (sub_nonneg.2 (mul_le_one ht\u2082.2 ht\u2081.1 ht\u2081.2))\u27e9,\n    _\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 \u2191(lineMap (\u2191(lineMap w (\u2191(lineMap w z) t\u2081)) t\u2082) z) ((t\u2081 - t\u2082 * t\u2081) / (1 - t\u2082 * t\u2081)) = \u2191(lineMap w z) t\u2081\n[PROOFSTEP]\nsimp only [lineMap_apply, smul_smul, \u2190 add_vadd, vsub_vadd_eq_vsub_sub, smul_sub, \u2190 sub_smul, \u2190 add_smul, vadd_vsub,\n  vadd_right_cancel_iff, div_mul_eq_mul_div, div_sub_div_same]\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 ((t\u2081 - t\u2082 * t\u2081 - (t\u2081 - t\u2082 * t\u2081) * (t\u2082 * t\u2081)) / (1 - t\u2082 * t\u2081) + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nnth_rw 1 [\u2190 mul_one (t\u2081 - t\u2082 * t\u2081)]\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 (((t\u2081 - t\u2082 * t\u2081) * 1 - (t\u2081 - t\u2082 * t\u2081) * (t\u2082 * t\u2081)) / (1 - t\u2082 * t\u2081) + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nrw [\u2190 mul_sub, mul_div_assoc]\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\n\u22a2 ((t\u2081 - t\u2082 * t\u2081) * ((1 - t\u2082 * t\u2081) / (1 - t\u2082 * t\u2081)) + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nby_cases h : 1 - t\u2082 * t\u2081 = 0\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : 1 - t\u2082 * t\u2081 = 0\n\u22a2 ((t\u2081 - t\u2082 * t\u2081) * ((1 - t\u2082 * t\u2081) / (1 - t\u2082 * t\u2081)) + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nrw [sub_eq_zero, eq_comm] at h \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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : t\u2082 * t\u2081 = 1\n\u22a2 ((t\u2081 - t\u2082 * t\u2081) * ((1 - t\u2082 * t\u2081) / (1 - t\u2082 * t\u2081)) + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nrw [h]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : t\u2082 * t\u2081 = 1\n\u22a2 ((t\u2081 - 1) * ((1 - 1) / (1 - 1)) + 1) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nsuffices t\u2081 = 1 by simp [this]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : t\u2082 * t\u2081 = 1\nthis : t\u2081 = 1\n\u22a2 ((t\u2081 - 1) * ((1 - 1) / (1 - 1)) + 1) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nsimp [this]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : t\u2082 * t\u2081 = 1\n\u22a2 t\u2081 = 1\n[PROOFSTEP]\nexact eq_of_le_of_not_lt ht\u2081.2 fun ht\u2081lt => (mul_lt_one_of_nonneg_of_lt_one_right ht\u2082.2 ht\u2081.1 ht\u2081lt).ne h\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : \u00ac1 - t\u2082 * t\u2081 = 0\n\u22a2 ((t\u2081 - t\u2082 * t\u2081) * ((1 - t\u2082 * t\u2081) / (1 - t\u2082 * t\u2081)) + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\n[PROOFSTEP]\nrw [div_self h]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw z : P\nt\u2081 : R\nht\u2081 : t\u2081 \u2208 Set.Icc 0 1\nt\u2082 : R\nht\u2082 : t\u2082 \u2208 Set.Icc 0 1\nh : \u00ac1 - t\u2082 * t\u2081 = 0\n\u22a2 ((t\u2081 - t\u2082 * t\u2081) * 1 + t\u2082 * t\u2081) \u2022 (z -\u1d65 w) = t\u2081 \u2022 (z -\u1d65 w)\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw x y z : P\nh\u2081 : Wbtw R w x z\nh\u2082 : Wbtw R x y z\n\u22a2 Wbtw R w x y\n[PROOFSTEP]\nrw [wbtw_comm] at *\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Wbtw R z y x\n\u22a2 Wbtw R y x w\n[PROOFSTEP]\nexact h\u2081.trans_left_right h\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 Collinear R {x, y, z}\n[PROOFSTEP]\nrw [collinear_iff_exists_forall_eq_smul_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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 \u2203 p\u2080 v, \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 p\u2080\n[PROOFSTEP]\nrefine' \u27e8x, z -\u1d65 x, _\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n\u22a2 \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 (z -\u1d65 x) +\u1d65 x\n[PROOFSTEP]\nintro p 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\np : P\nhp : p \u2208 {x, y, z}\n\u22a2 \u2203 r, p = r \u2022 (z -\u1d65 x) +\u1d65 x\n[PROOFSTEP]\nsimp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\np : P\nhp : p = x \u2228 p = y \u2228 p = z\n\u22a2 \u2203 r, p = r \u2022 (z -\u1d65 x) +\u1d65 x\n[PROOFSTEP]\nrcases hp with (rfl | rfl | rfl)\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\ny z p : P\nh : Wbtw R p y z\n\u22a2 \u2203 r, p = r \u2022 (z -\u1d65 p) +\u1d65 p\n[PROOFSTEP]\nrefine' \u27e80, _\u27e9\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\ny z p : P\nh : Wbtw R p y z\n\u22a2 p = 0 \u2022 (z -\u1d65 p) +\u1d65 p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z p : P\nh : Wbtw R x p z\n\u22a2 \u2203 r, p = r \u2022 (z -\u1d65 x) +\u1d65 x\n[PROOFSTEP]\nrcases h with \u27e8t, -, rfl\u27e9\n[GOAL]\ncase inr.inl.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nt : R\n\u22a2 \u2203 r, \u2191(lineMap x z) t = r \u2022 (z -\u1d65 x) +\u1d65 x\n[PROOFSTEP]\nexact \u27e8t, rfl\u27e9\n[GOAL]\ncase inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y p : P\nh : Wbtw R x y p\n\u22a2 \u2203 r, p = r \u2022 (p -\u1d65 x) +\u1d65 x\n[PROOFSTEP]\nrefine' \u27e81, _\u27e9\n[GOAL]\ncase inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y p : P\nh : Wbtw R x y p\n\u22a2 p = 1 \u2022 (p -\u1d65 x) +\u1d65 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Collinear R {x, y, z}\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y\n[PROOFSTEP]\nrw [collinear_iff_of_mem (Set.mem_insert _ _)] 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : \u2203 v, \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y\n[PROOFSTEP]\nrcases h with \u27e8v, h\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y\n[PROOFSTEP]\nsimp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y\n[PROOFSTEP]\nhave hy := h y (Or.inr (Or.inl rfl))\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy : \u2203 r, y = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y\n[PROOFSTEP]\nhave hz := h z (Or.inr (Or.inr rfl))\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy : \u2203 r, y = r \u2022 v +\u1d65 x\nhz : \u2203 r, z = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y\n[PROOFSTEP]\nrcases hy with \u27e8ty, rfl\u27e9\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nv : V\nhz : \u2203 r, z = r \u2022 v +\u1d65 x\nty : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) z \u2228 Wbtw R (ty \u2022 v +\u1d65 x) z x \u2228 Wbtw R z x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrcases hz with \u27e8tz, 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrcases lt_trichotomy ty 0 with (hy0 | rfl | hy0)\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrw [wbtw_comm (z := x)]\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrw [\u2190 or_assoc]\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 (Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x)) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonpos _ _ hy0.le hz0.le)\n[GOAL]\ncase intro.intro.intro.inl.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty : R\nhy0 : ty < 0\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = 0 \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) x \u2228 Wbtw R (0 \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.inl.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos _ _ hz0.le hy0.le))\n[GOAL]\ncase 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\ntz : R\nh : \u2200 (p : P), p = x \u2228 p = 0 \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (0 \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (0 \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (0 \u2022 v +\u1d65 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)\n[GOAL]\ncase intro.intro.intro.inr.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrefine' Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg _ _ hz0.le hy0.le))\n[GOAL]\ncase intro.intro.intro.inr.inr.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty : R\nhy0 : 0 < ty\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = 0 \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) x \u2228 Wbtw R (0 \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrw [wbtw_comm (z := x)]\n[GOAL]\ncase intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nrw [\u2190 or_assoc]\n[GOAL]\ncase intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 (Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x)) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n[PROOFSTEP]\nexact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonneg _ _ hy0.le hz0.le)\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 Wbtw R x y z \u2194 SameRay R (y -\u1d65 x) (z -\u1d65 y)\n[PROOFSTEP]\nrefine' \u27e8Wbtw.sameRay_vsub, fun 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : SameRay R (y -\u1d65 x) (z -\u1d65 y)\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrcases h with (h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9)\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : y -\u1d65 x = 0\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : y = x\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : z -\u1d65 y = 0\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : z = y\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\n\u22a2 Wbtw R x y z\n[PROOFSTEP]\nrefine'\n  \u27e8r\u2082 / (r\u2081 + r\u2082),\n    \u27e8div_nonneg hr\u2082.le (add_nonneg hr\u2081.le hr\u2082.le),\n      div_le_one_of_le (le_add_of_nonneg_left hr\u2081.le) (add_nonneg hr\u2081.le hr\u2082.le)\u27e9,\n    _\u27e9\n[GOAL]\ncase 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\n\u22a2 \u2191(lineMap x z) (r\u2082 / (r\u2081 + r\u2082)) = y\n[PROOFSTEP]\nhave h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y := by simp [h, hr\u2082.ne']\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\n\u22a2 z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y\n[PROOFSTEP]\nsimp [h, hr\u2082.ne']\n[GOAL]\ncase 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\nh' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y\n\u22a2 \u2191(lineMap x z) (r\u2082 / (r\u2081 + r\u2082)) = y\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\nh' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y\n\u22a2 y = \u2191(lineMap x z) (r\u2082 / (r\u2081 + r\u2082))\n[PROOFSTEP]\nsimp only [lineMap_apply, h', vadd_vsub_assoc, smul_smul, \u2190 add_smul, eq_vadd_iff_vsub_eq, smul_add]\n[GOAL]\ncase 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\nh' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y\n\u22a2 y -\u1d65 x = (r\u2082 / (r\u2081 + r\u2082) * (r\u2082\u207b\u00b9 * r\u2081) + r\u2082 / (r\u2081 + r\u2082)) \u2022 (y -\u1d65 x)\n[PROOFSTEP]\nconvert (one_smul R (y -\u1d65 x)).symm\n[GOAL]\ncase h.e'_3.h.e'_5\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\nh' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y\n\u22a2 r\u2082 / (r\u2081 + r\u2082) * (r\u2082\u207b\u00b9 * r\u2081) + r\u2082 / (r\u2081 + r\u2082) = 1\n[PROOFSTEP]\nfield_simp [(add_pos hr\u2081 hr\u2082).ne', hr\u2082.ne']\n[GOAL]\ncase h.e'_3.h.e'_5\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y)\nh' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y\n\u22a2 r\u2082 * r\u2081 * (r\u2081 + r\u2082) + r\u2082 * ((r\u2081 + r\u2082) * r\u2082) = (r\u2081 + r\u2082) * r\u2082 * (r\u2081 + r\u2082)\n[PROOFSTEP]\nring\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 Wbtw R y x (\u2191(pointReflection R x) y)\n[PROOFSTEP]\nrefine' \u27e82\u207b\u00b9, \u27e8by norm_num, by norm_num\u27e9, _\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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 0 \u2264 2\u207b\u00b9\n[PROOFSTEP]\nnorm_num\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 2\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nnorm_num\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 \u2191(lineMap y (\u2191(pointReflection R x) y)) 2\u207b\u00b9 = x\n[PROOFSTEP]\nrw [lineMap_apply, pointReflection_apply, vadd_vsub_assoc, \u2190 two_smul R (x -\u1d65 y)]\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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 2\u207b\u00b9 \u2022 2 \u2022 (x -\u1d65 y) +\u1d65 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh : x \u2260 y\n\u22a2 Sbtw R y x (\u2191(pointReflection R x) y)\n[PROOFSTEP]\nrefine' \u27e8wbtw_pointReflection _ _ _, 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh : x \u2260 y\n\u22a2 x \u2260 \u2191(pointReflection R x) y\n[PROOFSTEP]\nnth_rw 1 [\u2190 pointReflection_self 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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh : x \u2260 y\n\u22a2 \u2191(pointReflection R x) x \u2260 \u2191(pointReflection R x) y\n[PROOFSTEP]\nexact (pointReflection_involutive R x).injective.ne 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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 Wbtw R x (midpoint R x y) y\n[PROOFSTEP]\nconvert wbtw_pointReflection R (midpoint R x y) x\n[GOAL]\ncase h.e'_10\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 y = \u2191(pointReflection R (midpoint R x y)) x\n[PROOFSTEP]\nrw [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\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh : x \u2260 y\n\u22a2 Sbtw R x (midpoint R x y) y\n[PROOFSTEP]\nhave h : midpoint R x y \u2260 x := by simp [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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh : x \u2260 y\n\u22a2 midpoint R x y \u2260 x\n[PROOFSTEP]\nsimp [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 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh\u271d : x \u2260 y\nh : midpoint R x y \u2260 x\n\u22a2 Sbtw R x (midpoint R x y) y\n[PROOFSTEP]\nconvert sbtw_pointReflection_of_ne R h\n[GOAL]\ncase h.e'_10\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nh\u271d : x \u2260 y\nh : midpoint R x y \u2260 x\n\u22a2 y = \u2191(pointReflection R (midpoint R x y)) x\n[PROOFSTEP]\nrw [pointReflection_midpoint_left]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Between", "llama_tokens": 164742, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.7122321842389469, "lm_q1q2_score": 0.570278613555457}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b2 : Type u_3\nm : \u03b2 \u2192 (i : \u03b9) \u2192 \u03b1 i\nl : Filter \u03b2\n\u22a2 Tendsto m l (pi f) \u2194 \u2200 (i : \u03b9), Tendsto (fun x => m x i) l (f i)\n[PROOFSTEP]\nsimp only [pi, tendsto_iInf, tendsto_comap_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b2 : Type u_3\nm : \u03b2 \u2192 (i : \u03b9) \u2192 \u03b1 i\nl : Filter \u03b2\n\u22a2 (\u2200 (i : \u03b9), Tendsto (eval i \u2218 m) l (f i)) \u2194 \u2200 (i : \u03b9), Tendsto (fun x => m x i) l (f i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 s i \u2208 f i\n\u22a2 Set.pi I s \u2208 pi f\n[PROOFSTEP]\nrw [pi_def, biInter_eq_iInter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 s i \u2208 f i\n\u22a2 \u22c2 (x : \u2191I), eval \u2191x \u207b\u00b9' s \u2191x \u2208 pi f\n[PROOFSTEP]\nrefine' mem_iInf_of_iInter hI (fun i => _) Subset.rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 s i \u2208 f i\ni : \u2191I\n\u22a2 eval \u2191i \u207b\u00b9' s \u2191i \u2208 comap (eval \u2191i) (f \u2191i)\n[PROOFSTEP]\nexact preimage_mem_comap (h i i.2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 s \u2208 pi f \u2194 \u2203 I, Set.Finite I \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 Set.pi I t \u2286 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 s \u2208 pi f \u2192 \u2203 I, Set.Finite I \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 Set.pi I t \u2286 s\n[PROOFSTEP]\nsimp only [pi, mem_iInf', mem_comap, pi_def]\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 (\u2203 I,\n      Set.Finite I \u2227\n        \u2203 V,\n          (\u2200 (i : \u03b9), \u2203 t, t \u2208 f i \u2227 eval i \u207b\u00b9' t \u2286 V i) \u2227\n            (\u2200 (i : \u03b9), \u00aci \u2208 I \u2192 V i = univ) \u2227 s = \u22c2 (i : \u03b9) (_ : i \u2208 I), V i \u2227 s = \u22c2 (i : \u03b9), V i) \u2192\n    \u2203 I, Set.Finite I \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 \u22c2 (a : \u03b9) (_ : a \u2208 I), eval a \u207b\u00b9' t a \u2286 s\n[PROOFSTEP]\nrintro \u27e8I, If, V, hVf, -, rfl, -\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nI : Set \u03b9\nIf : Set.Finite I\nV : \u03b9 \u2192 Set ((i : \u03b9) \u2192 \u03b1 i)\nhVf : \u2200 (i : \u03b9), \u2203 t, t \u2208 f i \u2227 eval i \u207b\u00b9' t \u2286 V i\n\u22a2 \u2203 I_1,\n    Set.Finite I_1 \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 \u22c2 (a : \u03b9) (_ : a \u2208 I_1), eval a \u207b\u00b9' t a \u2286 \u22c2 (i : \u03b9) (_ : i \u2208 I), V i\n[PROOFSTEP]\nchoose t htf htV using hVf\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nI : Set \u03b9\nIf : Set.Finite I\nV : \u03b9 \u2192 Set ((i : \u03b9) \u2192 \u03b1 i)\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhtV : \u2200 (i : \u03b9), eval i \u207b\u00b9' t i \u2286 V i\n\u22a2 \u2203 I_1,\n    Set.Finite I_1 \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 \u22c2 (a : \u03b9) (_ : a \u2208 I_1), eval a \u207b\u00b9' t a \u2286 \u22c2 (i : \u03b9) (_ : i \u2208 I), V i\n[PROOFSTEP]\nexact \u27e8I, If, t, htf, iInter\u2082_mono fun i _ => htV i\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 (\u2203 I, Set.Finite I \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 Set.pi I t \u2286 s) \u2192 s \u2208 pi f\n[PROOFSTEP]\nrintro \u27e8I, If, t, htf, hts\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\nI : Set \u03b9\nIf : Set.Finite I\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I t \u2286 s\n\u22a2 s \u2208 pi f\n[PROOFSTEP]\nexact mem_of_superset (pi_mem_pi If fun i _ => htf i) hts\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\n\u22a2 s i \u2208 f i\n[PROOFSTEP]\nrcases mem_pi.1 h with \u27e8I', -, t, htf, hts\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\n\u22a2 s i \u2208 f i\n[PROOFSTEP]\nrefine' mem_of_superset (htf i) fun x hx => _\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\nx : \u03b1 i\nhx : x \u2208 t i\n\u22a2 x \u2208 s i\n[PROOFSTEP]\nhave : \u2200 i, (t i).Nonempty := fun i => nonempty_of_mem (htf i)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\nx : \u03b1 i\nhx : x \u2208 t i\nthis : \u2200 (i : \u03b9), Set.Nonempty (t i)\n\u22a2 x \u2208 s i\n[PROOFSTEP]\nchoose g hg using this\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\nx : \u03b1 i\nhx : x \u2208 t i\ng : (i : \u03b9) \u2192 \u03b1 i\nhg : \u2200 (i : \u03b9), g i \u2208 t i\n\u22a2 x \u2208 s i\n[PROOFSTEP]\nhave : update g i x \u2208 I'.pi t := fun j _ => by rcases eq_or_ne j i with (rfl | hne) <;> simp [*]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\nx : \u03b1 i\nhx : x \u2208 t i\ng : (i : \u03b9) \u2192 \u03b1 i\nhg : \u2200 (i : \u03b9), g i \u2208 t i\nj : \u03b9\nx\u271d : j \u2208 I'\n\u22a2 update g i x j \u2208 t j\n[PROOFSTEP]\nrcases eq_or_ne j i with (rfl | hne)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\ng : (i : \u03b9) \u2192 \u03b1 i\nhg : \u2200 (i : \u03b9), g i \u2208 t i\nj : \u03b9\nx\u271d : j \u2208 I'\nhi : j \u2208 I\nx : \u03b1 j\nhx : x \u2208 t j\n\u22a2 update g j x j \u2208 t j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\nx : \u03b1 i\nhx : x \u2208 t i\ng : (i : \u03b9) \u2192 \u03b1 i\nhg : \u2200 (i : \u03b9), g i \u2208 t i\nj : \u03b9\nx\u271d : j \u2208 I'\nhne : j \u2260 i\n\u22a2 update g i x j \u2208 t j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\nh : Set.pi I s \u2208 pi f\ni : \u03b9\nhi : i \u2208 I\nI' : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I' t \u2286 Set.pi I s\nx : \u03b1 i\nhx : x \u2208 t i\ng : (i : \u03b9) \u2192 \u03b1 i\nhg : \u2200 (i : \u03b9), g i \u2208 t i\nthis : update g i x \u2208 Set.pi I' t\n\u22a2 x \u2208 s i\n[PROOFSTEP]\nsimpa using hts this i hi\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b9' : \u03b9 \u2192 Type\ns : (i : \u03b9) \u2192 \u03b9' i \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop\nh : \u2200 (i : \u03b9), HasBasis (f i) (p i) (s i)\n\u22a2 HasBasis (pi f) (fun If => Set.Finite If.fst \u2227 \u2200 (i : \u03b9), i \u2208 If.fst \u2192 p i (Prod.snd If i)) fun If =>\n    Set.pi If.fst fun i => s i (Prod.snd If i)\n[PROOFSTEP]\nsimpa [Set.pi_def] using hasBasis_iInf' fun i => (h i).comap (eval i : (\u2200 j, \u03b1 j) \u2192 \u03b1 i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi f \u2293 \ud835\udcdf (Set.pi univ s) = \u22a5 \u2194 \u2203 i, f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi f \u2293 \ud835\udcdf (Set.pi univ s) = \u22a5 \u2192 \u2203 i, f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nsimp only [inf_principal_eq_bot, mem_pi]\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (\u2203 I, Set.Finite I \u2227 \u2203 t, (\u2200 (i : \u03b9), t i \u2208 f i) \u2227 Set.pi I t \u2286 (Set.pi univ s)\u1d9c) \u2192 \u2203 i, (s i)\u1d9c \u2208 f i\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (\u2200 (i : \u03b9), \u00ac(s i)\u1d9c \u2208 f i) \u2192\n    \u2200 (I : Set \u03b9), Set.Finite I \u2192 \u2200 (t : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), t i \u2208 f i) \u2192 \u00acSet.pi I t \u2286 (Set.pi univ s)\u1d9c\n[PROOFSTEP]\nrintro (hsf : \u2200 i, \u2203\u1da0 x in f i, x \u2208 s i) I - t htf hts\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhsf : \u2200 (i : \u03b9), \u2203\u1da0 (x : \u03b1 i) in f i, x \u2208 s i\nI : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I t \u2286 (Set.pi univ s)\u1d9c\n\u22a2 False\n[PROOFSTEP]\nhave : \u2200 i, (s i \u2229 t i).Nonempty := fun i => ((hsf i).and_eventually (htf i)).exists\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhsf : \u2200 (i : \u03b9), \u2203\u1da0 (x : \u03b1 i) in f i, x \u2208 s i\nI : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I t \u2286 (Set.pi univ s)\u1d9c\nthis : \u2200 (i : \u03b9), Set.Nonempty (s i \u2229 t i)\n\u22a2 False\n[PROOFSTEP]\nchoose x hxs hxt using this\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhsf : \u2200 (i : \u03b9), \u2203\u1da0 (x : \u03b1 i) in f i, x \u2208 s i\nI : Set \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhts : Set.pi I t \u2286 (Set.pi univ s)\u1d9c\nx : (i : \u03b9) \u2192 \u03b1 i\nhxs : \u2200 (i : \u03b9), x i \u2208 s i\nhxt : \u2200 (i : \u03b9), x i \u2208 t i\n\u22a2 False\n[PROOFSTEP]\nexact hts (fun i _ => hxt i) (mem_univ_pi.2 hxs)\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (\u2203 i, f i \u2293 \ud835\udcdf (s i) = \u22a5) \u2192 pi f \u2293 \ud835\udcdf (Set.pi univ s) = \u22a5\n[PROOFSTEP]\nsimp only [inf_principal_eq_bot]\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 (\u2203 i, (s i)\u1d9c \u2208 f i) \u2192 (Set.pi univ s)\u1d9c \u2208 pi f\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9\n[GOAL]\ncase mpr.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nhi : (s i)\u1d9c \u2208 f i\n\u22a2 (Set.pi univ s)\u1d9c \u2208 pi f\n[PROOFSTEP]\nfilter_upwards [mem_pi_of_mem i hi] with x using mt fun h => h i trivial\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\n\u22a2 pi f \u2293 \ud835\udcdf (Set.pi I s) = \u22a5 \u2194 \u2203 i, i \u2208 I \u2227 f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nrw [\u2190 univ_pi_piecewise_univ I, pi_inf_principal_univ_pi_eq_bot]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\n\u22a2 (\u2203 i, f i \u2293 \ud835\udcdf (piecewise I s (fun x => univ) i) = \u22a5) \u2194 \u2203 i, i \u2208 I \u2227 f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nrefine' exists_congr fun i => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\ni : \u03b9\n\u22a2 f i \u2293 \ud835\udcdf (piecewise I s (fun x => univ) i) = \u22a5 \u2194 i \u2208 I \u2227 f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nby_cases hi : i \u2208 I\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\ni : \u03b9\nhi : i \u2208 I\n\u22a2 f i \u2293 \ud835\udcdf (piecewise I s (fun x => univ) i) = \u22a5 \u2194 i \u2208 I \u2227 f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nsimp [hi, NeBot.ne']\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 f i \u2293 \ud835\udcdf (piecewise I s (fun x => univ) i) = \u22a5 \u2194 i \u2208 I \u2227 f i \u2293 \ud835\udcdf (s i) = \u22a5\n[PROOFSTEP]\nsimp [hi, NeBot.ne']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 NeBot (pi f \u2293 \ud835\udcdf (Set.pi univ s)) \u2194 \u2200 (i : \u03b9), NeBot (f i \u2293 \ud835\udcdf (s i))\n[PROOFSTEP]\nsimp [neBot_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\nI : Set \u03b9\n\u22a2 NeBot (pi f \u2293 \ud835\udcdf (Set.pi I s)) \u2194 \u2200 (i : \u03b9), i \u2208 I \u2192 NeBot (f i \u2293 \ud835\udcdf (s i))\n[PROOFSTEP]\nsimp [neBot_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi f = \u22a5 \u2194 \u2203 i, f i = \u22a5\n[PROOFSTEP]\nsimpa using @pi_inf_principal_univ_pi_eq_bot \u03b9 \u03b1 f fun _ => univ\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 NeBot (pi f) \u2194 \u2200 (i : \u03b9), NeBot (f i)\n[PROOFSTEP]\nsimp [neBot_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\ni : \u03b9\n\u22a2 map (eval i) (pi f) = f i\n[PROOFSTEP]\nrefine' le_antisymm (tendsto_eval_pi f i) fun s hs => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\n\u22a2 s \u2208 f i\n[PROOFSTEP]\nrcases mem_pi.1 (mem_map.1 hs) with \u27e8I, hIf, t, htf, hI\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : Set.Finite I\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : Set.pi I t \u2286 eval i \u207b\u00b9' s\n\u22a2 s \u2208 f i\n[PROOFSTEP]\nrw [\u2190 image_subset_iff] at hI \n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : Set.Finite I\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : eval i '' Set.pi I t \u2286 s\n\u22a2 s \u2208 f i\n[PROOFSTEP]\nrefine' mem_of_superset (htf i) ((subset_eval_image_pi _ _).trans hI)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : Set.Finite I\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : eval i '' Set.pi I t \u2286 s\n\u22a2 Set.Nonempty (Set.pi I t)\n[PROOFSTEP]\nexact nonempty_of_mem (pi_mem_pi hIf fun i _ => htf i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f\u2081 i)\n\u22a2 pi f\u2081 = pi f\u2082 \u2194 f\u2081 = f\u2082\n[PROOFSTEP]\nrefine' \u27e8fun h => _, congr_arg pi\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f\u2081 i)\nh : pi f\u2081 = pi f\u2082\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nhave hle : f\u2081 \u2264 f\u2082 := pi_le_pi.1 h.le\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f\u2081 i)\nh : pi f\u2081 = pi f\u2082\nhle : f\u2081 \u2264 f\u2082\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nhaveI : \u2200 i, NeBot (f\u2082 i) := fun i => neBot_of_le (hle i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NeBot (f\u2081 i)\nh : pi f\u2081 = pi f\u2082\nhle : f\u2081 \u2264 f\u2082\nthis : \u2200 (i : \u03b9), NeBot (f\u2082 i)\n\u22a2 f\u2081 = f\u2082\n[PROOFSTEP]\nexact hle.antisymm (pi_le_pi.1 h.ge)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 s \u2208 Filter.coprod\u1d62 f \u2194 \u2200 (i : \u03b9), \u2203 t\u2081, t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 s\n[PROOFSTEP]\nsimp [Filter.coprod\u1d62]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 s\u1d9c \u2208 Filter.coprod\u1d62 f \u2194 \u2200 (i : \u03b9), (eval i '' s)\u1d9c \u2208 f i\n[PROOFSTEP]\nsimp only [Filter.coprod\u1d62, mem_iSup, compl_mem_comap]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 NeBot (Filter.coprod\u1d62 f) \u2194 (\u2200 (i : \u03b9), Nonempty (\u03b1 i)) \u2227 \u2203 d, NeBot (f d)\n[PROOFSTEP]\nsimp only [Filter.coprod\u1d62, iSup_neBot, \u2190 exists_and_left, \u2190 comap_eval_neBot_iff']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 NeBot (Filter.coprod\u1d62 f) \u2194 \u2203 d, NeBot (f d)\n[PROOFSTEP]\nsimp [coprod\u1d62_neBot_iff', *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 Filter.coprod\u1d62 f = \u22a5 \u2194 (\u2203 i, IsEmpty (\u03b1 i)) \u2228 f = \u22a5\n[PROOFSTEP]\nsimpa only [not_neBot, not_and_or, funext_iff, not_forall, not_exists, not_nonempty_iff] using coprod\u1d62_neBot_iff'.not\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 Filter.coprod\u1d62 f = \u22a5 \u2194 f = \u22a5\n[PROOFSTEP]\nsimpa [funext_iff] using coprod\u1d62_neBot_iff.not\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b2 : \u03b9 \u2192 Type u_3\nm : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 map (fun k i => m i (k i)) (Filter.coprod\u1d62 f) \u2264 Filter.coprod\u1d62 fun i => map (m i) (f i)\n[PROOFSTEP]\nsimp only [le_def, mem_map, mem_coprod\u1d62_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b2 : \u03b9 \u2192 Type u_3\nm : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 \u2200 (x : Set ((i : \u03b9) \u2192 \u03b2 i)),\n    (\u2200 (i : \u03b9), \u2203 t\u2081, m i \u207b\u00b9' t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 x) \u2192\n      \u2200 (i : \u03b9), \u2203 t\u2081, t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 (fun k i => m i (k i)) \u207b\u00b9' x\n[PROOFSTEP]\nintro s h i\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b2 : \u03b9 \u2192 Type u_3\nm : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nh : \u2200 (i : \u03b9), \u2203 t\u2081, m i \u207b\u00b9' t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 s\ni : \u03b9\n\u22a2 \u2203 t\u2081, t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 (fun k i => m i (k i)) \u207b\u00b9' s\n[PROOFSTEP]\nobtain \u27e8t, H, hH\u27e9 := h i\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u03b2 : \u03b9 \u2192 Type u_3\nm : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nh : \u2200 (i : \u03b9), \u2203 t\u2081, m i \u207b\u00b9' t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 s\ni : \u03b9\nt : Set (\u03b2 i)\nH : m i \u207b\u00b9' t \u2208 f i\nhH : eval i \u207b\u00b9' t \u2286 s\n\u22a2 \u2203 t\u2081, t\u2081 \u2208 f i \u2227 eval i \u207b\u00b9' t\u2081 \u2286 (fun k i => m i (k i)) \u207b\u00b9' s\n[PROOFSTEP]\nexact \u27e8{x : \u03b1 i | m i x \u2208 t}, H, fun x hx => hH hx\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Pi", "llama_tokens": 10667, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527944504227, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.5698544125992085}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Even (card (Fin (bit0 n)))\n[PROOFSTEP]\nrw [Fintype.card_fin]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Even (bit0 n)\n[PROOFSTEP]\nexact even_bit0 _\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Parity", "llama_tokens": 87, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.5698544114095387}}
{"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u03b1 \u03b2 : A\u2081 \u27f6 A\u2082\n\u22a2 F.map (\u03b1.f + \u03b2.f) \u226b A\u2082.str = A\u2081.str \u226b (\u03b1.f + \u03b2.f)\n[PROOFSTEP]\nsimp only [Functor.map_add, add_comp, Endofunctor.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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a b c : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d b\u271d c\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d b\u271d c\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 F.map 0 \u226b A\u2082.str = A\u2081.str \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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn : \u2115\n\u03b1 : A\u2081 \u27f6 A\u2082\n\u22a2 F.map (n \u2022 \u03b1.f) \u226b A\u2082.str = A\u2081.str \u226b (n \u2022 \u03b1.f)\n[PROOFSTEP]\nrw [comp_nsmul, Functor.map_nsmul, nsmul_comp, Endofunctor.Algebra.Hom.h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (x : A\u2081 \u27f6 A\u2082), (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nx\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) 0 x\u271d = 0\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nx\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (n : \u2115) (x : A\u2081 \u27f6 A\u2082),\n    (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\nx\u271d : A\u2081 \u27f6 A\u2082\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]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\nx\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u03b1 : A\u2081 \u27f6 A\u2082\n\u22a2 F.map (-\u03b1.f) \u226b A\u2082.str = A\u2081.str \u226b (-\u03b1.f)\n[PROOFSTEP]\nsimp only [Functor.map_neg, neg_comp, Endofunctor.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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u03b1 \u03b2 : A\u2081 \u27f6 A\u2082\n\u22a2 F.map (\u03b1.f - \u03b2.f) \u226b A\u2082.str = A\u2081.str \u226b (\u03b1.f - \u03b2.f)\n[PROOFSTEP]\nsimp only [Functor.map_sub, sub_comp, Endofunctor.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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a b : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nr : \u2124\n\u03b1 : A\u2081 \u27f6 A\u2082\n\u22a2 F.map (r \u2022 \u03b1.f) \u226b A\u2082.str = A\u2081.str \u226b (r \u2022 \u03b1.f)\n[PROOFSTEP]\nrw [comp_zsmul, Functor.map_zsmul, zsmul_comp, Endofunctor.Algebra.Hom.h]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) 0 a\u271d = 0\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (n : \u2115) (a : A\u2081 \u27f6 A\u2082),\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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 f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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 f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (n : \u2115) (a : A\u2081 \u27f6 A\u2082),\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), -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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\n\u22a2 \u2200 (a b : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Algebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\n\u22a2 \u2200 (P Q R : Algebra F) (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Algebra F\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]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Algebra F\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\n\u22a2 \u2200 (P Q R : Algebra F) (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Algebra F\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]\napply Algebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Algebra F\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u03b1 \u03b2 : A\u2081 \u27f6 A\u2082\n\u22a2 A\u2081.str \u226b F.map (\u03b1.f + \u03b2.f) = (\u03b1.f + \u03b2.f) \u226b A\u2082.str\n[PROOFSTEP]\nsimp only [Functor.map_add, comp_add, Endofunctor.Coalgebra.Hom.h, add_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a b c : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d b\u271d c\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d b\u271d c\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 A\u2081.str \u226b F.map 0 = 0 \u226b A\u2082.str\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn : \u2115\n\u03b1 : A\u2081 \u27f6 A\u2082\n\u22a2 A\u2081.str \u226b F.map (n \u2022 \u03b1.f) = (n \u2022 \u03b1.f) \u226b A\u2082.str\n[PROOFSTEP]\nrw [Functor.map_nsmul, comp_nsmul, Endofunctor.Coalgebra.Hom.h, nsmul_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (x : A\u2081 \u27f6 A\u2082), (fun n \u03b1 => Coalgebra.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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nx\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) 0 x\u271d = 0\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nx\u271d : A\u2081 \u27f6 A\u2082\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (n : \u2115) (x : A\u2081 \u27f6 A\u2082),\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\nx\u271d : A\u2081 \u27f6 A\u2082\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]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\nx\u271d : A\u2081 \u27f6 A\u2082\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u03b1 : A\u2081 \u27f6 A\u2082\n\u22a2 A\u2081.str \u226b F.map (-\u03b1.f) = (-\u03b1.f) \u226b A\u2082.str\n[PROOFSTEP]\nsimp only [Functor.map_neg, comp_neg, Endofunctor.Coalgebra.Hom.h, neg_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u03b1 \u03b2 : A\u2081 \u27f6 A\u2082\n\u22a2 A\u2081.str \u226b F.map (\u03b1.f - \u03b2.f) = (\u03b1.f - \u03b2.f) \u226b A\u2082.str\n[PROOFSTEP]\nsimp only [Functor.map_sub, comp_sub, Endofunctor.Coalgebra.Hom.h, sub_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a b : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nr : \u2124\n\u03b1 : A\u2081 \u27f6 A\u2082\n\u22a2 A\u2081.str \u226b F.map (r \u2022 \u03b1.f) = (r \u2022 \u03b1.f) \u226b A\u2082.str\n[PROOFSTEP]\nrw [Functor.map_zsmul, comp_zsmul, Endofunctor.Coalgebra.Hom.h, zsmul_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), (fun r \u03b1 => Coalgebra.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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) 0 a\u271d = 0\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (n : \u2115) (a : A\u2081 \u27f6 A\u2082),\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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 f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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 f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (n : \u2115) (a : A\u2081 \u27f6 A\u2082),\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\nn\u271d : \u2115\na\u271d : A\u2081 \u27f6 A\u2082\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\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a : A\u2081 \u27f6 A\u2082), -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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\n\u22a2 \u2200 (a b : A\u2081 \u27f6 A\u2082), 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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nA\u2081 A\u2082 : Coalgebra F\na\u271d b\u271d : A\u2081 \u27f6 A\u2082\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\n\u22a2 \u2200 (P Q R : Coalgebra F) (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Coalgebra F\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]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Coalgebra F\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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\n\u22a2 \u2200 (P Q R : Coalgebra F) (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\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Coalgebra F\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]\napply Coalgebra.Hom.ext\n[GOAL]\ncase f\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nF : C \u2964 C\ninst\u271d : Functor.Additive F\nP\u271d Q\u271d R\u271d : Coalgebra F\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.EndoFunctor", "llama_tokens": 12962, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104788995148792, "lm_q2_score": 0.702530051167069, "lm_q1q2_score": 0.5693857827460178}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1\u271d D2\u271d : Derivation R A A\na\u271d : A\nD1 D2 : Derivation R A A\na b : A\n\u22a2 \u2191\u2045\u2191D1, \u2191D2\u2046 (a * b) = a \u2022 \u2191\u2045\u2191D1, \u2191D2\u2046 b + b \u2022 \u2191\u2045\u2191D1, \u2191D2\u2046 a\n[PROOFSTEP]\nsimp only [Ring.lie_def, map_add, Algebra.id.smul_eq_mul, LinearMap.mul_apply, leibniz, coeFn_coe, LinearMap.sub_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1\u271d D2\u271d : Derivation R A A\na\u271d : A\nD1 D2 : Derivation R A A\na b : A\n\u22a2 a * \u2191D1 (\u2191D2 b) + \u2191D2 b * \u2191D1 a + (b * \u2191D1 (\u2191D2 a) + \u2191D2 a * \u2191D1 b) -\n      (a * \u2191D2 (\u2191D1 b) + \u2191D1 b * \u2191D2 a + (b * \u2191D2 (\u2191D1 a) + \u2191D1 a * \u2191D2 b)) =\n    a * (\u2191D1 (\u2191D2 b) - \u2191D2 (\u2191D1 b)) + b * (\u2191D1 (\u2191D2 a) - \u2191D2 (\u2191D1 a))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na : A\nd e f : Derivation R A A\n\u22a2 \u2045d + e, f\u2046 = \u2045d, f\u2046 + \u2045e, f\u2046\n[PROOFSTEP]\next a\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd e f : Derivation R A A\na : A\n\u22a2 \u2191\u2045d + e, f\u2046 a = \u2191(\u2045d, f\u2046 + \u2045e, f\u2046) a\n[PROOFSTEP]\nsimp only [commutator_apply, add_apply, map_add]\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd e f : Derivation R A A\na : A\n\u22a2 \u2191d (\u2191f a) + \u2191e (\u2191f a) - (\u2191f (\u2191d a) + \u2191f (\u2191e a)) = \u2191d (\u2191f a) - \u2191f (\u2191d a) + (\u2191e (\u2191f a) - \u2191f (\u2191e a))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na : A\nd e f : Derivation R A A\n\u22a2 \u2045d, e + f\u2046 = \u2045d, e\u2046 + \u2045d, f\u2046\n[PROOFSTEP]\next a\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd e f : Derivation R A A\na : A\n\u22a2 \u2191\u2045d, e + f\u2046 a = \u2191(\u2045d, e\u2046 + \u2045d, f\u2046) a\n[PROOFSTEP]\nsimp only [commutator_apply, add_apply, map_add]\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd e f : Derivation R A A\na : A\n\u22a2 \u2191d (\u2191e a) + \u2191d (\u2191f a) - (\u2191e (\u2191d a) + \u2191f (\u2191d a)) = \u2191d (\u2191e a) - \u2191e (\u2191d a) + (\u2191d (\u2191f a) - \u2191f (\u2191d a))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na : A\nd : Derivation R A A\n\u22a2 \u2045d, d\u2046 = 0\n[PROOFSTEP]\next a\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd : Derivation R A A\na : A\n\u22a2 \u2191\u2045d, d\u2046 a = \u21910 a\n[PROOFSTEP]\nsimp only [commutator_apply, add_apply, map_add]\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd : Derivation R A A\na : A\n\u22a2 \u2191d (\u2191d a) - \u2191d (\u2191d a) = \u21910 a\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd : Derivation R A A\na : A\n\u22a2 0 = \u21910 a\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na : A\nd e f : Derivation R A A\n\u22a2 \u2045d, \u2045e, f\u2046\u2046 = \u2045\u2045d, e\u2046, f\u2046 + \u2045e, \u2045d, f\u2046\u2046\n[PROOFSTEP]\next a\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd e f : Derivation R A A\na : A\n\u22a2 \u2191\u2045d, \u2045e, f\u2046\u2046 a = \u2191(\u2045\u2045d, e\u2046, f\u2046 + \u2045e, \u2045d, f\u2046\u2046) a\n[PROOFSTEP]\nsimp only [commutator_apply, add_apply, sub_apply, map_sub]\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nd e f : Derivation R A A\na : A\n\u22a2 \u2191d (\u2191e (\u2191f a)) - \u2191d (\u2191f (\u2191e a)) - (\u2191e (\u2191f (\u2191d a)) - \u2191f (\u2191e (\u2191d a))) =\n    \u2191d (\u2191e (\u2191f a)) - \u2191e (\u2191d (\u2191f a)) - (\u2191f (\u2191d (\u2191e a)) - \u2191f (\u2191e (\u2191d a))) +\n      (\u2191e (\u2191d (\u2191f a)) - \u2191e (\u2191f (\u2191d a)) - (\u2191d (\u2191f (\u2191e a)) - \u2191f (\u2191d (\u2191e a))))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na : A\nsrc\u271d : Module R (Derivation R A A) := instModule\nr : R\nd e : Derivation R A A\n\u22a2 \u2045d, r \u2022 e\u2046 = r \u2022 \u2045d, e\u2046\n[PROOFSTEP]\next a\n[GOAL]\ncase H\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nA : Type u_2\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nD D1 D2 : Derivation R A A\na\u271d : A\nsrc\u271d : Module R (Derivation R A A) := instModule\nr : R\nd e : Derivation R A A\na : A\n\u22a2 \u2191\u2045d, r \u2022 e\u2046 a = \u2191(r \u2022 \u2045d, e\u2046) a\n[PROOFSTEP]\nsimp only [commutator_apply, map_smul, smul_sub, smul_apply]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Derivation.Lie", "llama_tokens": 2737, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267626522813, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.5689900614596868}}
{"text": "[GOAL]\n\u03b9 : Type u\nhi : Nonempty \u03b9\nR : Type v\ninst\u271d\u00b9 : Semiring R\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\ni : \u03b9\nh : \u2191x = 0\nj : \u03b9\n\u22a2 \u2191(Pi.evalRingHom (fun x => R) j) \u2191x = 0\n[PROOFSTEP]\nrw [map_natCast, h]\n[GOAL]\n\u03b9 : Type u\nhi : Nonempty \u03b9\nR : Type v\ninst\u271d\u00b9 : Semiring R\np : \u2115\ninst\u271d : CharP R p\nx : \u2115\ni : \u03b9\nh : \u2191x = 0\n\u22a2 \u2191(Pi.evalRingHom (fun x => R) i) \u2191x = 0\n[PROOFSTEP]\nrw [h, RingHom.map_zero]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.Pi", "llama_tokens": 223, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.837619947119304, "lm_q2_score": 0.6791787056691698, "lm_q1q2_score": 0.5688936315271673}}
{"text": "[GOAL]\nm n : \u2124\n\u22a2 \u00acn % 2 = 1 \u2194 n % 2 = 0\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one n with h h\n[GOAL]\ncase inl\nm n : \u2124\nh : n % 2 = 0\n\u22a2 \u00acn % 2 = 1 \u2194 n % 2 = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\nm n : \u2124\nh : n % 2 = 1\n\u22a2 \u00acn % 2 = 1 \u2194 n % 2 = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nm n : \u2124\n\u22a2 \u00acn % 2 = 0 \u2194 n % 2 = 1\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one n with h h\n[GOAL]\ncase inl\nm n : \u2124\nh : n % 2 = 0\n\u22a2 \u00acn % 2 = 0 \u2194 n % 2 = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\nm n : \u2124\nh : n % 2 = 1\n\u22a2 \u00acn % 2 = 0 \u2194 n % 2 = 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nm\u271d n : \u2124\nx\u271d : Even n\nm : \u2124\nhm : n = m + m\n\u22a2 n % 2 = 0\n[PROOFSTEP]\nsimp [\u2190 two_mul, hm]\n[GOAL]\nm n : \u2124\nh : n % 2 = 0\n\u22a2 n % 2 + 2 * (n / 2) = n / 2 + n / 2\n[PROOFSTEP]\nsimp [\u2190 two_mul, h]\n[GOAL]\nm\u271d n : \u2124\nx\u271d : Odd n\nm : \u2124\nhm : n = 2 * m + 1\n\u22a2 n % 2 = 1\n[PROOFSTEP]\nrw [hm, add_emod]\n[GOAL]\nm\u271d n : \u2124\nx\u271d : Odd n\nm : \u2124\nhm : n = 2 * m + 1\n\u22a2 (2 * m % 2 + 1 % 2) % 2 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n : \u2124\nh : n % 2 = 1\n\u22a2 n % 2 + 2 * (n / 2) = 2 * (n / 2) + 1\n[PROOFSTEP]\nrw [h]\n[GOAL]\nm n : \u2124\nh : n % 2 = 1\n\u22a2 1 + 2 * (n / 2) = 2 * (n / 2) + 1\n[PROOFSTEP]\nabel\n[GOAL]\nm n : \u2124\nh : n % 2 = 1\n\u22a2 1 + 2 * (n / 2) = 2 * (n / 2) + 1\n[PROOFSTEP]\nabel\n[GOAL]\nm n : \u2124\n\u22a2 \u00acEven n \u2194 n % 2 = 1\n[PROOFSTEP]\nrw [even_iff, emod_two_ne_zero]\n[GOAL]\nm n : \u2124\n\u22a2 \u00acOdd n \u2194 n % 2 = 0\n[PROOFSTEP]\nrw [odd_iff, emod_two_ne_one]\n[GOAL]\nm n : \u2124\n\u22a2 Even n \u2194 \u00acOdd n\n[PROOFSTEP]\nrw [not_odd_iff, even_iff]\n[GOAL]\nm n : \u2124\n\u22a2 Odd n \u2194 \u00acEven n\n[PROOFSTEP]\nrw [not_even_iff, odd_iff]\n[GOAL]\nm n : \u2124\n\u22a2 IsCompl {n | Even n} {n | Odd n}\n[PROOFSTEP]\nsimp [\u2190 Set.compl_setOf, isCompl_compl]\n[GOAL]\nm n\u271d n : \u2124\n\u22a2 \u2203 k, n = 2 * k \u2228 n = 2 * k + 1\n[PROOFSTEP]\nsimpa only [two_mul, exists_or, Odd, Even] using even_or_odd n\n[GOAL]\nm n\u271d n : \u2124\n\u22a2 Xor' (Even n) (Odd n)\n[PROOFSTEP]\ncases even_or_odd n with\n| inl h => exact Or.inl \u27e8h, even_iff_not_odd.mp h\u27e9\n| inr h => exact Or.inr \u27e8h, odd_iff_not_even.mp h\u27e9\n[GOAL]\nm n\u271d n : \u2124\nx\u271d : Even n \u2228 Odd n\n\u22a2 Xor' (Even n) (Odd n)\n[PROOFSTEP]\ncases even_or_odd n with\n| inl h => exact Or.inl \u27e8h, even_iff_not_odd.mp h\u27e9\n| inr h => exact Or.inr \u27e8h, odd_iff_not_even.mp h\u27e9\n[GOAL]\ncase inl\nm n\u271d n : \u2124\nh : Even n\n\u22a2 Xor' (Even n) (Odd n)\n[PROOFSTEP]\n\n| inl h => exact Or.inl \u27e8h, even_iff_not_odd.mp h\u27e9\n[GOAL]\ncase inl\nm n\u271d n : \u2124\nh : Even n\n\u22a2 Xor' (Even n) (Odd n)\n[PROOFSTEP]\nexact Or.inl \u27e8h, even_iff_not_odd.mp h\u27e9\n[GOAL]\ncase inr\nm n\u271d n : \u2124\nh : Odd n\n\u22a2 Xor' (Even n) (Odd n)\n[PROOFSTEP]\n\n| inr h => exact Or.inr \u27e8h, odd_iff_not_even.mp h\u27e9\n[GOAL]\ncase inr\nm n\u271d n : \u2124\nh : Odd n\n\u22a2 Xor' (Even n) (Odd n)\n[PROOFSTEP]\nexact Or.inr \u27e8h, odd_iff_not_even.mp h\u27e9\n[GOAL]\nm n\u271d n : \u2124\n\u22a2 \u2203 k, Xor' (n = 2 * k) (n = 2 * k + 1)\n[PROOFSTEP]\nrcases even_or_odd n with (\u27e8k, rfl\u27e9 | \u27e8k, rfl\u27e9)\n[GOAL]\ncase inl.intro\nm n k : \u2124\n\u22a2 \u2203 k_1, Xor' (k + k = 2 * k_1) (k + k = 2 * k_1 + 1)\n[PROOFSTEP]\nuse k\n[GOAL]\ncase inr.intro\nm n k : \u2124\n\u22a2 \u2203 k_1, Xor' (2 * k + 1 = 2 * k_1) (2 * k + 1 = 2 * k_1 + 1)\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nm n k : \u2124\n\u22a2 Xor' (k + k = 2 * k) (k + k = 2 * k + 1)\n[PROOFSTEP]\nsimpa only [\u2190 two_mul, Xor', true_and_iff, eq_self_iff_true, not_true, or_false_iff, and_false_iff] using\n  (succ_ne_self (2 * k)).symm\n[GOAL]\ncase h\nm n k : \u2124\n\u22a2 Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)\n[PROOFSTEP]\nsimp only [Xor', add_right_eq_self, false_or_iff, eq_self_iff_true, not_true, not_false_iff, one_ne_zero, and_self_iff]\n[GOAL]\nm n : \u2124\n\u22a2 \u00acEven 1\n[PROOFSTEP]\nrw [even_iff]\n[GOAL]\nm n : \u2124\n\u22a2 \u00ac1 % 2 = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n : \u2124\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one m with h\u2081 h\u2081\n[GOAL]\ncase inl\nm n : \u2124\nh\u2081 : m % 2 = 0\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one n with h\u2082 h\u2082\n[GOAL]\ncase inr\nm n : \u2124\nh\u2081 : m % 2 = 1\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one n with h\u2082 h\u2082\n[GOAL]\ncase inl.inl\nm n : \u2124\nh\u2081 : m % 2 = 0\nh\u2082 : n % 2 = 0\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.add_emod]\n[GOAL]\ncase inl.inr\nm n : \u2124\nh\u2081 : m % 2 = 0\nh\u2082 : n % 2 = 1\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.add_emod]\n[GOAL]\ncase inr.inl\nm n : \u2124\nh\u2081 : m % 2 = 1\nh\u2082 : n % 2 = 0\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.add_emod]\n[GOAL]\ncase inr.inr\nm n : \u2124\nh\u2081 : m % 2 = 1\nh\u2082 : n % 2 = 1\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.add_emod]\n[GOAL]\nm n : \u2124\n\u22a2 Even (m + n) \u2194 (Odd m \u2194 Odd n)\n[PROOFSTEP]\nrw [even_add, even_iff_not_odd, even_iff_not_odd, not_iff_not]\n[GOAL]\nm n\u271d n : \u2124\n\u22a2 \u00acEven (bit1 n)\n[PROOFSTEP]\nsimp [bit1, parity_simps]\n[GOAL]\nm n\u271d n : \u2124\n\u22a2 \u00ac2 \u2223 2 * n + 1\n[PROOFSTEP]\nsimp [add_emod]\n[GOAL]\nm n : \u2124\n\u22a2 Even (m - n) \u2194 (Even m \u2194 Even n)\n[PROOFSTEP]\nsimp [sub_eq_add_neg, parity_simps]\n[GOAL]\nm n : \u2124\n\u22a2 Even (m - n) \u2194 (Odd m \u2194 Odd n)\n[PROOFSTEP]\nrw [even_sub, even_iff_not_odd, even_iff_not_odd, not_iff_not]\n[GOAL]\nm n : \u2124\n\u22a2 Even (n + 1) \u2194 \u00acEven n\n[PROOFSTEP]\nsimp [even_add]\n[GOAL]\nm n : \u2124\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one m with h\u2081 h\u2081\n[GOAL]\ncase inl\nm n : \u2124\nh\u2081 : m % 2 = 0\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one n with h\u2082 h\u2082\n[GOAL]\ncase inr\nm n : \u2124\nh\u2081 : m % 2 = 1\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\ncases' emod_two_eq_zero_or_one n with h\u2082 h\u2082\n[GOAL]\ncase inl.inl\nm n : \u2124\nh\u2081 : m % 2 = 0\nh\u2082 : n % 2 = 0\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.mul_emod]\n[GOAL]\ncase inl.inr\nm n : \u2124\nh\u2081 : m % 2 = 0\nh\u2082 : n % 2 = 1\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.mul_emod]\n[GOAL]\ncase inr.inl\nm n : \u2124\nh\u2081 : m % 2 = 1\nh\u2082 : n % 2 = 0\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.mul_emod]\n[GOAL]\ncase inr.inr\nm n : \u2124\nh\u2081 : m % 2 = 1\nh\u2082 : n % 2 = 1\n\u22a2 Even (m * n) \u2194 Even m \u2228 Even n\n[PROOFSTEP]\nsimp [even_iff, h\u2081, h\u2082, Int.mul_emod]\n[GOAL]\nm n : \u2124\n\u22a2 Odd (m * n) \u2194 Odd m \u2227 Odd n\n[PROOFSTEP]\nsimp [not_or, parity_simps]\n[GOAL]\nm n\u271d : \u2124\nn : \u2115\n\u22a2 Even (m ^ n) \u2194 Even m \u2227 n \u2260 0\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nm n : \u2124\n\u22a2 Even (m ^ Nat.zero) \u2194 Even m \u2227 Nat.zero \u2260 0\n[PROOFSTEP]\nsimp [*, even_mul, pow_succ]\n[GOAL]\ncase succ\nm n\u271d : \u2124\nn : \u2115\nih : Even (m ^ n) \u2194 Even m \u2227 n \u2260 0\n\u22a2 Even (m ^ Nat.succ n) \u2194 Even m \u2227 Nat.succ n \u2260 0\n[PROOFSTEP]\nsimp [*, even_mul, pow_succ]\n[GOAL]\ncase succ\nm n\u271d : \u2124\nn : \u2115\nih : Even (m ^ n) \u2194 Even m \u2227 n \u2260 0\n\u22a2 Even m \u2192 \u00acn = 0 \u2192 Even m\n[PROOFSTEP]\ntauto\n[GOAL]\nm n\u271d : \u2124\nn : \u2115\n\u22a2 Odd (m ^ n) \u2194 Odd m \u2228 n = 0\n[PROOFSTEP]\nrw [\u2190 not_iff_not, \u2190 Int.even_iff_not_odd, not_or, \u2190 Int.even_iff_not_odd, Int.even_pow]\n[GOAL]\nm n : \u2124\n\u22a2 Odd (m + n) \u2194 (Odd m \u2194 Even n)\n[PROOFSTEP]\nrw [odd_iff_not_even, even_add, not_iff, odd_iff_not_even]\n[GOAL]\nm n : \u2124\n\u22a2 Odd (m + n) \u2194 (Odd n \u2194 Even m)\n[PROOFSTEP]\nrw [add_comm, odd_add]\n[GOAL]\nm n : \u2124\nh : Odd (m + n)\nhnot : m = n\n\u22a2 False\n[PROOFSTEP]\nsimp [hnot, parity_simps] at h \n[GOAL]\nm n : \u2124\n\u22a2 Odd (m - n) \u2194 (Odd m \u2194 Even n)\n[PROOFSTEP]\nrw [odd_iff_not_even, even_sub, not_iff, odd_iff_not_even]\n[GOAL]\nm n : \u2124\n\u22a2 Odd (m - n) \u2194 (Odd n \u2194 Even m)\n[PROOFSTEP]\nrw [odd_iff_not_even, even_sub, not_iff, not_iff_comm, odd_iff_not_even]\n[GOAL]\nm n\u271d n : \u2124\n\u22a2 Even (n * (n + 1))\n[PROOFSTEP]\nsimpa [even_mul, parity_simps] using n.even_or_odd\n[GOAL]\nm n\u271d : \u2124\nn : \u2115\n\u22a2 Even \u2191n \u2194 Even n\n[PROOFSTEP]\nrw_mod_cast [even_iff, Nat.even_iff]\n[GOAL]\nm n\u271d : \u2124\nn : \u2115\n\u22a2 Odd \u2191n \u2194 Odd n\n[PROOFSTEP]\nrw [odd_iff_not_even, Nat.odd_iff_not_even, even_coe_nat]\n[GOAL]\nm n : \u2124\n\u22a2 Even (natAbs n) \u2194 Even n\n[PROOFSTEP]\nsimp [even_iff_two_dvd, dvd_natAbs, coe_nat_dvd_left.symm]\n[GOAL]\nm n : \u2124\n\u22a2 Odd (natAbs n) \u2194 Odd n\n[PROOFSTEP]\nrw [odd_iff_not_even, Nat.odd_iff_not_even, natAbs_even]\n[GOAL]\nm n a b : \u2124\nha : Odd a\nhb : Odd b\n\u22a2 4 \u2223 a + b \u2228 4 \u2223 a - b\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := ha\n[GOAL]\ncase intro\nm\u271d n b : \u2124\nhb : Odd b\nm : \u2124\n\u22a2 4 \u2223 2 * m + 1 + b \u2228 4 \u2223 2 * m + 1 - b\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := hb\n[GOAL]\ncase intro.intro\nm\u271d n\u271d m n : \u2124\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)\n[PROOFSTEP]\nobtain h | h := Int.even_or_odd (m + n)\n[GOAL]\ncase intro.intro.inl\nm\u271d n\u271d m n : \u2124\nh : Even (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.intro.inl.h\nm\u271d n\u271d m n : \u2124\nh : Even (m + n)\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)\n[PROOFSTEP]\nrw [Int.even_add, \u2190 Int.even_sub] at h \n[GOAL]\ncase intro.intro.inl.h\nm\u271d n\u271d m n : \u2124\nh : Even (m - n)\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := h\n[GOAL]\ncase intro.intro.inl.h.intro\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)\n[PROOFSTEP]\nconvert dvd_mul_right 4 k using 1\n[GOAL]\ncase h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 2 * m + 1 - (2 * n + 1) = 4 * k\n[PROOFSTEP]\nrw [eq_add_of_sub_eq hk, mul_add, add_assoc, add_sub_cancel, \u2190 two_mul, \u2190 mul_assoc]\n[GOAL]\ncase h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 2 * 2 * k = 4 * k\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.inr\nm\u271d n\u271d m n : \u2124\nh : Odd (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)\n[PROOFSTEP]\nleft\n[GOAL]\ncase intro.intro.inr.h\nm\u271d n\u271d m n : \u2124\nh : Odd (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := h\n[GOAL]\ncase intro.intro.inr.h.intro\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1)\n[PROOFSTEP]\nconvert dvd_mul_right 4 (k + 1) using 1\n[GOAL]\ncase h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 2 * m + 1 + (2 * n + 1) = 4 * (k + 1)\n[PROOFSTEP]\nrw [eq_sub_of_add_eq hk, add_right_comm, \u2190 add_sub, mul_add, mul_sub, add_assoc, add_assoc, sub_add, add_assoc, \u2190\n  sub_sub (2 * n), sub_self, zero_sub, sub_neg_eq_add, \u2190 mul_assoc, mul_add]\n[GOAL]\ncase h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 2 * 2 * k + (2 * 1 + (1 + 1)) = 4 * k + 4 * 1\n[PROOFSTEP]\nrfl\n[GOAL]\nm n : \u2124\n\u22a2 Odd n \u2192 2 * (n / 2) + 1 = n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\nm c : \u2124\n\u22a2 2 * ((2 * c + 1) / 2) + 1 = 2 * c + 1\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase intro\nm c : \u2124\n\u22a2 (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1\n[PROOFSTEP]\nconvert Int.ediv_add_emod' (2 * c + 1) 2\n[GOAL]\ncase h.e'_2.h.e'_6\nm c : \u2124\n\u22a2 1 = (2 * c + 1) % 2\n[PROOFSTEP]\nsimp [Int.add_emod]\n[GOAL]\nm n : \u2124\n\u22a2 Odd n \u2192 n / 2 * 2 + 1 = n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\nm c : \u2124\n\u22a2 (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1\n[PROOFSTEP]\nconvert Int.ediv_add_emod' (2 * c + 1) 2\n[GOAL]\ncase h.e'_2.h.e'_6\nm c : \u2124\n\u22a2 1 = (2 * c + 1) % 2\n[PROOFSTEP]\nsimp [Int.add_emod]\n[GOAL]\nm n : \u2124\n\u22a2 Odd n \u2192 1 + n / 2 * 2 = n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\nm c : \u2124\n\u22a2 1 + (2 * c + 1) / 2 * 2 = 2 * c + 1\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase intro\nm c : \u2124\n\u22a2 (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1\n[PROOFSTEP]\nconvert Int.ediv_add_emod' (2 * c + 1) 2\n[GOAL]\ncase h.e'_2.h.e'_6\nm c : \u2124\n\u22a2 1 = (2 * c + 1) % 2\n[PROOFSTEP]\nsimp [Int.add_emod]\n[GOAL]\nm\u271d n\u271d m n : \u2124\nh : Even m\n\u22a2 \u00acEven (n + 3) \u2194 Even (m ^ 2 + m + n)\n[PROOFSTEP]\nsimp [*, (by decide : \u00ac2 = 0), parity_simps]\n[GOAL]\nm\u271d n\u271d m n : \u2124\nh : Even m\n\u22a2 \u00ac2 = 0\n[PROOFSTEP]\ndecide\n[GOAL]\nm n : \u2124\n\u22a2 \u00acEven 25394535\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Parity", "llama_tokens": 6533, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256472515683, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.5688759726425575}}
{"text": "[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np q : S\nh : Commute p q\nh\u2081 : IsIdempotentElem p\nh\u2082 : IsIdempotentElem q\n\u22a2 IsIdempotentElem (p * q)\n[PROOFSTEP]\nrw [IsIdempotentElem, mul_assoc, \u2190 mul_assoc q, \u2190 h.eq, mul_assoc p, h\u2082.eq, \u2190 mul_assoc, h\u2081.eq]\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : R\nh : IsIdempotentElem p\n\u22a2 IsIdempotentElem (1 - p)\n[PROOFSTEP]\nrw [IsIdempotentElem, mul_sub, mul_one, sub_mul, one_mul, h.eq, sub_self, sub_zero]\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n\u22a2 p ^ Nat.succ n * p ^ Nat.succ n = p ^ Nat.succ n\n[PROOFSTEP]\nconv_rhs =>\n  rw [\u2190 h.eq]\n    --Porting note: was `nth_rw 3 [\u2190 h.eq]`\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n| p ^ Nat.succ n\n[PROOFSTEP]\nrw [\u2190 h.eq]\n    --Porting note: was `nth_rw 3 [\u2190 h.eq]`\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n| p ^ Nat.succ n\n[PROOFSTEP]\nrw [\u2190 h.eq]\n    --Porting note: was `nth_rw 3 [\u2190 h.eq]`\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n| p ^ Nat.succ n\n[PROOFSTEP]\nrw [\u2190 h.eq]\n  --Porting note: was `nth_rw 3 [\u2190 h.eq]`\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n\u22a2 p ^ Nat.succ n * p ^ Nat.succ n = (p * p) ^ Nat.succ n\n[PROOFSTEP]\nrw [\u2190 sq, \u2190 sq, \u2190 pow_mul, \u2190 pow_mul']\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nih : p ^ (n + 1) = p\n\u22a2 p ^ (Nat.succ n + 1) = p\n[PROOFSTEP]\nrw [pow_succ, ih, h.eq]\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : G\u2080\n\u22a2 IsIdempotentElem p \u2194 p = 0 \u2228 p = 1\n[PROOFSTEP]\nrefine'\n  Iff.intro (fun h => or_iff_not_imp_left.mpr fun hp => _) fun h =>\n    h.elim (fun hp => hp.symm \u25b8 zero) fun hp => hp.symm \u25b8 one\n[GOAL]\nM : Type u_1\nN : Type u_2\nS : Type u_3\nM\u2080 : Type u_4\nM\u2081 : Type u_5\nR : Type u_6\nG : Type u_7\nG\u2080 : Type u_8\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : G\u2080\nh : IsIdempotentElem p\nhp : \u00acp = 0\n\u22a2 p = 1\n[PROOFSTEP]\nexact mul_left_cancel\u2080 hp (h.trans (mul_one p).symm)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Idempotents", "llama_tokens": 2480, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333246035907933, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.5688054974124825}}
{"text": "[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nX Y : C\nf : X \u27f6 Y\n\u22a2 inv (inv f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nX Y : C\nf : Y \u27f6 X\n\u22a2 inv (inv f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nX Y a\u271d b\u271d : C\nf : a\u271d \u27f6 b\u271d\n\u22a2 Quiver.reverse (Quiver.reverse f) = f\n[PROOFSTEP]\ndsimp [Quiver.reverse]\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nX Y a\u271d b\u271d : C\nf : a\u271d \u27f6 b\u271d\n\u22a2 Groupoid.inv (Groupoid.inv f) = f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Groupoid C\nX Y : C\nD : Type u_1\ninst\u271d : Groupoid D\nF : C \u2964 D\nu\u271d v\u271d : C\nf : u\u271d \u27f6 v\u271d\n\u22a2 F.map (Quiver.reverse f) = Quiver.reverse (F.map f)\n[PROOFSTEP]\nsimp only [Quiver.reverse, Quiver.HasReverse.reverse', Groupoid.inv_eq_inv, Functor.map_inv]\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nX Y : C\nf : X \u27f6 Y\n\u22a2 f \u226b inv f = \ud835\udfd9 X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nX Y : C\nf : X \u27f6 Y\n\u22a2 inv f \u226b f = \ud835\udfd9 Y\n[PROOFSTEP]\naesop_cat\n[GOAL]\nI : Type u\nJ : I \u2192 Type u\u2082\ninst\u271d : (i : I) \u2192 Groupoid (J i)\nX\u271d Y\u271d : (i : I) \u2192 J i\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (fun {X Y} f i => Groupoid.inv (f i)) f \u226b f = \ud835\udfd9 Y\u271d\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type u\nJ : I \u2192 Type u\u2082\ninst\u271d : (i : I) \u2192 Groupoid (J i)\nX\u271d Y\u271d : (i : I) \u2192 J i\nf : X\u271d \u27f6 Y\u271d\ni : I\n\u22a2 ((fun {X Y} f i => Groupoid.inv (f i)) f \u226b f) i = \ud835\udfd9 Y\u271d i\n[PROOFSTEP]\napply Groupoid.inv_comp\n[GOAL]\nI : Type u\nJ : I \u2192 Type u\u2082\ninst\u271d : (i : I) \u2192 Groupoid (J i)\nX\u271d Y\u271d : (i : I) \u2192 J i\nf : X\u271d \u27f6 Y\u271d\n\u22a2 f \u226b (fun {X Y} f i => Groupoid.inv (f i)) f = \ud835\udfd9 X\u271d\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase h\nI : Type u\nJ : I \u2192 Type u\u2082\ninst\u271d : (i : I) \u2192 Groupoid (J i)\nX\u271d Y\u271d : (i : I) \u2192 J i\nf : X\u271d \u27f6 Y\u271d\ni : I\n\u22a2 (f \u226b (fun {X Y} f i => Groupoid.inv (f i)) f) i = \ud835\udfd9 X\u271d i\n[PROOFSTEP]\napply Groupoid.comp_inv\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Groupoid", "llama_tokens": 1012, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619263765707, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.5683062261527533}}
{"text": "[GOAL]\nm : \u211d\nhm : 1 < m\nk : \u2115\n\u22a2 Summable fun i => 1 / m ^ (i + (k + 1))!\n[PROOFSTEP]\nconvert (summable_nat_add_iff (k + 1)).2 (LiouvilleNumber.summable hm)\n[GOAL]\nm : \u211d\nhm : 1 < m\nk x\u271d : \u2115\n\u22a2 0 \u2264 1 / m ^ (x\u271d + (k + 1))!\n[PROOFSTEP]\npositivity\n[GOAL]\nm : \u211d\nhm : 1 < m\nk : \u2115\n\u22a2 0 < 1 / m ^ (0 + (k + 1))!\n[PROOFSTEP]\npositivity\n[GOAL]\nn : \u2115\nm : \u211d\nm1 : 1 < m\nm0 : 0 < m\nmi : 1 / m < 1\n\u22a2 \u2211' (i : \u2115), 1 / m ^ (i + (n + 1)!) = \u2211' (i : \u2115), (1 / m) ^ i * (1 / m ^ (n + 1)!)\n[PROOFSTEP]\nsimp only [pow_add, one_div, mul_inv, inv_pow]\n  -- factor the constant `(1 / m ^ (n + 1)!)` out of the series\n[GOAL]\nn : \u2115\nm : \u211d\nm1 : 1 < m\nm0 : 0 < m\nmi : 1 / m < 1\n\u22a2 (\u2211' (i : \u2115), (1 / m) ^ i) * (1 / m ^ (n + 1)!) = (1 - 1 / m)\u207b\u00b9 * (1 / m ^ (n + 1)!)\n[PROOFSTEP]\nrw [tsum_geometric_of_lt_1 (by positivity) mi]\n[GOAL]\nn : \u2115\nm : \u211d\nm1 : 1 < m\nm0 : 0 < m\nmi : 1 / m < 1\n\u22a2 0 \u2264 1 / m\n[PROOFSTEP]\npositivity\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 \u2264 1 / m ^ (n + 1)!\n[PROOFSTEP]\npositivity\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 2 / m ^ (n ! * (n + 1)) \u2264 1 / m ^ (n ! * n)\n[PROOFSTEP]\napply (div_le_div_iff _ _).mpr\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 2 * m ^ (n ! * n) \u2264 1 * m ^ (n ! * (n + 1))\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * (n + 1))\nn : \u2115 m : \u211d hm : 2 \u2264 m \u22a2 0 < m ^ (n ! * n)\n[PROOFSTEP]\nconv_rhs =>\n  rw [one_mul, mul_add, pow_add, mul_one, pow_mul, mul_comm, \u2190 pow_mul]\n    -- the second factors coincide, so we prove the inequality of the first factors*\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n| 1 * m ^ (n ! * (n + 1))\n[PROOFSTEP]\nrw [one_mul, mul_add, pow_add, mul_one, pow_mul, mul_comm, \u2190 pow_mul]\n    -- the second factors coincide, so we prove the inequality of the first factors*\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n| 1 * m ^ (n ! * (n + 1))\n[PROOFSTEP]\nrw [one_mul, mul_add, pow_add, mul_one, pow_mul, mul_comm, \u2190 pow_mul]\n    -- the second factors coincide, so we prove the inequality of the first factors*\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n| 1 * m ^ (n ! * (n + 1))\n[PROOFSTEP]\nrw [one_mul, mul_add, pow_add, mul_one, pow_mul, mul_comm, \u2190 pow_mul]\n  -- the second factors coincide, so we prove the inequality of the first factors*\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 2 * m ^ (n ! * n) \u2264 m ^ n ! * m ^ (n ! * n)\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * (n + 1))\nn : \u2115 m : \u211d hm : 2 \u2264 m \u22a2 0 < m ^ (n ! * n)\n[PROOFSTEP]\nrefine'\n  (mul_le_mul_right _).mpr\n    _\n      -- solve all the inequalities `0 < m ^ ??`\n[GOAL]\ncase refine'_1\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * n)\ncase refine'_2\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 2 \u2264 m ^ n !\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * (n + 1))\nn : \u2115 m : \u211d hm : 2 \u2264 m \u22a2 0 < m ^ (n ! * n)\n[PROOFSTEP]\nany_goals\n  exact\n    pow_pos (zero_lt_two.trans_le hm)\n      _\n        -- `2 \u2264 m ^ n!` is a consequence of monotonicity of exponentiation at `2 \u2264 m`.\n[GOAL]\ncase refine'_1\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * n)\n[PROOFSTEP]\nexact\n  pow_pos (zero_lt_two.trans_le hm)\n    _\n      -- `2 \u2264 m ^ n!` is a consequence of monotonicity of exponentiation at `2 \u2264 m`.\n[GOAL]\ncase refine'_2\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 2 \u2264 m ^ n !\n[PROOFSTEP]\nexact\n  pow_pos (zero_lt_two.trans_le hm)\n    _\n      -- `2 \u2264 m ^ n!` is a consequence of monotonicity of exponentiation at `2 \u2264 m`.\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * (n + 1))\n[PROOFSTEP]\nexact\n  pow_pos (zero_lt_two.trans_le hm)\n    _\n      -- `2 \u2264 m ^ n!` is a consequence of monotonicity of exponentiation at `2 \u2264 m`.\n[GOAL]\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 0 < m ^ (n ! * n)\n[PROOFSTEP]\nexact\n  pow_pos (zero_lt_two.trans_le hm)\n    _\n      -- `2 \u2264 m ^ n!` is a consequence of monotonicity of exponentiation at `2 \u2264 m`.\n[GOAL]\ncase refine'_2\nn : \u2115\nm : \u211d\nhm : 2 \u2264 m\n\u22a2 2 \u2264 m ^ n !\n[PROOFSTEP]\nexact _root_.trans (_root_.trans hm (pow_one _).symm.le) (pow_mono (one_le_two.trans hm) n.factorial_pos)\n[GOAL]\nm : \u2115\nhm : 0 < m\nk : \u2115\n\u22a2 \u2203 p, partialSum (\u2191m) k = \u2191p / \u2191(m ^ k !)\n[PROOFSTEP]\ninduction' k with k h\n[GOAL]\ncase zero\nm : \u2115\nhm : 0 < m\n\u22a2 \u2203 p, partialSum (\u2191m) Nat.zero = \u2191p / \u2191(m ^ Nat.zero !)\n[PROOFSTEP]\nexact \u27e81, by rw [partialSum, range_one, sum_singleton, Nat.cast_one, Nat.factorial, pow_one, pow_one]\u27e9\n[GOAL]\nm : \u2115\nhm : 0 < m\n\u22a2 partialSum (\u2191m) Nat.zero = \u21911 / \u2191(m ^ Nat.zero !)\n[PROOFSTEP]\nrw [partialSum, range_one, sum_singleton, Nat.cast_one, Nat.factorial, pow_one, pow_one]\n[GOAL]\ncase succ\nm : \u2115\nhm : 0 < m\nk : \u2115\nh : \u2203 p, partialSum (\u2191m) k = \u2191p / \u2191(m ^ k !)\n\u22a2 \u2203 p, partialSum (\u2191m) (Nat.succ k) = \u2191p / \u2191(m ^ (Nat.succ k)!)\n[PROOFSTEP]\nrcases h with \u27e8p_k, h_k\u27e9\n[GOAL]\ncase succ.intro\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2203 p, partialSum (\u2191m) (Nat.succ k) = \u2191p / \u2191(m ^ (Nat.succ k)!)\n[PROOFSTEP]\nuse p_k * m ^ ((k + 1)! - k !) + 1\n[GOAL]\ncase h\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 partialSum (\u2191m) (Nat.succ k) = \u2191(p_k * m ^ ((k + 1)! - k !) + 1) / \u2191(m ^ (Nat.succ k)!)\n[PROOFSTEP]\nrw [partialSum_succ, h_k, div_add_div, div_eq_div_iff, add_mul]\n[GOAL]\ncase h\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191p_k * \u2191m ^ (k + 1)! * \u2191(m ^ (Nat.succ k)!) + \u2191(m ^ k !) * 1 * \u2191(m ^ (Nat.succ k)!) =\n    \u2191(p_k * m ^ ((k + 1)! - k !) + 1) * (\u2191(m ^ k !) * \u2191m ^ (k + 1)!)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 p_k * m ^ (k + 1)! * m ^ (Nat.succ k)! + m ^ k ! * 1 * m ^ (Nat.succ k)! =\n    (p_k * m ^ ((k + 1)! - k !) + 1) * (m ^ k ! * m ^ (k + 1)!)\n[PROOFSTEP]\nrw [add_mul, one_mul, Nat.factorial_succ, add_mul, one_mul, add_tsub_cancel_right, pow_add]\n[GOAL]\ncase h\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 p_k * (m ^ (k * k !) * m ^ k !) * (m ^ (k * k !) * m ^ k !) + m ^ k ! * 1 * (m ^ (k * k !) * m ^ k !) =\n    p_k * m ^ (k * k !) * (m ^ k ! * (m ^ (k * k !) * m ^ k !)) + m ^ k ! * (m ^ (k * k !) * m ^ k !)\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\ncase h.hb\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191(m ^ k !) * \u2191m ^ (k + 1)! \u2260 0\ncase h.hd\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191(m ^ (Nat.succ k)!) \u2260 0\ncase h.hb\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191(m ^ k !) \u2260 0\ncase h.hd m : \u2115 hm : 0 < m k p_k : \u2115 h_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !) \u22a2 \u2191m ^ (k + 1)! \u2260 0\n[PROOFSTEP]\nall_goals positivity\n[GOAL]\ncase h.hb\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191(m ^ k !) * \u2191m ^ (k + 1)! \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h.hd\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191(m ^ (Nat.succ k)!) \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h.hb\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191(m ^ k !) \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h.hd\nm : \u2115\nhm : 0 < m\nk p_k : \u2115\nh_k : partialSum (\u2191m) k = \u2191p_k / \u2191(m ^ k !)\n\u22a2 \u2191m ^ (k + 1)! \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\n\u22a2 Liouville (liouvilleNumber \u2191m)\n[PROOFSTEP]\nhave mZ1 : 1 < (m : \u2124) := by norm_cast\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\n\u22a2 1 < \u2191m\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\n\u22a2 Liouville (liouvilleNumber \u2191m)\n[PROOFSTEP]\nhave m1 : 1 < (m : \u211d) := by norm_cast\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\n\u22a2 1 < \u2191m\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\n\u22a2 Liouville (liouvilleNumber \u2191m)\n[PROOFSTEP]\nintro n\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn : \u2115\n\u22a2 \u2203 a b, 1 < b \u2227 liouvilleNumber \u2191m \u2260 \u2191a / \u2191b \u2227 |liouvilleNumber \u2191m - \u2191a / \u2191b| < 1 / \u2191b ^ n\n[PROOFSTEP]\nrcases partialSum_eq_rat (zero_lt_two.trans_le hm) n with \u27e8p, hp\u27e9\n[GOAL]\ncase intro\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191(m ^ n !)\n\u22a2 \u2203 a b, 1 < b \u2227 liouvilleNumber \u2191m \u2260 \u2191a / \u2191b \u2227 |liouvilleNumber \u2191m - \u2191a / \u2191b| < 1 / \u2191b ^ n\n[PROOFSTEP]\nrefine' \u27e8p, m ^ n !, by rw [Nat.cast_pow]; exact one_lt_pow mZ1 n.factorial_ne_zero, _\u27e9\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191(m ^ n !)\n\u22a2 1 < \u2191(m ^ n !)\n[PROOFSTEP]\nrw [Nat.cast_pow]\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191(m ^ n !)\n\u22a2 1 < \u2191m ^ n !\n[PROOFSTEP]\nexact one_lt_pow mZ1 n.factorial_ne_zero\n[GOAL]\ncase intro\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191(m ^ n !)\n\u22a2 liouvilleNumber \u2191m \u2260 \u2191\u2191p / \u2191\u2191(m ^ n !) \u2227 |liouvilleNumber \u2191m - \u2191\u2191p / \u2191\u2191(m ^ n !)| < 1 / \u2191\u2191(m ^ n !) ^ n\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase intro\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191(m ^ n !)\n\u22a2 liouvilleNumber \u2191m \u2260 \u2191p / \u2191m ^ n ! \u2227 |liouvilleNumber \u2191m - \u2191p / \u2191m ^ n !| < 1 / (\u2191m ^ n !) ^ n\n[PROOFSTEP]\nrw [Nat.cast_pow] at hp \n[GOAL]\ncase intro\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191m ^ n !\n\u22a2 liouvilleNumber \u2191m \u2260 \u2191p / \u2191m ^ n ! \u2227 |liouvilleNumber \u2191m - \u2191p / \u2191m ^ n !| < 1 / (\u2191m ^ n !) ^ n\n[PROOFSTEP]\nrw [\u2190 partialSum_add_remainder m1 n, \u2190 hp]\n[GOAL]\ncase intro\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191m ^ n !\n\u22a2 partialSum (\u2191m) n + remainder (\u2191m) n \u2260 partialSum (\u2191m) n \u2227\n    |partialSum (\u2191m) n + remainder (\u2191m) n - partialSum (\u2191m) n| < 1 / (\u2191m ^ n !) ^ n\n[PROOFSTEP]\nhave hpos := remainder_pos m1 n\n[GOAL]\ncase intro\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191m ^ n !\nhpos : 0 < remainder (\u2191m) n\n\u22a2 partialSum (\u2191m) n + remainder (\u2191m) n \u2260 partialSum (\u2191m) n \u2227\n    |partialSum (\u2191m) n + remainder (\u2191m) n - partialSum (\u2191m) n| < 1 / (\u2191m ^ n !) ^ n\n[PROOFSTEP]\nsimpa [abs_of_pos hpos, hpos.ne'] using @remainder_lt n m (by assumption_mod_cast)\n[GOAL]\nm : \u2115\nhm : 2 \u2264 m\nmZ1 : 1 < \u2191m\nm1 : 1 < \u2191m\nn p : \u2115\nhp : partialSum (\u2191m) n = \u2191p / \u2191m ^ n !\nhpos : 0 < remainder (\u2191m) n\n\u22a2 2 \u2264 \u2191m\n[PROOFSTEP]\nassumption_mod_cast\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Liouville.LiouvilleNumber", "llama_tokens": 5507, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.7185943985973772, "lm_q1q2_score": 0.5680706589174278}}
{"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 \ud835\udd5c G\n\u22a2 \u2200 (x : E), \u2016\u21910 x\u2016 \u2264 0 * \u2016x\u2016\n[PROOFSTEP]\nsimp [le_refl]\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 \ud835\udd5c G\n\u22a2 \u2200 (x : E), \u2016\u2191LinearMap.id x\u2016 \u2264 1 * \u2016x\u2016\n[PROOFSTEP]\nsimp [le_refl]\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 \ud835\udd5c G\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun x => x.fst\n[PROOFSTEP]\nrefine' (LinearMap.fst \ud835\udd5c E F).isLinear.with_bound 1 fun x => _\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 \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016\u2191(LinearMap.fst \ud835\udd5c E F) x\u2016 \u2264 1 * \u2016x\u2016\n[PROOFSTEP]\nrw [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 \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016\u2191(LinearMap.fst \ud835\udd5c E F) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nexact le_max_left _ _\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 \ud835\udd5c G\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun x => x.snd\n[PROOFSTEP]\nrefine' (LinearMap.snd \ud835\udd5c E F).isLinear.with_bound 1 fun x => _\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 \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016\u2191(LinearMap.snd \ud835\udd5c E F) x\u2016 \u2264 1 * \u2016x\u2016\n[PROOFSTEP]\nrw [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 \ud835\udd5c G\nx : E \u00d7 F\n\u22a2 \u2016\u2191(LinearMap.snd \ud835\udd5c E F) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nexact le_max_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\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\nhf : IsBoundedLinearMap \ud835\udd5c f\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun e => -f e\n[PROOFSTEP]\nrw [show (fun e => -f e) = fun e => (-1 : \ud835\udd5c) \u2022 f e by funext; simp]\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 \ud835\udd5c G\nf g : E \u2192 F\nhf : IsBoundedLinearMap \ud835\udd5c f\n\u22a2 (fun e => -f e) = fun e => -1 \u2022 f e\n[PROOFSTEP]\nfunext\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 \ud835\udd5c G\nf g : E \u2192 F\nhf : IsBoundedLinearMap \ud835\udd5c f\nx\u271d : E\n\u22a2 -f x\u271d = -1 \u2022 f x\u271d\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 \ud835\udd5c G\nf g : E \u2192 F\nhf : IsBoundedLinearMap \ud835\udd5c f\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun e => -1 \u2022 f e\n[PROOFSTEP]\nexact smul (-1) hf\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 \ud835\udd5c G\nf g : E \u2192 F\nhf : IsBoundedLinearMap \ud835\udd5c f\nhg : IsBoundedLinearMap \ud835\udd5c g\nhlf : IsLinearMap \ud835\udd5c f\nMf : \u211d\nleft\u271d\u00b9 : 0 < Mf\nhMf : \u2200 (x : E), \u2016f x\u2016 \u2264 Mf * \u2016x\u2016\nhlg : IsLinearMap \ud835\udd5c g\nMg : \u211d\nleft\u271d : 0 < Mg\nhMg : \u2200 (x : E), \u2016g x\u2016 \u2264 Mg * \u2016x\u2016\nx : E\n\u22a2 Mf * \u2016x\u2016 + Mg * \u2016x\u2016 \u2264 (Mf + Mg) * \u2016x\u2016\n[PROOFSTEP]\nrw [add_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 \ud835\udd5c G\nf g : E \u2192 F\nhf : IsBoundedLinearMap \ud835\udd5c f\nhg : IsBoundedLinearMap \ud835\udd5c g\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun e => f e - g e\n[PROOFSTEP]\nsimpa [sub_eq_add_neg] using add hf (neg hg)\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 \ud835\udd5c G\nf g : E \u2192 F\nx : E\nhf\u271d : IsBoundedLinearMap \ud835\udd5c f\nhf : IsLinearMap \ud835\udd5c f\nM : \u211d\nleft\u271d : 0 < M\nhM : \u2200 (x : E), \u2016f x\u2016 \u2264 M * \u2016x\u2016\ne : E\n\u22a2 \u2016f e - f x\u2016 = \u2016\u2191(IsLinearMap.mk' f hf) (e - x)\u2016\n[PROOFSTEP]\nrw [(hf.mk' _).map_sub e x]\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 \ud835\udd5c G\nf g : E \u2192 F\nx : E\nhf\u271d : IsBoundedLinearMap \ud835\udd5c f\nhf : IsLinearMap \ud835\udd5c f\nM : \u211d\nleft\u271d : 0 < M\nhM : \u2200 (x : E), \u2016f x\u2016 \u2264 M * \u2016x\u2016\ne : E\n\u22a2 \u2016f e - f x\u2016 = \u2016\u2191(IsLinearMap.mk' f hf) e - \u2191(IsLinearMap.mk' f hf) x\u2016\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\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\nx : E\nhf\u271d : IsBoundedLinearMap \ud835\udd5c f\nhf : IsLinearMap \ud835\udd5c f\nM : \u211d\nleft\u271d : 0 < M\nhM : \u2200 (x : E), \u2016f x\u2016 \u2264 M * \u2016x\u2016\nthis : Tendsto (fun e => M * \u2016e - x\u2016) (\ud835\udcdd x) (\ud835\udcdd (M * 0))\n\u22a2 Tendsto (fun e => M * \u2016e - x\u2016) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np\u2081 p\u2082 : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 ContinuousMultilinearMap.prod (p\u2081 + p\u2082).fst (p\u2081 + p\u2082).snd =\n    ContinuousMultilinearMap.prod p\u2081.fst p\u2081.snd + ContinuousMultilinearMap.prod p\u2082.fst p\u2082.snd\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np\u2081 p\u2082 : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(ContinuousMultilinearMap.prod (p\u2081 + p\u2082).fst (p\u2081 + p\u2082).snd) m =\n    \u2191(ContinuousMultilinearMap.prod p\u2081.fst p\u2081.snd + ContinuousMultilinearMap.prod p\u2082.fst p\u2082.snd) m\n[PROOFSTEP]\nrfl\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\nc : \ud835\udd5c\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 ContinuousMultilinearMap.prod (c \u2022 p).fst (c \u2022 p).snd = c \u2022 ContinuousMultilinearMap.prod p.fst p.snd\n[PROOFSTEP]\next1 m\n[GOAL]\ncase H\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\nc : \ud835\udd5c\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2191(ContinuousMultilinearMap.prod (c \u2022 p).fst (c \u2022 p).snd) m = \u2191(c \u2022 ContinuousMultilinearMap.prod p.fst p.snd) m\n[PROOFSTEP]\nrfl\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2016ContinuousMultilinearMap.prod p.fst p.snd\u2016 \u2264 1 * \u2016p\u2016\n[PROOFSTEP]\nrw [one_mul]\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2016ContinuousMultilinearMap.prod p.fst p.snd\u2016 \u2264 \u2016p\u2016\n[PROOFSTEP]\napply ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) _\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191(ContinuousMultilinearMap.prod p.fst p.snd) m\u2016 \u2264 \u2016p\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nintro m\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191(ContinuousMultilinearMap.prod p.fst p.snd) m\u2016 \u2264 \u2016p\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nrw [ContinuousMultilinearMap.prod_apply, norm_prod_le_iff]\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016(\u2191p.fst m, \u2191p.snd m).fst\u2016 \u2264 \u2016p\u2016 * \u220f i : \u03b9, \u2016m i\u2016 \u2227 \u2016(\u2191p.fst m, \u2191p.snd m).snd\u2016 \u2264 \u2016p\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016(\u2191p.fst m, \u2191p.snd m).fst\u2016 \u2264 \u2016p\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact\n  (p.1.le_op_norm m).trans (mul_le_mul_of_nonneg_right (norm_fst_le p) (Finset.prod_nonneg fun i _ => norm_nonneg _))\n[GOAL]\ncase right\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 \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Fintype \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)\np : ContinuousMultilinearMap \ud835\udd5c E F \u00d7 ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 \u2016(\u2191p.fst m, \u2191p.snd m).snd\u2016 \u2264 \u2016p\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nexact\n  (p.2.le_op_norm m).trans (mul_le_mul_of_nonneg_right (norm_snd_le p) (Finset.prod_nonneg fun i _ => norm_nonneg _))\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\n\u22a2 IsBoundedLinearMap \ud835\udd5c fun f => ContinuousMultilinearMap.compContinuousLinearMap f fun x => g\n[PROOFSTEP]\nrefine' IsLinearMap.with_bound \u27e8fun f\u2081 f\u2082 => by ext; rfl, fun c f => by ext; rfl\u27e9 (\u2016g\u2016 ^ Fintype.card \u03b9) fun f => _\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\n\u22a2 (ContinuousMultilinearMap.compContinuousLinearMap (f\u2081 + f\u2082) fun x => g) =\n    (ContinuousMultilinearMap.compContinuousLinearMap f\u2081 fun x => g) +\n      ContinuousMultilinearMap.compContinuousLinearMap f\u2082 fun x => g\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf\u2081 f\u2082 : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nx\u271d : \u03b9 \u2192 G\n\u22a2 \u2191(ContinuousMultilinearMap.compContinuousLinearMap (f\u2081 + f\u2082) fun x => g) x\u271d =\n    \u2191((ContinuousMultilinearMap.compContinuousLinearMap f\u2081 fun x => g) +\n          ContinuousMultilinearMap.compContinuousLinearMap f\u2082 fun x => g)\n      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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nc : \ud835\udd5c\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\n\u22a2 (ContinuousMultilinearMap.compContinuousLinearMap (c \u2022 f) fun x => g) =\n    c \u2022 ContinuousMultilinearMap.compContinuousLinearMap f fun x => g\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nc : \ud835\udd5c\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nx\u271d : \u03b9 \u2192 G\n\u22a2 \u2191(ContinuousMultilinearMap.compContinuousLinearMap (c \u2022 f) fun x => g) x\u271d =\n    \u2191(c \u2022 ContinuousMultilinearMap.compContinuousLinearMap f fun x => g) 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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\n\u22a2 \u2016ContinuousMultilinearMap.compContinuousLinearMap f fun x => g\u2016 \u2264 \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016\n[PROOFSTEP]\napply ContinuousMultilinearMap.op_norm_le_bound _ _ _\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\n\u22a2 0 \u2264 \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016\n[PROOFSTEP]\napply_rules [mul_nonneg, pow_nonneg, norm_nonneg]\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\n\u22a2 \u2200 (m : \u03b9 \u2192 G),\n    \u2016\u2191(ContinuousMultilinearMap.compContinuousLinearMap f fun x => g) m\u2016 \u2264 \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nintro m\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nm : \u03b9 \u2192 G\n\u22a2 \u2016\u2191(ContinuousMultilinearMap.compContinuousLinearMap f fun x => g) m\u2016 \u2264 \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\ncalc\n  \u2016f (g \u2218 m)\u2016 \u2264 \u2016f\u2016 * \u220f i, \u2016g (m i)\u2016 := f.le_op_norm _\n  _ \u2264 \u2016f\u2016 * \u220f i, \u2016g\u2016 * \u2016m i\u2016 := by\n    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)\n    exact Finset.prod_le_prod (fun i _ => norm_nonneg _) fun i _ => g.le_op_norm _\n  _ = \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016 * \u220f i, \u2016m i\u2016 :=\n    by\n    simp [Finset.prod_mul_distrib, Finset.card_univ]\n    ring\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nm : \u03b9 \u2192 G\n\u22a2 \u2016f\u2016 * \u220f i : \u03b9, \u2016\u2191g (m i)\u2016 \u2264 \u2016f\u2016 * \u220f i : \u03b9, \u2016g\u2016 * \u2016m i\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left _ (norm_nonneg _)\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nm : \u03b9 \u2192 G\n\u22a2 \u220f i : \u03b9, \u2016\u2191g (m i)\u2016 \u2264 \u220f i : \u03b9, \u2016g\u2016 * \u2016m i\u2016\n[PROOFSTEP]\nexact Finset.prod_le_prod (fun i _ => norm_nonneg _) fun i _ => g.le_op_norm _\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nm : \u03b9 \u2192 G\n\u22a2 \u2016f\u2016 * \u220f i : \u03b9, \u2016g\u2016 * \u2016m i\u2016 = \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016 * \u220f i : \u03b9, \u2016m i\u2016\n[PROOFSTEP]\nsimp [Finset.prod_mul_distrib, Finset.card_univ]\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\n\u03b9 : Type u_5\ninst\u271d : Fintype \u03b9\ng : G \u2192L[\ud835\udd5c] E\nf : ContinuousMultilinearMap \ud835\udd5c (fun x => E) F\nm : \u03b9 \u2192 G\n\u22a2 \u2016f\u2016 * (\u2016g\u2016 ^ Fintype.card \u03b9 * \u220f x : \u03b9, \u2016m x\u2016) = \u2016g\u2016 ^ Fintype.card \u03b9 * \u2016f\u2016 * \u220f x : \u03b9, \u2016m x\u2016\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup G\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c G\nR : Type u_5\n\ud835\udd5c\u2082 : Type u_6\n\ud835\udd5c' : Type u_7\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2082\nM : Type u_8\ninst\u271d\u2077 : TopologicalSpace M\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nG' : Type u_9\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2082 G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G'\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c\u2082 \ud835\udd5c' G'\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c1\u2081\u2082 : R \u2192+* \ud835\udd5c'\nf : M \u2192SL[\u03c1\u2081\u2082] F \u2192SL[\u03c3\u2081\u2082] G'\nx x' : M\ny : F\n\u22a2 \u2191(\u2191f (x + x')) y = \u2191(\u2191f x) y + \u2191(\u2191f x') y\n[PROOFSTEP]\nrw [f.map_add, add_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup G\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c G\nR : Type u_5\n\ud835\udd5c\u2082 : Type u_6\n\ud835\udd5c' : Type u_7\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2082\nM : Type u_8\ninst\u271d\u2077 : TopologicalSpace M\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nG' : Type u_9\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2082 G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G'\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c\u2082 \ud835\udd5c' G'\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c1\u2081\u2082 : R \u2192+* \ud835\udd5c'\nf : M \u2192SL[\u03c1\u2081\u2082] F \u2192SL[\u03c3\u2081\u2082] G'\ny : F\n\u22a2 \u2191(\u2191f 0) y = 0\n[PROOFSTEP]\nrw [f.map_zero, zero_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup G\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c G\nR : Type u_5\n\ud835\udd5c\u2082 : Type u_6\n\ud835\udd5c' : Type u_7\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2082\nM : Type u_8\ninst\u271d\u2077 : TopologicalSpace M\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nG' : Type u_9\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2082 G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G'\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c\u2082 \ud835\udd5c' G'\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u03c1\u2081\u2082 : R \u2192+* \ud835\udd5c'\nf : M \u2192SL[\u03c1\u2081\u2082] F \u2192SL[\u03c3\u2081\u2082] G'\nc : R\nx : M\ny : F\n\u22a2 \u2191(\u2191f (c \u2022 x)) y = \u2191\u03c1\u2081\u2082 c \u2022 \u2191(\u2191f x) y\n[PROOFSTEP]\nrw [f.map_smul\u209b\u2097, smul_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup G\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c G\nR : Type u_5\n\ud835\udd5c\u2082 : Type u_6\n\ud835\udd5c' : Type u_7\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2082\nM : Type u_8\ninst\u271d\u2077 : TopologicalSpace M\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nG' : Type u_9\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2082 G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G'\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c\u2082 \ud835\udd5c' G'\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c1\u2081\u2082 : R \u2192+* \ud835\udd5c'\nf : M \u2192SL[\u03c1\u2081\u2082] F \u2192SL[\u03c3\u2081\u2082] G'\nx x' : M\ny : F\n\u22a2 \u2191(\u2191f (x - x')) y = \u2191(\u2191f x) y - \u2191(\u2191f x') y\n[PROOFSTEP]\nrw [f.map_sub, sub_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup G\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c G\nR : Type u_5\n\ud835\udd5c\u2082 : Type u_6\n\ud835\udd5c' : Type u_7\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2082\nM : Type u_8\ninst\u271d\u2077 : TopologicalSpace M\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nG' : Type u_9\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2082 G'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c' G'\ninst\u271d\u00b3 : SMulCommClass \ud835\udd5c\u2082 \ud835\udd5c' G'\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03c1\u2081\u2082 : R \u2192+* \ud835\udd5c'\nf : M \u2192SL[\u03c1\u2081\u2082] F \u2192SL[\u03c3\u2081\u2082] G'\nx : M\ny : F\n\u22a2 \u2191(\u2191f (-x)) y = -\u2191(\u2191f x) y\n[PROOFSTEP]\nrw [f.map_neg, neg_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\nR : Type u_5\n\ud835\udd5c\u2082 : Type u_6\n\ud835\udd5c' : Type u_7\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u2082\nM : Type u_8\ninst\u271d\u2074 : TopologicalSpace M\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nG' : Type u_9\ninst\u271d\u00b3 : NormedAddCommGroup G'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c\u2082 G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G'\ninst\u271d : SMulCommClass \ud835\udd5c\u2082 \ud835\udd5c' G'\nf : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nc : \ud835\udd5c\nx : E\ny : F\n\u22a2 \u2191(\u2191f (c \u2022 x)) y = c \u2022 \u2191(\u2191f x) y\n[PROOFSTEP]\nrw [f.map_smul, smul_apply]\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 \ud835\udd5c G\nf\u271d : E \u00d7 F \u2192 G\nf : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nx : E\ny : F\n\u22a2 \u2016f\u2016 * \u2016x\u2016 * \u2016y\u2016 \u2264 max \u2016f\u2016 1 * \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\napply_rules [mul_le_mul_of_nonneg_right, norm_nonneg, le_max_left]\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nC : \u211d\nCpos : C > 0\nhC : \u2200 (x : E) (y : F), \u2016f (x, y)\u2016 \u2264 C * \u2016x\u2016 * \u2016y\u2016\nx\u271d : E \u00d7 F\nx : E\ny : F\n\u22a2 \u2016f (x, y)\u2016 \u2264 ?m.264130 h C Cpos hC * \u2016\u2016(x, y).fst\u2016 * \u2016(x, y).snd\u2016\u2016\n[PROOFSTEP]\nsimpa [mul_assoc] using hC x 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\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\n\u22a2 Continuous f\n[PROOFSTEP]\nrefine continuous_iff_continuousAt.2 fun x \u21a6 tendsto_sub_nhds_zero_iff.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\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 Tendsto (fun n => f n - f x) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices : Tendsto (\u03bb y : E \u00d7 F \u21a6 f (y.1 - x.1, y.2) + f (x.1, y.2 - x.2)) (\ud835\udcdd x) (\ud835\udcdd (0 + 0))\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\nthis : Tendsto (fun y => f (y.fst - x.fst, y.snd) + f (x.fst, y.snd - x.snd)) (\ud835\udcdd x) (\ud835\udcdd (0 + 0))\n\u22a2 Tendsto (fun n => f n - f x) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [h.map_sub_left, h.map_sub_right, sub_add_sub_cancel, zero_add] using this\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 Tendsto (fun y => f (y.fst - x.fst, y.snd) + f (x.fst, y.snd - x.snd)) (\ud835\udcdd x) (\ud835\udcdd (0 + 0))\n[PROOFSTEP]\napply Tendsto.add\n[GOAL]\ncase this.hf\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 Tendsto (fun x_1 => f (x_1.fst - x.fst, x_1.snd)) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 isLittleO_one_iff \u211d, \u2190 one_mul 1]\n[GOAL]\ncase this.hf\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 (fun x_1 => f (x_1.fst - x.fst, x_1.snd)) =o[\ud835\udcdd x] fun _x => 1 * 1\n[PROOFSTEP]\nrefine h.isBigO_comp.trans_isLittleO ?_\n[GOAL]\ncase this.hf\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 (fun x_1 => \u2016x_1.fst - x.fst\u2016 * \u2016x_1.snd\u2016) =o[\ud835\udcdd x] fun _x => 1 * 1\n[PROOFSTEP]\nrefine (IsLittleO.norm_left ?_).mul_isBigO (IsBigO.norm_left ?_)\n[GOAL]\ncase this.hf.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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 (fun x_1 => x_1.fst - x.fst) =o[\ud835\udcdd x] fun _x => 1\n[PROOFSTEP]\nexact (isLittleO_one_iff _).2 (tendsto_sub_nhds_zero_iff.2 (continuous_fst.tendsto _))\n[GOAL]\ncase this.hf.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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 (fun x => x.snd) =O[\ud835\udcdd x] fun _x => 1\n[PROOFSTEP]\nexact (continuous_snd.tendsto _).isBigO_one \u211d\n[GOAL]\ncase this.hg\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 Tendsto (fun x_1 => f (x.fst, x_1.snd - x.snd)) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine Continuous.tendsto' ?_ _ _ (by rw [h.map_sub_right, sub_self])\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 f (x.fst, x.snd - x.snd) = 0\n[PROOFSTEP]\nrw [h.map_sub_right, sub_self]\n[GOAL]\ncase this.hg\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\nh : IsBoundedBilinearMap \ud835\udd5c f\nx : E \u00d7 F\n\u22a2 Continuous fun x_1 => f (x.fst, x_1.snd - x.snd)\n[PROOFSTEP]\nexact ((h.toContinuousLinearMap x.1).continuous).comp (continuous_snd.sub continuous_const)\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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\n\u22a2 IsBoundedBilinearMap \ud835\udd5c fun p => p.fst * p.snd\n[PROOFSTEP]\nsimp_rw [\u2190 smul_eq_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 \ud835\udd5c G\nf : E \u00d7 F \u2192 G\n\u22a2 IsBoundedBilinearMap \ud835\udd5c fun p => p.fst \u2022 p.snd\n[PROOFSTEP]\nexact isBoundedBilinearMap_smul\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\n\u22a2 IsOpen (range toContinuousLinearMap)\n[PROOFSTEP]\nrw [isOpen_iff_mem_nhds, forall_range_iff]\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\n\u22a2 \u2200 (i : E \u2243L[\ud835\udd5c] F), range toContinuousLinearMap \u2208 \ud835\udcdd \u2191i\n[PROOFSTEP]\nrefine' fun e => IsOpen.mem_nhds _ (mem_range_self _)\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\n\u22a2 IsOpen (range toContinuousLinearMap)\n[PROOFSTEP]\nlet O : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => (e.symm : F \u2192L[\ud835\udd5c] E).comp f\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\n\u22a2 IsOpen (range toContinuousLinearMap)\n[PROOFSTEP]\nhave h_O : Continuous O := isBoundedBilinearMap_comp.continuous_right\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\n\u22a2 IsOpen (range toContinuousLinearMap)\n[PROOFSTEP]\nconvert show IsOpen (O \u207b\u00b9' {x | IsUnit x}) from Units.isOpen.preimage h_O using 1\n[GOAL]\ncase h.e'_3\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\n\u22a2 range toContinuousLinearMap = O \u207b\u00b9' {x | IsUnit x}\n[PROOFSTEP]\next f'\n[GOAL]\ncase h.e'_3.h\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\nf' : E \u2192L[\ud835\udd5c] F\n\u22a2 f' \u2208 range toContinuousLinearMap \u2194 f' \u2208 O \u207b\u00b9' {x | IsUnit x}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_3.h.mp\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\nf' : E \u2192L[\ud835\udd5c] F\n\u22a2 f' \u2208 range toContinuousLinearMap \u2192 f' \u2208 O \u207b\u00b9' {x | IsUnit x}\n[PROOFSTEP]\nrintro \u27e8e', rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\ne' : E \u2243L[\ud835\udd5c] F\n\u22a2 \u2191e' \u2208 O \u207b\u00b9' {x | IsUnit x}\n[PROOFSTEP]\nexact \u27e8(e'.trans e.symm).toUnit, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mpr\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\nf' : E \u2192L[\ud835\udd5c] F\n\u22a2 f' \u2208 O \u207b\u00b9' {x | IsUnit x} \u2192 f' \u2208 range toContinuousLinearMap\n[PROOFSTEP]\nrintro \u27e8w, hw\u27e9\n[GOAL]\ncase h.e'_3.h.mpr.intro\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\nf' : E \u2192L[\ud835\udd5c] F\nw : (E \u2192L[\ud835\udd5c] E)\u02e3\nhw : \u2191w = O f'\n\u22a2 f' \u2208 range toContinuousLinearMap\n[PROOFSTEP]\nuse(unitsEquiv \ud835\udd5c E w).trans e\n[GOAL]\ncase h\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\nf' : E \u2192L[\ud835\udd5c] F\nw : (E \u2192L[\ud835\udd5c] E)\u02e3\nhw : \u2191w = O f'\n\u22a2 \u2191(ContinuousLinearEquiv.trans (\u2191(unitsEquiv \ud835\udd5c E) w) e) = f'\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\nO : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] E := fun f => comp (\u2191(ContinuousLinearEquiv.symm e)) f\nh_O : Continuous O\nf' : E \u2192L[\ud835\udd5c] F\nw : (E \u2192L[\ud835\udd5c] E)\u02e3\nhw : \u2191w = O f'\nx : E\n\u22a2 \u2191\u2191(ContinuousLinearEquiv.trans (\u2191(unitsEquiv \ud835\udd5c E) w) e) x = \u2191f' x\n[PROOFSTEP]\nsimp [hw]\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\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace E\ne : E \u2243L[\ud835\udd5c] F\n\u22a2 \u2191e \u2208 range toContinuousLinearMap\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "llama_tokens": 20096, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.6757646075489392, "lm_q1q2_score": 0.5674624951095444}}
{"text": "[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nclassical\nby_cases hF : ringChar F = 2\nfocus\n  have h := FiniteField.even_card_of_char_two hF\n  simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]\nrotate_left\nfocus\n  have h := FiniteField.odd_card_of_char_ne_two hF\n  rw [\u2190 quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticChar_two hF, \u03c7\u2088_nat_eq_if_mod_eight]\n  simp only [h, Nat.one_ne_zero, if_false, ite_eq_left_iff, Ne.def, (by decide : (-1 : \u2124) \u2260 1), imp_false,\n    Classical.not_not]\nall_goals\n  rw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n  have h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n  revert h\u2081 h\n  generalize Fintype.card F % 8 = n\n  intros;\n  interval_cases n <;>\n    simp_all\n      -- Porting note: was `decide!`\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nby_cases hF : ringChar F = 2\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nfocus\n  have h := FiniteField.even_card_of_char_two hF\n  simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nhave h := FiniteField.even_card_of_char_two hF\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nsimp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nfocus\n  have h := FiniteField.odd_card_of_char_ne_two hF\n  rw [\u2190 quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticChar_two hF, \u03c7\u2088_nat_eq_if_mod_eight]\n  simp only [h, Nat.one_ne_zero, if_false, ite_eq_left_iff, Ne.def, (by decide : (-1 : \u2124) \u2260 1), imp_false,\n    Classical.not_not]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nhave h := FiniteField.odd_card_of_char_ne_two hF\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 IsSquare 2 \u2194 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nrw [\u2190 quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticChar_two hF, \u03c7\u2088_nat_eq_if_mod_eight]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 (if Fintype.card F % 2 = 0 then 0 else if Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 7 then 1 else -1) = 1 \u2194\n    Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nsimp only [h, Nat.one_ne_zero, if_false, ite_eq_left_iff, Ne.def, (by decide : (-1 : \u2124) \u2260 1), imp_false,\n  Classical.not_not]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 7 \u2194 \u00acFintype.card F % 8 = 3 \u2227 \u00acFintype.card F % 8 = 5\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nall_goals\n  rw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n  have h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n  revert h\u2081 h\n  generalize Fintype.card F % 8 = n\n  intros;\n  interval_cases n <;>\n    simp_all\n      -- Porting note: was `decide!`\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 7 \u2194 \u00acFintype.card F % 8 = 3 \u2227 \u00acFintype.card F % 8 = 5\n[PROOFSTEP]\nrw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 2 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 8 % 2 = 1\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 7 \u2194 \u00acFintype.card F % 8 = 3 \u2227 \u00acFintype.card F % 8 = 5\n[PROOFSTEP]\nhave h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 8 % 2 = 1\n\u22a2 0 < 8\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 8 % 2 = 1\nh\u2081 : Fintype.card F % 8 < 8\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 7 \u2194 \u00acFintype.card F % 8 = 3 \u2227 \u00acFintype.card F % 8 = 5\n[PROOFSTEP]\nrevert h\u2081 h\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 Fintype.card F % 8 % 2 = 1 \u2192\n    Fintype.card F % 8 < 8 \u2192\n      (Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 7 \u2194 \u00acFintype.card F % 8 = 3 \u2227 \u00acFintype.card F % 8 = 5)\n[PROOFSTEP]\ngeneralize Fintype.card F % 8 = n\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\n\u22a2 n % 2 = 1 \u2192 n < 8 \u2192 (n = 1 \u2228 n = 7 \u2194 \u00acn = 3 \u2227 \u00acn = 5)\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : n % 2 = 1\nh\u2081\u271d : n < 8\n\u22a2 n = 1 \u2228 n = 7 \u2194 \u00acn = 3 \u2227 \u00acn = 5\n[PROOFSTEP]\ninterval_cases n\n[GOAL]\ncase neg.\u00ab0\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 0 % 2 = 1\nh\u2081\u271d : 0 < 8\n\u22a2 0 = 1 \u2228 0 = 7 \u2194 \u00ac0 = 3 \u2227 \u00ac0 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab1\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 1 % 2 = 1\nh\u2081\u271d : 1 < 8\n\u22a2 1 = 1 \u2228 1 = 7 \u2194 \u00ac1 = 3 \u2227 \u00ac1 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab2\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 2 % 2 = 1\nh\u2081\u271d : 2 < 8\n\u22a2 2 = 1 \u2228 2 = 7 \u2194 \u00ac2 = 3 \u2227 \u00ac2 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab3\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 3 % 2 = 1\nh\u2081\u271d : 3 < 8\n\u22a2 3 = 1 \u2228 3 = 7 \u2194 \u00ac3 = 3 \u2227 \u00ac3 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab4\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 4 % 2 = 1\nh\u2081\u271d : 4 < 8\n\u22a2 4 = 1 \u2228 4 = 7 \u2194 \u00ac4 = 3 \u2227 \u00ac4 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab5\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 5 % 2 = 1\nh\u2081\u271d : 5 < 8\n\u22a2 5 = 1 \u2228 5 = 7 \u2194 \u00ac5 = 3 \u2227 \u00ac5 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab6\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 6 % 2 = 1\nh\u2081\u271d : 6 < 8\n\u22a2 6 = 1 \u2228 6 = 7 \u2194 \u00ac6 = 3 \u2227 \u00ac6 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab7\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 7 % 2 = 1\nh\u2081\u271d : 7 < 8\n\u22a2 7 = 1 \u2228 7 = 7 \u2194 \u00ac7 = 3 \u2227 \u00ac7 = 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nrw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 2 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 8 % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nhave h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 8 % 2 = 0\n\u22a2 0 < 8\n[PROOFSTEP]\ndecide\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 8 % 2 = 0\nh\u2081 : Fintype.card F % 8 < 8\n\u22a2 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\nrevert h\u2081 h\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\n\u22a2 Fintype.card F % 8 % 2 = 0 \u2192 Fintype.card F % 8 < 8 \u2192 Fintype.card F % 8 \u2260 3 \u2227 Fintype.card F % 8 \u2260 5\n[PROOFSTEP]\ngeneralize Fintype.card F % 8 = n\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\n\u22a2 n % 2 = 0 \u2192 n < 8 \u2192 n \u2260 3 \u2227 n \u2260 5\n[PROOFSTEP]\nintros\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : n % 2 = 0\nh\u2081\u271d : n < 8\n\u22a2 n \u2260 3 \u2227 n \u2260 5\n[PROOFSTEP]\ninterval_cases n\n[GOAL]\ncase pos.\u00ab0\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 0 % 2 = 0\nh\u2081\u271d : 0 < 8\n\u22a2 0 \u2260 3 \u2227 0 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab1\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 1 % 2 = 0\nh\u2081\u271d : 1 < 8\n\u22a2 1 \u2260 3 \u2227 1 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab2\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 2 % 2 = 0\nh\u2081\u271d : 2 < 8\n\u22a2 2 \u2260 3 \u2227 2 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab3\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 3 % 2 = 0\nh\u2081\u271d : 3 < 8\n\u22a2 3 \u2260 3 \u2227 3 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab4\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 4 % 2 = 0\nh\u2081\u271d : 4 < 8\n\u22a2 4 \u2260 3 \u2227 4 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab5\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 5 % 2 = 0\nh\u2081\u271d : 5 < 8\n\u22a2 5 \u2260 3 \u2227 5 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab6\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 6 % 2 = 0\nh\u2081\u271d : 6 < 8\n\u22a2 6 \u2260 3 \u2227 6 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab7\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 7 % 2 = 0\nh\u2081\u271d : 7 < 8\n\u22a2 7 \u2260 3 \u2227 7 \u2260 5\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\ninst\u271d : DecidableEq F\nhF : ringChar F \u2260 2\n\u22a2 \u2191(quadraticChar F) (-2) = \u2191\u03c7\u2088' \u2191(Fintype.card F)\n[PROOFSTEP]\nrw [(by norm_num : (-2 : F) = -1 * 2), map_mul, \u03c7\u2088'_eq_\u03c7\u2084_mul_\u03c7\u2088, quadraticChar_neg_one hF, quadraticChar_two hF,\n  @cast_nat_cast _ (ZMod 4) _ _ _ (by norm_num : 4 \u2223 8)]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\ninst\u271d : DecidableEq F\nhF : ringChar F \u2260 2\n\u22a2 -2 = -1 * 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\ninst\u271d : DecidableEq F\nhF : ringChar F \u2260 2\n\u22a2 4 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nclassical\nby_cases hF : ringChar F = 2\nfocus\n  have h := FiniteField.even_card_of_char_two hF\n  simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]\nrotate_left\nfocus\n  have h := FiniteField.odd_card_of_char_ne_two hF\n  rw [\u2190 quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)), quadraticChar_neg_two hF,\n    \u03c7\u2088'_nat_eq_if_mod_eight]\n  simp only [h, Nat.one_ne_zero, if_false, ite_eq_left_iff, Ne.def, (by decide : (-1 : \u2124) \u2260 1), imp_false,\n    Classical.not_not]\nall_goals\n  rw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n  have h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n  revert h\u2081 h\n  generalize Fintype.card F % 8 = n\n  intros;\n  interval_cases n <;>\n    simp_all\n      -- Porting note: was `decide!`\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nby_cases hF : ringChar F = 2\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nfocus\n  have h := FiniteField.even_card_of_char_two hF\n  simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nhave h := FiniteField.even_card_of_char_two hF\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nsimp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nfocus\n  have h := FiniteField.odd_card_of_char_ne_two hF\n  rw [\u2190 quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)), quadraticChar_neg_two hF,\n    \u03c7\u2088'_nat_eq_if_mod_eight]\n  simp only [h, Nat.one_ne_zero, if_false, ite_eq_left_iff, Ne.def, (by decide : (-1 : \u2124) \u2260 1), imp_false,\n    Classical.not_not]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nhave h := FiniteField.odd_card_of_char_ne_two hF\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 IsSquare (-2) \u2194 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nrw [\u2190 quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)), quadraticChar_neg_two hF,\n  \u03c7\u2088'_nat_eq_if_mod_eight]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 (if Fintype.card F % 2 = 0 then 0 else if Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 3 then 1 else -1) = 1 \u2194\n    Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nsimp only [h, Nat.one_ne_zero, if_false, ite_eq_left_iff, Ne.def, (by decide : (-1 : \u2124) \u2260 1), imp_false,\n  Classical.not_not]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 3 \u2194 \u00acFintype.card F % 8 = 5 \u2227 \u00acFintype.card F % 8 = 7\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nall_goals\n  rw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n  have h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n  revert h\u2081 h\n  generalize Fintype.card F % 8 = n\n  intros;\n  interval_cases n <;>\n    simp_all\n      -- Porting note: was `decide!`\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 3 \u2194 \u00acFintype.card F % 8 = 5 \u2227 \u00acFintype.card F % 8 = 7\n[PROOFSTEP]\nrw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 2 = 1\n\u22a2 2 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 8 % 2 = 1\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 3 \u2194 \u00acFintype.card F % 8 = 5 \u2227 \u00acFintype.card F % 8 = 7\n[PROOFSTEP]\nhave h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 8 % 2 = 1\n\u22a2 0 < 8\n[PROOFSTEP]\ndecide\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nh : Fintype.card F % 8 % 2 = 1\nh\u2081 : Fintype.card F % 8 < 8\n\u22a2 Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 3 \u2194 \u00acFintype.card F % 8 = 5 \u2227 \u00acFintype.card F % 8 = 7\n[PROOFSTEP]\nrevert h\u2081 h\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\n\u22a2 Fintype.card F % 8 % 2 = 1 \u2192\n    Fintype.card F % 8 < 8 \u2192\n      (Fintype.card F % 8 = 1 \u2228 Fintype.card F % 8 = 3 \u2194 \u00acFintype.card F % 8 = 5 \u2227 \u00acFintype.card F % 8 = 7)\n[PROOFSTEP]\ngeneralize Fintype.card F % 8 = n\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\n\u22a2 n % 2 = 1 \u2192 n < 8 \u2192 (n = 1 \u2228 n = 3 \u2194 \u00acn = 5 \u2227 \u00acn = 7)\n[PROOFSTEP]\nintros\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : n % 2 = 1\nh\u2081\u271d : n < 8\n\u22a2 n = 1 \u2228 n = 3 \u2194 \u00acn = 5 \u2227 \u00acn = 7\n[PROOFSTEP]\ninterval_cases n\n[GOAL]\ncase neg.\u00ab0\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 0 % 2 = 1\nh\u2081\u271d : 0 < 8\n\u22a2 0 = 1 \u2228 0 = 3 \u2194 \u00ac0 = 5 \u2227 \u00ac0 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab1\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 1 % 2 = 1\nh\u2081\u271d : 1 < 8\n\u22a2 1 = 1 \u2228 1 = 3 \u2194 \u00ac1 = 5 \u2227 \u00ac1 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab2\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 2 % 2 = 1\nh\u2081\u271d : 2 < 8\n\u22a2 2 = 1 \u2228 2 = 3 \u2194 \u00ac2 = 5 \u2227 \u00ac2 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab3\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 3 % 2 = 1\nh\u2081\u271d : 3 < 8\n\u22a2 3 = 1 \u2228 3 = 3 \u2194 \u00ac3 = 5 \u2227 \u00ac3 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab4\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 4 % 2 = 1\nh\u2081\u271d : 4 < 8\n\u22a2 4 = 1 \u2228 4 = 3 \u2194 \u00ac4 = 5 \u2227 \u00ac4 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab5\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 5 % 2 = 1\nh\u2081\u271d : 5 < 8\n\u22a2 5 = 1 \u2228 5 = 3 \u2194 \u00ac5 = 5 \u2227 \u00ac5 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab6\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 6 % 2 = 1\nh\u2081\u271d : 6 < 8\n\u22a2 6 = 1 \u2228 6 = 3 \u2194 \u00ac6 = 5 \u2227 \u00ac6 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase neg.\u00ab7\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : \u00acringChar F = 2\nn : \u2115\nh\u271d : 7 % 2 = 1\nh\u2081\u271d : 7 < 8\n\u22a2 7 = 1 \u2228 7 = 3 \u2194 \u00ac7 = 5 \u2227 \u00ac7 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nrw [\u2190 Nat.mod_mod_of_dvd _ (by norm_num : 2 \u2223 8)] at h \n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 2 = 0\n\u22a2 2 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 8 % 2 = 0\n\u22a2 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nhave h\u2081 := Nat.mod_lt (Fintype.card F) (by decide : 0 < 8)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 8 % 2 = 0\n\u22a2 0 < 8\n[PROOFSTEP]\ndecide\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nh : Fintype.card F % 8 % 2 = 0\nh\u2081 : Fintype.card F % 8 < 8\n\u22a2 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\nrevert h\u2081 h\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\n\u22a2 Fintype.card F % 8 % 2 = 0 \u2192 Fintype.card F % 8 < 8 \u2192 Fintype.card F % 8 \u2260 5 \u2227 Fintype.card F % 8 \u2260 7\n[PROOFSTEP]\ngeneralize Fintype.card F % 8 = n\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\n\u22a2 n % 2 = 0 \u2192 n < 8 \u2192 n \u2260 5 \u2227 n \u2260 7\n[PROOFSTEP]\nintros\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : n % 2 = 0\nh\u2081\u271d : n < 8\n\u22a2 n \u2260 5 \u2227 n \u2260 7\n[PROOFSTEP]\ninterval_cases n\n[GOAL]\ncase pos.\u00ab0\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 0 % 2 = 0\nh\u2081\u271d : 0 < 8\n\u22a2 0 \u2260 5 \u2227 0 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab1\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 1 % 2 = 0\nh\u2081\u271d : 1 < 8\n\u22a2 1 \u2260 5 \u2227 1 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab2\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 2 % 2 = 0\nh\u2081\u271d : 2 < 8\n\u22a2 2 \u2260 5 \u2227 2 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab3\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 3 % 2 = 0\nh\u2081\u271d : 3 < 8\n\u22a2 3 \u2260 5 \u2227 3 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab4\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 4 % 2 = 0\nh\u2081\u271d : 4 < 8\n\u22a2 4 \u2260 5 \u2227 4 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab5\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 5 % 2 = 0\nh\u2081\u271d : 5 < 8\n\u22a2 5 \u2260 5 \u2227 5 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab6\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 6 % 2 = 0\nh\u2081\u271d : 6 < 8\n\u22a2 6 \u2260 5 \u2227 6 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase pos.\u00ab7\u00bb\nF : Type u_1\ninst\u271d\u00b9 : Field F\ninst\u271d : Fintype F\nhF : ringChar F = 2\nn : \u2115\nh\u271d : 7 % 2 = 0\nh\u2081\u271d : 7 < 8\n\u22a2 7 \u2260 5 \u2227 7 \u2260 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u22a2 \u2191(quadraticChar F) \u2191(Fintype.card F') = \u2191(quadraticChar F') (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nlet \u03c7 := (quadraticChar F).ringHomComp (algebraMap \u2124 F')\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\n\u22a2 \u2191(quadraticChar F) \u2191(Fintype.card F') = \u2191(quadraticChar F') (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nhave h\u03c7\u2081 : \u03c7.IsNontrivial := by\n  obtain \u27e8a, ha\u27e9 := quadraticChar_exists_neg_one hF\n  have hu : IsUnit a := by\n    contrapose ha\n    exact ne_of_eq_of_ne (map_nonunit (quadraticChar F) ha) (mt zero_eq_neg.mp one_ne_zero)\n  use hu.unit\n  simp only [IsUnit.unit_spec, ringHomComp_apply, eq_intCast, Ne.def, ha]\n  rw [Int.cast_neg, Int.cast_one]\n  exact Ring.neg_one_ne_one_of_char_ne_two hF'\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\n\u22a2 IsNontrivial \u03c7\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := quadraticChar_exists_neg_one hF\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u2191(quadraticChar F) a = -1\n\u22a2 IsNontrivial \u03c7\n[PROOFSTEP]\nhave hu : IsUnit a := by\n  contrapose ha\n  exact ne_of_eq_of_ne (map_nonunit (quadraticChar F) ha) (mt zero_eq_neg.mp one_ne_zero)\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u2191(quadraticChar F) a = -1\n\u22a2 IsUnit a\n[PROOFSTEP]\ncontrapose ha\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u00acIsUnit a\n\u22a2 \u00ac\u2191(quadraticChar F) a = -1\n[PROOFSTEP]\nexact ne_of_eq_of_ne (map_nonunit (quadraticChar F) ha) (mt zero_eq_neg.mp one_ne_zero)\n[GOAL]\ncase intro\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u2191(quadraticChar F) a = -1\nhu : IsUnit a\n\u22a2 IsNontrivial \u03c7\n[PROOFSTEP]\nuse hu.unit\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u2191(quadraticChar F) a = -1\nhu : IsUnit a\n\u22a2 \u2191\u03c7 \u2191(IsUnit.unit hu) \u2260 1\n[PROOFSTEP]\nsimp only [IsUnit.unit_spec, ringHomComp_apply, eq_intCast, Ne.def, ha]\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u2191(quadraticChar F) a = -1\nhu : IsUnit a\n\u22a2 \u00ac\u2191(-1) = 1\n[PROOFSTEP]\nrw [Int.cast_neg, Int.cast_one]\n[GOAL]\ncase h\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\na : F\nha : \u2191(quadraticChar F) a = -1\nhu : IsUnit a\n\u22a2 \u00ac-1 = 1\n[PROOFSTEP]\nexact Ring.neg_one_ne_one_of_char_ne_two hF'\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\nh\u03c7\u2081 : IsNontrivial \u03c7\n\u22a2 \u2191(quadraticChar F) \u2191(Fintype.card F') = \u2191(quadraticChar F') (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nhave h\u03c7\u2082 : \u03c7.IsQuadratic := IsQuadratic.comp (quadraticChar_isQuadratic F) _\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\nh\u03c7\u2081 : IsNontrivial \u03c7\nh\u03c7\u2082 : IsQuadratic \u03c7\n\u22a2 \u2191(quadraticChar F) \u2191(Fintype.card F') = \u2191(quadraticChar F') (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nhave h := Char.card_pow_card h\u03c7\u2081 h\u03c7\u2082 h hF'\n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh\u271d : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\nh\u03c7\u2081 : IsNontrivial \u03c7\nh\u03c7\u2082 : IsQuadratic \u03c7\nh : (\u2191\u03c7 (-1) * \u2191(Fintype.card F)) ^ (Fintype.card F' / 2) = \u2191\u03c7 \u2191(Fintype.card F')\n\u22a2 \u2191(quadraticChar F) \u2191(Fintype.card F') = \u2191(quadraticChar F') (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nrw [\u2190 quadraticChar_eq_pow_of_char_ne_two' hF'] at h \n[GOAL]\nF : Type u_1\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Fintype F\ninst\u271d\u00b3 : DecidableEq F\nhF : ringChar F \u2260 2\nF' : Type u_2\ninst\u271d\u00b2 : Field F'\ninst\u271d\u00b9 : Fintype F'\ninst\u271d : DecidableEq F'\nhF' : ringChar F' \u2260 2\nh\u271d : ringChar F' \u2260 ringChar F\n\u03c7 : MulChar F F' := ringHomComp (quadraticChar F) (algebraMap \u2124 F')\nh\u03c7\u2081 : IsNontrivial \u03c7\nh\u03c7\u2082 : IsQuadratic \u03c7\nh : \u2191(\u2191(quadraticChar F') (\u2191\u03c7 (-1) * \u2191(Fintype.card F))) = \u2191\u03c7 \u2191(Fintype.card F')\n\u22a2 \u2191(quadraticChar F) \u2191(Fintype.card F') = \u2191(quadraticChar F') (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nexact (IsQuadratic.eq_of_eq_coe (quadraticChar_isQuadratic F') (quadraticChar_isQuadratic F) hF' h).symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Fintype F\ninst\u271d\u00b9 : DecidableEq F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp\u2081 : p \u2260 2\nhp\u2082 : ringChar F \u2260 p\n\u22a2 \u2191(quadraticChar F) \u2191p = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nrw [\u2190 quadraticChar_neg_one hF]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Fintype F\ninst\u271d\u00b9 : DecidableEq F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp\u2081 : p \u2260 2\nhp\u2082 : ringChar F \u2260 p\n\u22a2 \u2191(quadraticChar F) \u2191p = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nhave h :=\n  quadraticChar_card_card hF (ne_of_eq_of_ne (ringChar_zmod_n p) hp\u2081) (ne_of_eq_of_ne (ringChar_zmod_n p) hp\u2082.symm)\n[GOAL]\nF : Type u_1\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Fintype F\ninst\u271d\u00b9 : DecidableEq F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp\u2081 : p \u2260 2\nhp\u2082 : ringChar F \u2260 p\nh :\n  \u2191(quadraticChar F) \u2191(Fintype.card (ZMod p)) =\n    \u2191(quadraticChar (ZMod p)) (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n\u22a2 \u2191(quadraticChar F) \u2191p = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F))\n[PROOFSTEP]\nrwa [card p] at h \n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 IsSquare \u2191p \u2194 \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F)) \u2260 -1\n[PROOFSTEP]\nclassical\nby_cases hFp : ringChar F = p\n\u00b7 rw [show (p : F) = 0 by rw [\u2190 hFp]; exact ringChar.Nat.cast_ringChar]\n  simp only [isSquare_zero, Ne.def, true_iff_iff, map_mul]\n  obtain \u27e8n, _, hc\u27e9 := FiniteField.card F (ringChar F)\n  have hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n p).symm\n  conv => enter [1, 1, 2]; rw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]\n  simp only [zero_pow n.pos, mul_zero, zero_eq_neg, one_ne_zero, not_false_iff]\n\u00b7 rw [\u2190 Iff.not_left (@quadraticChar_neg_one_iff_not_isSquare F _ _ _ _), quadraticChar_odd_prime hF hp]\n  exact hFp\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 IsSquare \u2191p \u2194 \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F)) \u2260 -1\n[PROOFSTEP]\nby_cases hFp : ringChar F = p\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\n\u22a2 IsSquare \u2191p \u2194 \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F)) \u2260 -1\n[PROOFSTEP]\nrw [show (p : F) = 0 by rw [\u2190 hFp]; exact ringChar.Nat.cast_ringChar]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\n\u22a2 \u2191p = 0\n[PROOFSTEP]\nrw [\u2190 hFp]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\n\u22a2 \u2191(ringChar F) = 0\n[PROOFSTEP]\nexact ringChar.Nat.cast_ringChar\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\n\u22a2 IsSquare 0 \u2194 \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F)) \u2260 -1\n[PROOFSTEP]\nsimp only [isSquare_zero, Ne.def, true_iff_iff, map_mul]\n[GOAL]\ncase pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\n\u22a2 \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F) = -1\n[PROOFSTEP]\nobtain \u27e8n, _, hc\u27e9 := FiniteField.card F (ringChar F)\n[GOAL]\ncase pos.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F) = -1\n[PROOFSTEP]\nhave hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n p).symm\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 ringChar F = ringChar (ZMod p)\n[PROOFSTEP]\nrw [hFp]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\n\u22a2 p = ringChar (ZMod p)\n[PROOFSTEP]\nexact (ringChar_zmod_n p).symm\n[GOAL]\ncase pos.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\nhchar : ringChar F = ringChar (ZMod p)\n\u22a2 \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F) = -1\n[PROOFSTEP]\nconv => enter [1, 1, 2]; rw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\nhchar : ringChar F = ringChar (ZMod p)\n| \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F) = -1\n[PROOFSTEP]\nenter [1, 1, 2]; rw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\nhchar : ringChar F = ringChar (ZMod p)\n| \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F) = -1\n[PROOFSTEP]\nenter [1, 1, 2]; rw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\nhchar : ringChar F = ringChar (ZMod p)\n| \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F) = -1\n[PROOFSTEP]\nenter [1, 1, 2]\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\nhchar : ringChar F = ringChar (ZMod p)\n| \u2191(quadraticChar (ZMod p)) \u2191(Fintype.card F)\n[PROOFSTEP]\nrw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]\n[GOAL]\ncase pos.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : ringChar F = p\nn : \u2115+\nleft\u271d : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ \u2191n\nhchar : ringChar F = ringChar (ZMod p)\n\u22a2 \u00ac\u2191(quadraticChar (ZMod p)) \u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * 0 ^ \u2191n = -1\n[PROOFSTEP]\nsimp only [zero_pow n.pos, mul_zero, zero_eq_neg, one_ne_zero, not_false_iff]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : \u00acringChar F = p\n\u22a2 IsSquare \u2191p \u2194 \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F)) \u2260 -1\n[PROOFSTEP]\nrw [\u2190 Iff.not_left (@quadraticChar_neg_one_iff_not_isSquare F _ _ _ _), quadraticChar_odd_prime hF hp]\n[GOAL]\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Fintype F\nhF : ringChar F \u2260 2\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nhFp : \u00acringChar F = p\n\u22a2 ringChar F \u2260 p\n[PROOFSTEP]\nexact hFp\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "llama_tokens": 20165, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677660619633, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.5671889001497485}}
{"text": "[GOAL]\n\u22a2 catalan 0 = 1\n[PROOFSTEP]\nrw [catalan]\n[GOAL]\nn : \u2115\n\u22a2 catalan (n + 1) = \u2211 i : Fin (Nat.succ n), catalan \u2191i * catalan (n - \u2191i)\n[PROOFSTEP]\nrw [catalan]\n[GOAL]\nn : \u2115\n\u22a2 catalan (n + 1) = \u2211 ij in Nat.antidiagonal n, catalan ij.fst * catalan ij.snd\n[PROOFSTEP]\nrw [catalan_succ, Nat.sum_antidiagonal_eq_sum_range_succ (fun x y => catalan x * catalan y) n, sum_range]\n[GOAL]\n\u22a2 catalan 1 = 1\n[PROOFSTEP]\nsimp [catalan_succ]\n[GOAL]\nn i : \u2115\nh : i \u2264 n\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave l\u2081 : (n : \u211a) + 1 \u2260 0 := by norm_cast; exact n.succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\n\u22a2 \u2191n + 1 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn i : \u2115\nh : i \u2264 n\n\u22a2 \u00acn + 1 = 0\n[PROOFSTEP]\nexact n.succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave l\u2082 : (n : \u211a) + 1 + 1 \u2260 0 := by norm_cast; exact (n + 1).succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\n\u22a2 \u2191n + 1 + 1 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\n\u22a2 \u00acn + 1 + 1 = 0\n[PROOFSTEP]\nexact (n + 1).succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave l\u2083 : (i : \u211a) + 1 \u2260 0 := by norm_cast; exact i.succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\n\u22a2 \u2191i + 1 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\n\u22a2 \u00aci + 1 = 0\n[PROOFSTEP]\nexact i.succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave l\u2084 : (n : \u211a) - i + 1 \u2260 0 := by norm_cast; exact (n - i).succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\n\u22a2 \u2191n - \u2191i + 1 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\n\u22a2 \u00acn - i + 1 = 0\n[PROOFSTEP]\nexact (n - i).succ_ne_zero\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave h\u2081 := (mul_div_cancel_left (\u2191(Nat.centralBinom (i + 1))) l\u2083).symm\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave h\u2082 := (mul_div_cancel_left (\u2191(Nat.centralBinom (n - i + 1))) l\u2084).symm\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave h\u2083 : ((i : \u211a) + 1) * (i + 1).centralBinom = 2 * (2 * i + 1) * i.centralBinom := by\n  exact_mod_cast Nat.succ_mul_centralBinom_succ i\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\n\u22a2 (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\n[PROOFSTEP]\nexact_mod_cast Nat.succ_mul_centralBinom_succ i\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nhave h\u2084 : ((n : \u211a) - i + 1) * (n - i + 1).centralBinom = 2 * (2 * (n - i) + 1) * (n - i).centralBinom := by\n  exact_mod_cast Nat.succ_mul_centralBinom_succ (n - i)\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\n\u22a2 (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n[PROOFSTEP]\nexact_mod_cast Nat.succ_mul_centralBinom_succ (n - i)\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nsimp only [gosperCatalan]\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 \u2191(Nat.centralBinom (i + 1)) * \u2191(Nat.centralBinom (n + 1 - (i + 1))) * (2 * \u2191(i + 1) - \u2191(n + 1)) /\n        (2 * \u2191(n + 1) * (\u2191(n + 1) + 1)) -\n      \u2191(Nat.centralBinom i) * \u2191(Nat.centralBinom (n + 1 - i)) * (2 * \u2191i - \u2191(n + 1)) / (2 * \u2191(n + 1) * (\u2191(n + 1) + 1)) =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\npush_cast\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 \u2191(Nat.centralBinom (i + 1)) * \u2191(Nat.centralBinom (n + 1 - (i + 1))) * (2 * (\u2191i + 1) - (\u2191n + 1)) /\n        (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) -\n      \u2191(Nat.centralBinom i) * \u2191(Nat.centralBinom (n + 1 - i)) * (2 * \u2191i - (\u2191n + 1)) / (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nrw [show n + 1 - i = n - i + 1 by rw [Nat.add_comm (n - i) 1, \u2190 (Nat.add_sub_assoc h 1), add_comm]]\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 n + 1 - i = n - i + 1\n[PROOFSTEP]\nrw [Nat.add_comm (n - i) 1, \u2190 (Nat.add_sub_assoc h 1), add_comm]\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 \u2191(Nat.centralBinom (i + 1)) * \u2191(Nat.centralBinom (n + 1 - (i + 1))) * (2 * (\u2191i + 1) - (\u2191n + 1)) /\n        (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) -\n      \u2191(Nat.centralBinom i) * \u2191(Nat.centralBinom (n - i + 1)) * (2 * \u2191i - (\u2191n + 1)) / (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nrw [h\u2081, h\u2082, h\u2083, h\u2084]\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n + 1 - (i + 1))) *\n          (2 * (\u2191i + 1) - (\u2191n + 1)) /\n        (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) -\n      \u2191(Nat.centralBinom i) * (2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)) *\n          (2 * \u2191i - (\u2191n + 1)) /\n        (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) =\n    \u2191(Nat.centralBinom i) / (\u2191i + 1) * \u2191(Nat.centralBinom (n - i)) / (\u2191n - \u2191i + 1)\n[PROOFSTEP]\nfield_simp\n[GOAL]\nn i : \u2115\nh : i \u2264 n\nl\u2081 : \u2191n + 1 \u2260 0\nl\u2082 : \u2191n + 1 + 1 \u2260 0\nl\u2083 : \u2191i + 1 \u2260 0\nl\u2084 : \u2191n - \u2191i + 1 \u2260 0\nh\u2081 : \u2191(Nat.centralBinom (i + 1)) = (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) / (\u2191i + 1)\nh\u2082 : \u2191(Nat.centralBinom (n - i + 1)) = (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) / (\u2191n - \u2191i + 1)\nh\u2083 : (\u2191i + 1) * \u2191(Nat.centralBinom (i + 1)) = 2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i)\nh\u2084 : (\u2191n - \u2191i + 1) * \u2191(Nat.centralBinom (n - i + 1)) = 2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))\n\u22a2 (2 * (2 * \u2191i + 1) * \u2191(Nat.centralBinom i) * \u2191(Nat.centralBinom (n - i)) * (2 * (\u2191i + 1) - (\u2191n + 1)) *\n          ((\u2191n - \u2191i + 1) * (2 * (\u2191n + 1) * (\u2191n + 1 + 1))) -\n        (\u2191i + 1) * (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) *\n          (\u2191(Nat.centralBinom i) * (2 * (2 * (\u2191n - \u2191i) + 1) * \u2191(Nat.centralBinom (n - i))) * (2 * \u2191i - (\u2191n + 1)))) *\n      ((\u2191i + 1) * (\u2191n - \u2191i + 1)) =\n    \u2191(Nat.centralBinom i) * \u2191(Nat.centralBinom (n - i)) *\n      ((\u2191i + 1) * (2 * (\u2191n + 1) * (\u2191n + 1 + 1)) * ((\u2191n - \u2191i + 1) * (2 * (\u2191n + 1) * (\u2191n + 1 + 1))))\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = \u2191(Nat.centralBinom (n + 1)) / (\u2191n + 2)\n[PROOFSTEP]\nhave : (n : \u211a) + 1 \u2260 0 := by norm_cast; exact n.succ_ne_zero\n[GOAL]\nn : \u2115\n\u22a2 \u2191n + 1 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn : \u2115\n\u22a2 \u00acn + 1 = 0\n[PROOFSTEP]\nexact n.succ_ne_zero\n[GOAL]\nn : \u2115\nthis : \u2191n + 1 \u2260 0\n\u22a2 gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = \u2191(Nat.centralBinom (n + 1)) / (\u2191n + 2)\n[PROOFSTEP]\nhave : (n : \u211a) + 1 + 1 \u2260 0 := by norm_cast; exact (n + 1).succ_ne_zero\n[GOAL]\nn : \u2115\nthis : \u2191n + 1 \u2260 0\n\u22a2 \u2191n + 1 + 1 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn : \u2115\nthis : \u2191n + 1 \u2260 0\n\u22a2 \u00acn + 1 + 1 = 0\n[PROOFSTEP]\nexact (n + 1).succ_ne_zero\n[GOAL]\nn : \u2115\nthis\u271d : \u2191n + 1 \u2260 0\nthis : \u2191n + 1 + 1 \u2260 0\n\u22a2 gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = \u2191(Nat.centralBinom (n + 1)) / (\u2191n + 2)\n[PROOFSTEP]\nhave h : (n : \u211a) + 2 \u2260 0 := by norm_cast; exact (n + 1).succ_ne_zero\n[GOAL]\nn : \u2115\nthis\u271d : \u2191n + 1 \u2260 0\nthis : \u2191n + 1 + 1 \u2260 0\n\u22a2 \u2191n + 2 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn : \u2115\nthis\u271d : \u2191n + 1 \u2260 0\nthis : \u2191n + 1 + 1 \u2260 0\n\u22a2 \u00acn + 2 = 0\n[PROOFSTEP]\nexact (n + 1).succ_ne_zero\n[GOAL]\nn : \u2115\nthis\u271d : \u2191n + 1 \u2260 0\nthis : \u2191n + 1 + 1 \u2260 0\nh : \u2191n + 2 \u2260 0\n\u22a2 gosperCatalan (n + 1) (n + 1) - gosperCatalan (n + 1) 0 = \u2191(Nat.centralBinom (n + 1)) / (\u2191n + 2)\n[PROOFSTEP]\nsimp only [gosperCatalan, Nat.sub_zero, Nat.centralBinom_zero, Nat.sub_self]\n[GOAL]\nn : \u2115\nthis\u271d : \u2191n + 1 \u2260 0\nthis : \u2191n + 1 + 1 \u2260 0\nh : \u2191n + 2 \u2260 0\n\u22a2 \u2191(Nat.centralBinom (n + 1)) * \u21911 * (2 * \u2191(n + 1) - \u2191(n + 1)) / (2 * \u2191(n + 1) * (\u2191(n + 1) + 1)) -\n      \u21911 * \u2191(Nat.centralBinom (n + 1)) * (2 * \u21910 - \u2191(n + 1)) / (2 * \u2191(n + 1) * (\u2191(n + 1) + 1)) =\n    \u2191(Nat.centralBinom (n + 1)) / (\u2191n + 2)\n[PROOFSTEP]\nfield_simp\n[GOAL]\nn : \u2115\nthis\u271d : \u2191n + 1 \u2260 0\nthis : \u2191n + 1 + 1 \u2260 0\nh : \u2191n + 2 \u2260 0\n\u22a2 (\u2191(Nat.centralBinom (n + 1)) * (2 * (\u2191n + 1) - (\u2191n + 1)) - \u2191(Nat.centralBinom (n + 1)) * (-1 + -\u2191n)) * (\u2191n + 2) =\n    \u2191(Nat.centralBinom (n + 1)) * (2 * (\u2191n + 1) * (\u2191n + 1 + 1))\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 catalan n = Nat.centralBinom n / (n + 1)\n[PROOFSTEP]\nsuffices (catalan n : \u211a) = Nat.centralBinom n / (n + 1)\n  by\n  have h := Nat.succ_dvd_centralBinom n\n  exact_mod_cast this\n[GOAL]\nn : \u2115\nthis : \u2191(catalan n) = \u2191(Nat.centralBinom n) / (\u2191n + 1)\n\u22a2 catalan n = Nat.centralBinom n / (n + 1)\n[PROOFSTEP]\nhave h := Nat.succ_dvd_centralBinom n\n[GOAL]\nn : \u2115\nthis : \u2191(catalan n) = \u2191(Nat.centralBinom n) / (\u2191n + 1)\nh : n + 1 \u2223 Nat.centralBinom n\n\u22a2 catalan n = Nat.centralBinom n / (n + 1)\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\nn : \u2115\n\u22a2 \u2191(catalan n) = \u2191(Nat.centralBinom n) / (\u2191n + 1)\n[PROOFSTEP]\ninduction' n using Nat.case_strong_induction_on with d hd\n[GOAL]\ncase hz\n\u22a2 \u2191(catalan 0) = \u2191(Nat.centralBinom 0) / (\u21910 + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hi\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2191(catalan (Nat.succ d)) = \u2191(Nat.centralBinom (Nat.succ d)) / (\u2191(Nat.succ d) + 1)\n[PROOFSTEP]\nsimp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]\n[GOAL]\ncase hi\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2211 x : Fin (Nat.succ d), \u2191(catalan \u2191x) * \u2191(catalan (d - \u2191x)) = \u2191(Nat.centralBinom (Nat.succ d)) / (\u2191(Nat.succ d) + 1)\n[PROOFSTEP]\ntrans (\u2211 i : Fin d.succ, Nat.centralBinom i / (i + 1) * (Nat.centralBinom (d - i) / (d - i + 1)) : \u211a)\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2211 x : Fin (Nat.succ d), \u2191(catalan \u2191x) * \u2191(catalan (d - \u2191x)) =\n    \u2211 i : Fin (Nat.succ d), \u2191(Nat.centralBinom \u2191i) / (\u2191\u2191i + 1) * (\u2191(Nat.centralBinom (d - \u2191i)) / (\u2191d - \u2191\u2191i + 1))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 (fun x => \u2191(catalan \u2191x) * \u2191(catalan (d - \u2191x))) = fun i =>\n    \u2191(Nat.centralBinom \u2191i) / (\u2191\u2191i + 1) * (\u2191(Nat.centralBinom (d - \u2191i)) / (\u2191d - \u2191\u2191i + 1))\n[PROOFSTEP]\next1 x\n[GOAL]\ncase e_f.h\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\nx : Fin (Nat.succ d)\n\u22a2 \u2191(catalan \u2191x) * \u2191(catalan (d - \u2191x)) =\n    \u2191(Nat.centralBinom \u2191x) / (\u2191\u2191x + 1) * (\u2191(Nat.centralBinom (d - \u2191x)) / (\u2191d - \u2191\u2191x + 1))\n[PROOFSTEP]\nhave m_le_d : x.val \u2264 d := by apply Nat.le_of_lt_succ; apply x.2\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\nx : Fin (Nat.succ d)\n\u22a2 \u2191x \u2264 d\n[PROOFSTEP]\napply Nat.le_of_lt_succ\n[GOAL]\ncase a\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\nx : Fin (Nat.succ d)\n\u22a2 \u2191x < Nat.succ d\n[PROOFSTEP]\napply x.2\n[GOAL]\ncase e_f.h\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\nx : Fin (Nat.succ d)\nm_le_d : \u2191x \u2264 d\n\u22a2 \u2191(catalan \u2191x) * \u2191(catalan (d - \u2191x)) =\n    \u2191(Nat.centralBinom \u2191x) / (\u2191\u2191x + 1) * (\u2191(Nat.centralBinom (d - \u2191x)) / (\u2191d - \u2191\u2191x + 1))\n[PROOFSTEP]\nhave d_minus_x_le_d : (d - x.val) \u2264 d := tsub_le_self\n[GOAL]\ncase e_f.h\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\nx : Fin (Nat.succ d)\nm_le_d : \u2191x \u2264 d\nd_minus_x_le_d : d - \u2191x \u2264 d\n\u22a2 \u2191(catalan \u2191x) * \u2191(catalan (d - \u2191x)) =\n    \u2191(Nat.centralBinom \u2191x) / (\u2191\u2191x + 1) * (\u2191(Nat.centralBinom (d - \u2191x)) / (\u2191d - \u2191\u2191x + 1))\n[PROOFSTEP]\nrw [hd _ m_le_d, hd _ d_minus_x_le_d]\n[GOAL]\ncase e_f.h\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\nx : Fin (Nat.succ d)\nm_le_d : \u2191x \u2264 d\nd_minus_x_le_d : d - \u2191x \u2264 d\n\u22a2 \u2191(Nat.centralBinom \u2191x) / (\u2191\u2191x + 1) * (\u2191(Nat.centralBinom (d - \u2191x)) / (\u2191(d - \u2191x) + 1)) =\n    \u2191(Nat.centralBinom \u2191x) / (\u2191\u2191x + 1) * (\u2191(Nat.centralBinom (d - \u2191x)) / (\u2191d - \u2191\u2191x + 1))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2211 i : Fin (Nat.succ d), \u2191(Nat.centralBinom \u2191i) / (\u2191\u2191i + 1) * (\u2191(Nat.centralBinom (d - \u2191i)) / (\u2191d - \u2191\u2191i + 1)) =\n    \u2191(Nat.centralBinom (Nat.succ d)) / (\u2191(Nat.succ d) + 1)\n[PROOFSTEP]\ntrans (\u2211 i : Fin d.succ, (gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i))\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2211 i : Fin (Nat.succ d), \u2191(Nat.centralBinom \u2191i) / (\u2191\u2191i + 1) * (\u2191(Nat.centralBinom (d - \u2191i)) / (\u2191d - \u2191\u2191i + 1)) =\n    \u2211 i : Fin (Nat.succ d), (gosperCatalan (d + 1) (\u2191i + 1) - gosperCatalan (d + 1) \u2191i)\n[PROOFSTEP]\nrefine' sum_congr rfl fun i _ => _\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\ni : Fin (Nat.succ d)\nx\u271d : i \u2208 univ\n\u22a2 \u2191(Nat.centralBinom \u2191i) / (\u2191\u2191i + 1) * (\u2191(Nat.centralBinom (d - \u2191i)) / (\u2191d - \u2191\u2191i + 1)) =\n    gosperCatalan (d + 1) (\u2191i + 1) - gosperCatalan (d + 1) \u2191i\n[PROOFSTEP]\nrw [gosper_trick i.is_le, mul_div]\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2211 i : Fin (Nat.succ d), (gosperCatalan (d + 1) (\u2191i + 1) - gosperCatalan (d + 1) \u2191i) =\n    \u2191(Nat.centralBinom (Nat.succ d)) / (\u2191(Nat.succ d) + 1)\n[PROOFSTEP]\nrw [\u2190 sum_range fun i => gosperCatalan (d + 1) (i + 1) - gosperCatalan (d + 1) i, sum_range_sub, Nat.succ_eq_add_one]\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 gosperCatalan (d + 1) (d + 1) - gosperCatalan (d + 1) 0 = \u2191(Nat.centralBinom (d + 1)) / (\u2191(d + 1) + 1)\n[PROOFSTEP]\nrw [gosper_catalan_sub_eq_central_binom_div d]\n[GOAL]\nd : \u2115\nhd : \u2200 (m : \u2115), m \u2264 d \u2192 \u2191(catalan m) = \u2191(Nat.centralBinom m) / (\u2191m + 1)\n\u22a2 \u2191(Nat.centralBinom (d + 1)) / (\u2191d + 2) = \u2191(Nat.centralBinom (d + 1)) / (\u2191(d + 1) + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u22a2 catalan 2 = 2\n[PROOFSTEP]\nunfold catalan\n[GOAL]\n\u22a2 \u2211 i : Fin (Nat.succ 1), catalan \u2191i * catalan (1 - \u2191i) = 2\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 catalan 3 = 5\n[PROOFSTEP]\nunfold catalan\n[GOAL]\n\u22a2 \u2211 i : Fin (Nat.succ 2), catalan \u2191i * catalan (2 - \u2191i) = 5\n[PROOFSTEP]\nrfl\n[GOAL]\na b : Finset (Tree Unit)\nx\u271d\u00b9 x\u271d : Tree Unit \u00d7 Tree Unit\nx\u2081 x\u2082 y\u2081 y\u2082 : Tree Unit\nh : (fun x => node () x.fst x.snd) (x\u2081, x\u2082) = (fun x => node () x.fst x.snd) (y\u2081, y\u2082)\n\u22a2 (x\u2081, x\u2082) = (y\u2081, y\u2082)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (invImage (fun a => sizeOf a) instWellFoundedRelation).1 (\u2191ijh).fst (Nat.succ n)\n[PROOFSTEP]\nsimp_wf\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (\u2191ijh).fst < Nat.succ n\n[PROOFSTEP]\ntry exact Nat.lt_succ_of_le (fst_le ijh.2)\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (\u2191ijh).fst < Nat.succ n\n[PROOFSTEP]\nexact Nat.lt_succ_of_le (fst_le ijh.2)\n[GOAL]\n\n[PROOFSTEP]\ntry exact Nat.lt_succ_of_le (snd_le ijh.2)\n[GOAL]\n\n[PROOFSTEP]\nexact Nat.lt_succ_of_le (snd_le ijh.2)\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (invImage (fun a => sizeOf a) instWellFoundedRelation).1 (\u2191ijh).snd (Nat.succ n)\n[PROOFSTEP]\nsimp_wf\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (\u2191ijh).snd < Nat.succ n\n[PROOFSTEP]\ntry exact Nat.lt_succ_of_le (fst_le ijh.2)\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (\u2191ijh).snd < Nat.succ n\n[PROOFSTEP]\nexact Nat.lt_succ_of_le (fst_le ijh.2)\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (\u2191ijh).snd < Nat.succ n\n[PROOFSTEP]\ntry exact Nat.lt_succ_of_le (snd_le ijh.2)\n[GOAL]\nn : \u2115\nijh : { x // x \u2208 Nat.antidiagonal n }\n\u22a2 (\u2191ijh).snd < Nat.succ n\n[PROOFSTEP]\nexact Nat.lt_succ_of_le (snd_le ijh.2)\n[GOAL]\n\u22a2 treesOfNumNodesEq 0 = {nil}\n[PROOFSTEP]\nrw [treesOfNumNodesEq]\n[GOAL]\nn : \u2115\n\u22a2 treesOfNumNodesEq (n + 1) =\n    Finset.biUnion (Nat.antidiagonal n) fun ij => pairwiseNode (treesOfNumNodesEq ij.fst) (treesOfNumNodesEq ij.snd)\n[PROOFSTEP]\nrw [treesOfNumNodesEq]\n[GOAL]\nn : \u2115\n\u22a2 (Finset.biUnion (attach (Nat.antidiagonal n)) fun ijh =>\n      pairwiseNode (treesOfNumNodesEq (\u2191ijh).fst) (treesOfNumNodesEq (\u2191ijh).snd)) =\n    Finset.biUnion (Nat.antidiagonal n) fun ij => pairwiseNode (treesOfNumNodesEq ij.fst) (treesOfNumNodesEq ij.snd)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : \u2115\na\u271d : Tree Unit\n\u22a2 (a\u271d \u2208\n      Finset.biUnion (attach (Nat.antidiagonal n)) fun ijh =>\n        pairwiseNode (treesOfNumNodesEq (\u2191ijh).fst) (treesOfNumNodesEq (\u2191ijh).snd)) \u2194\n    a\u271d \u2208\n      Finset.biUnion (Nat.antidiagonal n) fun ij => pairwiseNode (treesOfNumNodesEq ij.fst) (treesOfNumNodesEq ij.snd)\n[PROOFSTEP]\nsimp\n[GOAL]\nx : Tree Unit\nn : \u2115\n\u22a2 x \u2208 treesOfNumNodesEq n \u2194 numNodes x = n\n[PROOFSTEP]\ninduction x using Tree.unitRecOn generalizing n\n[GOAL]\ncase base\nn : \u2115\n\u22a2 nil \u2208 treesOfNumNodesEq n \u2194 numNodes nil = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ind\nx\u271d y\u271d : Tree Unit\na\u271d\u00b9 : \u2200 {n : \u2115}, x\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes x\u271d = n\na\u271d : \u2200 {n : \u2115}, y\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes y\u271d = n\nn : \u2115\n\u22a2 node () x\u271d y\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes (node () x\u271d y\u271d) = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase base.zero\n\u22a2 nil \u2208 treesOfNumNodesEq Nat.zero \u2194 numNodes nil = Nat.zero\n[PROOFSTEP]\nsimp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]\n[GOAL]\ncase base.succ\nn\u271d : \u2115\n\u22a2 nil \u2208 treesOfNumNodesEq (Nat.succ n\u271d) \u2194 numNodes nil = Nat.succ n\u271d\n[PROOFSTEP]\nsimp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]\n[GOAL]\ncase ind.zero\nx\u271d y\u271d : Tree Unit\na\u271d\u00b9 : \u2200 {n : \u2115}, x\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes x\u271d = n\na\u271d : \u2200 {n : \u2115}, y\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes y\u271d = n\n\u22a2 node () x\u271d y\u271d \u2208 treesOfNumNodesEq Nat.zero \u2194 numNodes (node () x\u271d y\u271d) = Nat.zero\n[PROOFSTEP]\nsimp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]\n[GOAL]\ncase ind.succ\nx\u271d y\u271d : Tree Unit\na\u271d\u00b9 : \u2200 {n : \u2115}, x\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes x\u271d = n\na\u271d : \u2200 {n : \u2115}, y\u271d \u2208 treesOfNumNodesEq n \u2194 numNodes y\u271d = n\nn\u271d : \u2115\n\u22a2 node () x\u271d y\u271d \u2208 treesOfNumNodesEq (Nat.succ n\u271d) \u2194 numNodes (node () x\u271d y\u271d) = Nat.succ n\u271d\n[PROOFSTEP]\nsimp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]\n[GOAL]\ncase base.succ\nn\u271d : \u2115\n\u22a2 \u00ac0 = n\u271d + 1\n[PROOFSTEP]\nexact (Nat.succ_ne_zero _).symm\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (x : Tree Unit), x \u2208 \u2191(treesOfNumNodesEq n) \u2194 x \u2208 {x | numNodes x = n}\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 card (treesOfNumNodesEq n) = catalan n\n[PROOFSTEP]\ninduction' n using Nat.case_strong_induction_on with n ih\n[GOAL]\ncase hz\n\u22a2 card (treesOfNumNodesEq 0) = catalan 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hi\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\n\u22a2 card (treesOfNumNodesEq (Nat.succ n)) = catalan (Nat.succ n)\n[PROOFSTEP]\nrw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']\n[GOAL]\ncase hi\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\n\u22a2 \u2211 u in Nat.antidiagonal n, card (pairwiseNode (treesOfNumNodesEq u.fst) (treesOfNumNodesEq u.snd)) =\n    \u2211 ij in Nat.antidiagonal n, catalan ij.fst * catalan ij.snd\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase hi\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115),\n    x \u2208 Nat.antidiagonal n \u2192\n      card (pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd)) = catalan x.fst * catalan x.snd\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 H\n[GOAL]\ncase hi.mk\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\nH : (i, j) \u2208 Nat.antidiagonal n\n\u22a2 card (pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)) =\n    catalan (i, j).fst * catalan (i, j).snd\n[PROOFSTEP]\nrw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]\n[GOAL]\ncase hi\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115),\n    x \u2208 Nat.antidiagonal n \u2192\n      \u2200 (y : \u2115 \u00d7 \u2115),\n        y \u2208 Nat.antidiagonal n \u2192\n          x \u2260 y \u2192\n            Disjoint (pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd))\n              (pairwiseNode (treesOfNumNodesEq y.fst) (treesOfNumNodesEq y.snd))\n[PROOFSTEP]\nsimp_rw [disjoint_left]\n[GOAL]\ncase hi\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115),\n    x \u2208 Nat.antidiagonal n \u2192\n      \u2200 (y : \u2115 \u00d7 \u2115),\n        y \u2208 Nat.antidiagonal n \u2192\n          x \u2260 y \u2192\n            \u2200 \u2983a : Tree Unit\u2984,\n              a \u2208 pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd) \u2192\n                \u00aca \u2208 pairwiseNode (treesOfNumNodesEq y.fst) (treesOfNumNodesEq y.snd)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 _ \u27e8i', j'\u27e9\n  _\n      -- Porting note: was clear * -; tidy\n[GOAL]\ncase hi.mk.mk\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\n\u22a2 (i, j) \u2260 (i', j') \u2192\n    \u2200 \u2983a : Tree Unit\u2984,\n      a \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) \u2192\n        \u00aca \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n[PROOFSTEP]\nintros h a\n[GOAL]\ncase hi.mk.mk\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Tree Unit\n\u22a2 a \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) \u2192\n    \u00aca \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n[PROOFSTEP]\ncases' a with a l r\n[GOAL]\ncase hi.mk.mk.nil\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\n\u22a2 nil \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) \u2192\n    \u00acnil \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase hi.mk.mk.nil\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh\u271d : (i, j) \u2260 (i', j')\nh : nil \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\n\u22a2 \u00acnil \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase hi.mk.mk.node\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\n\u22a2 node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) \u2192\n    \u00acnode a l r \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n[PROOFSTEP]\nintro h1 h2\n[GOAL]\ncase hi.mk.mk.node\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase hi.mk.mk.node\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n\u22a2 (i, j) = (i', j')\n[PROOFSTEP]\ntrans (numNodes l, numNodes r)\n[GOAL]\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n\u22a2 (i, j) = (numNodes l, numNodes r)\n[PROOFSTEP]\nsimp at h1 \n[GOAL]\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\nh2 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\nh1 : numNodes l = i \u2227 numNodes r = j\n\u22a2 (i, j) = (numNodes l, numNodes r)\n[PROOFSTEP]\nsimp [h1]\n[GOAL]\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n\u22a2 (numNodes l, numNodes r) = (i', j')\n[PROOFSTEP]\nsimp at h2 \n[GOAL]\nn : \u2115\nih : \u2200 (m : \u2115), m \u2264 n \u2192 card (treesOfNumNodesEq m) = catalan m\ni j : \u2115\na\u271d\u00b9 : (i, j) \u2208 Nat.antidiagonal n\ni' j' : \u2115\na\u271d : (i', j') \u2208 Nat.antidiagonal n\nh : (i, j) \u2260 (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r \u2208 pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : numNodes l = i' \u2227 numNodes r = j'\n\u22a2 (numNodes l, numNodes r) = (i', j')\n[PROOFSTEP]\nsimp [h2]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Catalan", "llama_tokens": 17049, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240825770432, "lm_q2_score": 0.6548947155710234, "lm_q1q2_score": 0.5666306794644924}}
{"text": "[GOAL]\nm n k : \u2115\n\u22a2 gcd m (n + k * m) = gcd m n\n[PROOFSTEP]\nsimp [gcd_rec m (n + k * m), gcd_rec m n]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd m (n + m * k) = gcd m n\n[PROOFSTEP]\nsimp [gcd_rec m (n + m * k), gcd_rec m n]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd m (k * m + n) = gcd m n\n[PROOFSTEP]\nsimp [add_comm _ n]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd m (m * k + n) = gcd m n\n[PROOFSTEP]\nsimp [add_comm _ n]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd (m + k * n) n = gcd m n\n[PROOFSTEP]\nrw [gcd_comm, gcd_add_mul_right_right, gcd_comm]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd (m + n * k) n = gcd m n\n[PROOFSTEP]\nrw [gcd_comm, gcd_add_mul_left_right, gcd_comm]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd (k * n + m) n = gcd m n\n[PROOFSTEP]\nrw [gcd_comm, gcd_mul_right_add_right, gcd_comm]\n[GOAL]\nm n k : \u2115\n\u22a2 gcd (n * k + m) n = gcd m n\n[PROOFSTEP]\nrw [gcd_comm, gcd_mul_left_add_right, gcd_comm]\n[GOAL]\nm n : \u2115\n\u22a2 gcd m (n + m) = gcd m (n + 1 * m)\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\nm n : \u2115\n\u22a2 gcd (m + n) n = gcd m n\n[PROOFSTEP]\nrw [gcd_comm, gcd_add_self_right, gcd_comm]\n[GOAL]\nm n : \u2115\n\u22a2 gcd (m + n) m = gcd n m\n[PROOFSTEP]\nrw [add_comm, gcd_add_self_left]\n[GOAL]\nm n : \u2115\n\u22a2 gcd m (m + n) = gcd m n\n[PROOFSTEP]\nrw [add_comm, gcd_add_self_right]\n[GOAL]\nm n : \u2115\n\u22a2 0 < m \u2192 0 < n \u2192 0 < lcm m n\n[PROOFSTEP]\nsimp_rw [pos_iff_ne_zero]\n[GOAL]\nm n : \u2115\n\u22a2 m \u2260 0 \u2192 n \u2260 0 \u2192 lcm m n \u2260 0\n[PROOFSTEP]\nexact lcm_ne_zero\n[GOAL]\nm n : \u2115\nh : coprime m n\n\u22a2 lcm m n = m * n\n[PROOFSTEP]\nrw [\u2190 one_mul (lcm m n), \u2190 h.gcd_eq_one, gcd_mul_lcm]\n[GOAL]\nm n : \u2115\n\u22a2 coprime m (n + m) \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_add_self_right]\n[GOAL]\nm n : \u2115\n\u22a2 coprime m (m + n) \u2194 coprime m n\n[PROOFSTEP]\nrw [add_comm, coprime_add_self_right]\n[GOAL]\nm n : \u2115\n\u22a2 coprime (m + n) n \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_add_self_left]\n[GOAL]\nm n : \u2115\n\u22a2 coprime (m + n) m \u2194 coprime n m\n[PROOFSTEP]\nrw [coprime, coprime, gcd_self_add_left]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime m (n + k * m) \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_add_mul_right_right]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime m (n + m * k) \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_add_mul_left_right]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime m (k * m + n) \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_mul_right_add_right]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime m (m * k + n) \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_mul_left_add_right]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime (m + k * n) n \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_add_mul_right_left]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime (m + n * k) n \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_add_mul_left_left]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime (k * n + m) n \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_mul_right_add_left]\n[GOAL]\nm n k : \u2115\n\u22a2 coprime (n * k + m) n \u2194 coprime m n\n[PROOFSTEP]\nrw [coprime, coprime, gcd_mul_left_add_left]\n[GOAL]\nn : \u2115\nhn : 0 < n\na b : \u2115\n\u22a2 coprime (a ^ n) b \u2194 coprime a b\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := exists_eq_succ_of_ne_zero hn.ne'\n[GOAL]\ncase intro\na b n : \u2115\nhn : 0 < succ n\n\u22a2 coprime (a ^ succ n) b \u2194 coprime a b\n[PROOFSTEP]\nrw [pow_succ, Nat.coprime_mul_iff_left]\n[GOAL]\ncase intro\na b n : \u2115\nhn : 0 < succ n\n\u22a2 coprime (a ^ n) b \u2227 coprime a b \u2194 coprime a b\n[PROOFSTEP]\nexact \u27e8And.right, fun hab => \u27e8hab.pow_left _, hab\u27e9\u27e9\n[GOAL]\nn : \u2115\nhn : 0 < n\na b : \u2115\n\u22a2 coprime a (b ^ n) \u2194 coprime a b\n[PROOFSTEP]\nrw [Nat.coprime_comm, coprime_pow_left_iff hn, Nat.coprime_comm]\n[GOAL]\n\u22a2 \u00accoprime 0 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 coprime 1 n \u2194 True\n[PROOFSTEP]\nsimp [coprime]\n[GOAL]\nn : \u2115\n\u22a2 coprime n 1 \u2194 True\n[PROOFSTEP]\nsimp [coprime]\n[GOAL]\na b c : \u2115\nhac : coprime a c\nb_dvd_c : b \u2223 c\n\u22a2 gcd (a * b) c = b\n[PROOFSTEP]\nrcases exists_eq_mul_left_of_dvd b_dvd_c with \u27e8d, rfl\u27e9\n[GOAL]\ncase intro\na b d : \u2115\nhac : coprime a (d * b)\nb_dvd_c : b \u2223 d * b\n\u22a2 gcd (a * b) (d * b) = b\n[PROOFSTEP]\nrw [gcd_mul_right]\n[GOAL]\ncase intro\na b d : \u2115\nhac : coprime a (d * b)\nb_dvd_c : b \u2223 d * b\n\u22a2 gcd a d * b = b\n[PROOFSTEP]\nconvert one_mul b\n[GOAL]\ncase h.e'_2.h.e'_5\na b d : \u2115\nhac : coprime a (d * b)\nb_dvd_c : b \u2223 d * b\n\u22a2 gcd a d = 1\n[PROOFSTEP]\nexact coprime.coprime_mul_right_right hac\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\ncases h0 : gcd k m\n[GOAL]\ncase zero\nm n k : \u2115\nH : k \u2223 m * n\nh0 : gcd k m = zero\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\ncase succ m n k : \u2115 H : k \u2223 m * n n\u271d : \u2115 h0 : gcd k m = succ n\u271d \u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\ncase zero =>\n  obtain rfl : k = 0 := eq_zero_of_gcd_eq_zero_left h0\n  obtain rfl : m = 0 := eq_zero_of_gcd_eq_zero_right h0\n  exact \u27e8\u27e8\u27e80, dvd_refl 0\u27e9, \u27e8n, dvd_refl n\u27e9\u27e9, (zero_mul n).symm\u27e9\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nh0 : gcd k m = zero\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\ncase zero =>\n  obtain rfl : k = 0 := eq_zero_of_gcd_eq_zero_left h0\n  obtain rfl : m = 0 := eq_zero_of_gcd_eq_zero_right h0\n  exact \u27e8\u27e8\u27e80, dvd_refl 0\u27e9, \u27e8n, dvd_refl n\u27e9\u27e9, (zero_mul n).symm\u27e9\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nh0 : gcd k m = zero\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nobtain rfl : k = 0 := eq_zero_of_gcd_eq_zero_left h0\n[GOAL]\nm n : \u2115\nH : 0 \u2223 m * n\nh0 : gcd 0 m = zero\n\u22a2 { d // 0 = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nobtain rfl : m = 0 := eq_zero_of_gcd_eq_zero_right h0\n[GOAL]\nn : \u2115\nH : 0 \u2223 0 * n\nh0 : gcd 0 0 = zero\n\u22a2 { d // 0 = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e80, dvd_refl 0\u27e9, \u27e8n, dvd_refl n\u27e9\u27e9, (zero_mul n).symm\u27e9\n[GOAL]\ncase succ\nm n k : \u2115\nH : k \u2223 m * n\nn\u271d : \u2115\nh0 : gcd k m = succ n\u271d\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\ncase succ tmp =>\n  have hpos : 0 < gcd k m := h0.symm \u25b8 Nat.zero_lt_succ _; clear h0 tmp\n  have hd : gcd k m * (k / gcd k m) = k := Nat.mul_div_cancel' (gcd_dvd_left k m)\n  refine' \u27e8\u27e8\u27e8gcd k m, gcd_dvd_right k m\u27e9, \u27e8k / gcd k m, _\u27e9\u27e9, hd.symm\u27e9\n  apply Nat.dvd_of_mul_dvd_mul_left hpos\n  rw [hd, \u2190 gcd_mul_right]\n  exact dvd_gcd (dvd_mul_right _ _) H\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\ntmp : \u2115\nh0 : gcd k m = succ tmp\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\ncase succ tmp =>\n  have hpos : 0 < gcd k m := h0.symm \u25b8 Nat.zero_lt_succ _; clear h0 tmp\n  have hd : gcd k m * (k / gcd k m) = k := Nat.mul_div_cancel' (gcd_dvd_left k m)\n  refine' \u27e8\u27e8\u27e8gcd k m, gcd_dvd_right k m\u27e9, \u27e8k / gcd k m, _\u27e9\u27e9, hd.symm\u27e9\n  apply Nat.dvd_of_mul_dvd_mul_left hpos\n  rw [hd, \u2190 gcd_mul_right]\n  exact dvd_gcd (dvd_mul_right _ _) H\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\ntmp : \u2115\nh0 : gcd k m = succ tmp\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nhave hpos : 0 < gcd k m := h0.symm \u25b8 Nat.zero_lt_succ _\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\ntmp : \u2115\nh0 : gcd k m = succ tmp\nhpos : 0 < gcd k m\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nclear h0 tmp\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nhpos : 0 < gcd k m\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nhave hd : gcd k m * (k / gcd k m) = k := Nat.mul_div_cancel' (gcd_dvd_left k m)\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nhpos : 0 < gcd k m\nhd : gcd k m * (k / gcd k m) = k\n\u22a2 { d // k = \u2191d.fst * \u2191d.snd }\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8gcd k m, gcd_dvd_right k m\u27e9, \u27e8k / gcd k m, _\u27e9\u27e9, hd.symm\u27e9\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nhpos : 0 < gcd k m\nhd : gcd k m * (k / gcd k m) = k\n\u22a2 k / gcd k m \u2223 n\n[PROOFSTEP]\napply Nat.dvd_of_mul_dvd_mul_left hpos\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nhpos : 0 < gcd k m\nhd : gcd k m * (k / gcd k m) = k\n\u22a2 gcd k m * (k / gcd k m) \u2223 gcd k m * n\n[PROOFSTEP]\nrw [hd, \u2190 gcd_mul_right]\n[GOAL]\nm n k : \u2115\nH : k \u2223 m * n\nhpos : 0 < gcd k m\nhd : gcd k m * (k / gcd k m) = k\n\u22a2 k \u2223 gcd (k * n) (m * n)\n[PROOFSTEP]\nexact dvd_gcd (dvd_mul_right _ _) H\n[GOAL]\nx m n : \u2115\n\u22a2 x \u2223 m * n \u2194 \u2203 y z, y \u2223 m \u2227 z \u2223 n \u2227 y * z = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx m n : \u2115\n\u22a2 x \u2223 m * n \u2192 \u2203 y z, y \u2223 m \u2227 z \u2223 n \u2227 y * z = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nx m n : \u2115\nh : x \u2223 m * n\n\u22a2 \u2203 y z, y \u2223 m \u2227 z \u2223 n \u2227 y * z = x\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8y, hy\u27e9, \u27e8z, hz\u27e9\u27e9, rfl\u27e9 := prod_dvd_and_dvd_of_dvd_prod h\n[GOAL]\ncase mp.mk.mk.mk.mk\nm n y : \u2115\nhy : y \u2223 m\nz : \u2115\nhz : z \u2223 n\nh :\n  \u2191({ val := y, property := hy }, { val := z, property := hz }).fst *\n      \u2191({ val := y, property := hy }, { val := z, property := hz }).snd \u2223\n    m * n\n\u22a2 \u2203 y_1 z_1,\n    y_1 \u2223 m \u2227\n      z_1 \u2223 n \u2227\n        y_1 * z_1 =\n          \u2191({ val := y, property := hy }, { val := z, property := hz }).fst *\n            \u2191({ val := y, property := hy }, { val := z, property := hz }).snd\n[PROOFSTEP]\nexact \u27e8y, z, hy, hz, rfl\u27e9\n[GOAL]\ncase mpr\nx m n : \u2115\n\u22a2 (\u2203 y z, y \u2223 m \u2227 z \u2223 n \u2227 y * z = x) \u2192 x \u2223 m * n\n[PROOFSTEP]\nrintro \u27e8y, z, hy, hz, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nm n y z : \u2115\nhy : y \u2223 m\nhz : z \u2223 n\n\u22a2 y * z \u2223 m * n\n[PROOFSTEP]\nexact mul_dvd_mul hy hz\n[GOAL]\na b n : \u2115\nn0 : 0 < n\n\u22a2 a ^ n \u2223 b ^ n \u2194 a \u2223 b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => pow_dvd_pow_of_dvd h _\u27e9\n[GOAL]\na b n : \u2115\nn0 : 0 < n\nh : a ^ n \u2223 b ^ n\n\u22a2 a \u2223 b\n[PROOFSTEP]\ncases' Nat.eq_zero_or_pos (gcd a b) with g0 g0\n[GOAL]\ncase inl\na b n : \u2115\nn0 : 0 < n\nh : a ^ n \u2223 b ^ n\ng0 : gcd a b = 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nsimp [eq_zero_of_gcd_eq_zero_right g0]\n[GOAL]\ncase inr\na b n : \u2115\nn0 : 0 < n\nh : a ^ n \u2223 b ^ n\ng0 : gcd a b > 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrcases exists_coprime' g0 with \u27e8g, a', b', g0', co, rfl, rfl\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\nn : \u2115\nn0 : 0 < n\ng : \u2115\na' b' : \u2115\ng0' : 0 < g\nco : coprime a' b'\nh : (a' * g) ^ n \u2223 (b' * g) ^ n\ng0 : gcd (a' * g) (b' * g) > 0\n\u22a2 a' * g \u2223 b' * g\n[PROOFSTEP]\nrw [mul_pow, mul_pow] at h \n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\nn : \u2115\nn0 : 0 < n\ng : \u2115\na' b' : \u2115\ng0' : 0 < g\nco : coprime a' b'\nh : a' ^ n * g ^ n \u2223 b' ^ n * g ^ n\ng0 : gcd (a' * g) (b' * g) > 0\n\u22a2 a' * g \u2223 b' * g\n[PROOFSTEP]\nreplace h := Nat.dvd_of_mul_dvd_mul_right (pow_pos g0' _) h\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\nn : \u2115\nn0 : 0 < n\ng : \u2115\na' b' : \u2115\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n \u2223 b' ^ n\n\u22a2 a' * g \u2223 b' * g\n[PROOFSTEP]\nhave := pow_dvd_pow a' n0\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\nn : \u2115\nn0 : 0 < n\ng : \u2115\na' b' : \u2115\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n \u2223 b' ^ n\nthis : a' ^ succ 0 \u2223 a' ^ n\n\u22a2 a' * g \u2223 b' * g\n[PROOFSTEP]\nrw [pow_one, (co.pow n n).eq_one_of_dvd h] at this \n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\nn : \u2115\nn0 : 0 < n\ng : \u2115\na' b' : \u2115\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n \u2223 b' ^ n\nthis : a' \u2223 1\n\u22a2 a' * g \u2223 b' * g\n[PROOFSTEP]\nsimp [eq_one_of_dvd_one this]\n[GOAL]\na b k : \u2115\nh_ab_coprime : coprime a b\nhka : k \u2223 a\nhkb : k \u2223 b\n\u22a2 k = 1\n[PROOFSTEP]\nrw [coprime_iff_gcd_eq_one] at h_ab_coprime \n[GOAL]\na b k : \u2115\nh_ab_coprime : gcd a b = 1\nhka : k \u2223 a\nhkb : k \u2223 b\n\u22a2 k = 1\n[PROOFSTEP]\nhave h1 := dvd_gcd hka hkb\n[GOAL]\na b k : \u2115\nh_ab_coprime : gcd a b = 1\nhka : k \u2223 a\nhkb : k \u2223 b\nh1 : k \u2223 gcd a b\n\u22a2 k = 1\n[PROOFSTEP]\nrw [h_ab_coprime] at h1 \n[GOAL]\na b k : \u2115\nh_ab_coprime : gcd a b = 1\nhka : k \u2223 a\nhkb : k \u2223 b\nh1 : k \u2223 1\n\u22a2 k = 1\n[PROOFSTEP]\nexact Nat.dvd_one.mp h1\n[GOAL]\nm n a b : \u2115\ncop : coprime m n\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a * m + b * n \u2260 m * n\n[PROOFSTEP]\nintro h\n[GOAL]\nm n a b : \u2115\ncop : coprime m n\nha : a \u2260 0\nhb : b \u2260 0\nh : a * m + b * n = m * n\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 : n \u2223 a :=\n  cop.symm.dvd_of_dvd_mul_right\n    ((Nat.dvd_add_iff_left (Nat.dvd_mul_left n b)).mpr ((congr_arg _ h).mpr (Nat.dvd_mul_left n m)))\n[GOAL]\ncase intro\nm n b : \u2115\ncop : coprime m n\nhb : b \u2260 0\nx : \u2115\nha : n * x \u2260 0\nh : n * x * m + b * n = m * n\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 : m \u2223 b :=\n  cop.dvd_of_dvd_mul_right\n    ((Nat.dvd_add_iff_right (Nat.dvd_mul_left m (n * x))).mpr ((congr_arg _ h).mpr (Nat.dvd_mul_right m n)))\n[GOAL]\ncase intro.intro\nm n : \u2115\ncop : coprime m n\nx : \u2115\nha : n * x \u2260 0\ny : \u2115\nhb : m * y \u2260 0\nh : n * x * m + m * y * n = m * n\n\u22a2 False\n[PROOFSTEP]\nrw [mul_comm, mul_ne_zero_iff, \u2190 one_le_iff_ne_zero] at ha hb \n[GOAL]\ncase intro.intro\nm n : \u2115\ncop : coprime m n\nx : \u2115\nha : 1 \u2264 x \u2227 n \u2260 0\ny : \u2115\nhb : 1 \u2264 y \u2227 m \u2260 0\nh : n * x * m + m * y * n = m * n\n\u22a2 False\n[PROOFSTEP]\nrefine' mul_ne_zero hb.2 ha.2 (eq_zero_of_mul_eq_self_left (ne_of_gt (add_le_add ha.1 hb.1)) _)\n[GOAL]\ncase intro.intro\nm n : \u2115\ncop : coprime m n\nx : \u2115\nha : 1 \u2264 x \u2227 n \u2260 0\ny : \u2115\nhb : 1 \u2264 y \u2227 m \u2260 0\nh : n * x * m + m * y * n = m * n\n\u22a2 (x + y) * (m * n) = m * n\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 h, add_mul, add_mul, mul_comm _ n, \u2190 mul_assoc, mul_comm y]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.GCD.Basic", "llama_tokens": 6789, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744761936437, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.5662864065511084}}
{"text": "[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\nr : R\n\u22a2 \u2191(\u03b9 (Q' Q)) (m, r) = \u2191(v Q) m + r \u2022 e0 Q\n[PROOFSTEP]\nrw [e0, v, LinearMap.comp_apply, LinearMap.inl_apply, \u2190 LinearMap.map_smul, Prod.smul_mk, smul_zero, smul_eq_mul,\n  mul_one, \u2190 LinearMap.map_add, Prod.mk_add_mk, zero_add, add_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (\u2191(Q' Q) (0, 1)) = -1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (\u2191(Q' Q) (\u2191(LinearMap.inl R M R) m)) =\n    \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (\u2191Q m)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 -(e0 Q * \u2191(v Q) m) = \u2191(v Q) m * e0 Q\n[PROOFSTEP]\nrefine' neg_eq_of_add_eq_zero_right ((\u03b9_mul_\u03b9_add_swap _ _).trans _)\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (QuadraticForm.polar \u2191(Q' Q) (0, 1) (\u2191(LinearMap.inl R M R) m)) = 0\n[PROOFSTEP]\ndsimp [QuadraticForm.polar]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (\u2191Q (0 + m) + -((1 + 0) * (1 + 0)) - (\u2191Q 0 + -(1 * 1)) - (\u2191Q m + -(0 * 0))) =\n    0\n[PROOFSTEP]\nsimp only [add_zero, mul_zero, mul_one, zero_add, neg_zero, QuadraticForm.map_zero, add_sub_cancel, sub_self, map_zero,\n  zero_sub]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 -(\u2191(v Q) m * e0 Q) = e0 Q * \u2191(v Q) m\n[PROOFSTEP]\nrw [neg_eq_iff_eq_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(v Q) m * e0 Q = -(e0 Q * \u2191(v Q) m)\n[PROOFSTEP]\nexact (neg_e0_mul_v _ m).symm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 e0 Q * \u2191(v Q) m * e0 Q = \u2191(v Q) m\n[PROOFSTEP]\nrw [\u2190 neg_v_mul_e0, \u2190 neg_mul, mul_assoc, e0_mul_e0, mul_neg_one, neg_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 CliffordAlgebra Q \u2192\u2090[R] { x // x \u2208 even (Q' Q) }\n[PROOFSTEP]\nrefine' CliffordAlgebra.lift Q \u27e8_, fun m => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 M \u2192\u2097[R] { x // x \u2208 even (Q' Q) }\n[PROOFSTEP]\nrefine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem \u27e82, rfl\u27e9 _\n[GOAL]\ncase refine'_1.refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 M \u2192\u2097[R] CliffordAlgebra (Q' Q)\ncase refine'_1.refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191?refine'_1.refine'_1 m \u2208 LinearMap.range (\u03b9 (Q' Q)) ^ \u2191{ val := 2, property := (_ : \u21912 = \u21912) }\n[PROOFSTEP]\nexact (LinearMap.mulLeft R <| e0 Q).comp (v Q)\n[GOAL]\ncase refine'_1.refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n    LinearMap.range (\u03b9 (Q' Q)) ^ \u2191{ val := 2, property := (_ : \u21912 = \u21912) }\n[PROOFSTEP]\nrw [Subtype.coe_mk, pow_two]\n[GOAL]\ncase refine'_1.refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208 LinearMap.range (\u03b9 (Q' Q)) * LinearMap.range (\u03b9 (Q' Q))\n[PROOFSTEP]\nexact Submodule.mul_mem_mul (LinearMap.mem_range_self _ _) (LinearMap.mem_range_self _ _)\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n            (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n            (_ :\n              \u2200 (m : M),\n                \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                  \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n        m *\n      \u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n            (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n            (_ :\n              \u2200 (m : M),\n                \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                  \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n        m =\n    \u2191(algebraMap R { x // x \u2208 even (Q' Q) }) (\u2191Q m)\n[PROOFSTEP]\next1\n[GOAL]\ncase refine'_2.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(\u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n              (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n              (_ :\n                \u2200 (m : M),\n                  \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                    \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n          m *\n        \u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n              (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n              (_ :\n                \u2200 (m : M),\n                  \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                    \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n          m) =\n    \u2191(\u2191(algebraMap R { x // x \u2208 even (Q' Q) }) (\u2191Q m))\n[PROOFSTEP]\nrw [Subalgebra.coe_mul]\n  -- porting note: was part of the `dsimp only` below\n[GOAL]\ncase refine'_2.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(\u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n              (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n              (_ :\n                \u2200 (m : M),\n                  \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                    \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n          m) *\n      \u2191(\u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n              (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n              (_ :\n                \u2200 (m : M),\n                  \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                    \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n          m) =\n    \u2191(\u2191(algebraMap R { x // x \u2208 even (Q' Q) }) (\u2191Q m))\n[PROOFSTEP]\nerw [LinearMap.codRestrict_apply]\n  -- porting note: was part of the `dsimp only` below\n[GOAL]\ncase refine'_2.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m * \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m =\n    \u2191(\u2191(algebraMap R { x // x \u2208 even (Q' Q) }) (\u2191Q m))\n[PROOFSTEP]\ndsimp only [LinearMap.comp_apply, LinearMap.mulLeft_apply, Subalgebra.coe_algebraMap]\n[GOAL]\ncase refine'_2.a\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 e0 Q * \u2191(v Q) m * (e0 Q * \u2191(v Q) m) = \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (\u2191Q m)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, e0_mul_v_mul_e0, v_sq_scalar]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m)) = e0 Q * \u2191(v Q) m\n[PROOFSTEP]\nrw [toEven, CliffordAlgebra.lift_\u03b9_apply]\n  -- porting note: was `rw`\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(\u2191(LinearMap.codRestrict (\u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i)\n            (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))\n            (_ :\n              \u2200 (m : M),\n                \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m \u2208\n                  \u2a06 (i : { n // \u2191n = 0 }), LinearMap.range (\u03b9 (Q' Q)) ^ \u2191i))\n        m) =\n    e0 Q * \u2191(v Q) m\n[PROOFSTEP]\nerw [LinearMap.codRestrict_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m = e0 Q * \u2191(v Q) m\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\n\u22a2 { x // x \u2208 even (Q' Q) } \u2192\u2090[R] CliffordAlgebra Q\n[PROOFSTEP]\nlet f : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  ((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap.comp <|\n        (\u03b9 Q).comp (LinearMap.fst _ _ _) + (Algebra.linearMap R _).comp (LinearMap.snd _ _ _)).compl\u2082\n    ((\u03b9 Q).comp (LinearMap.fst _ _ _) - (Algebra.linearMap R _).comp (LinearMap.snd _ _ _))\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\n\u22a2 { x // x \u2208 even (Q' Q) } \u2192\u2090[R] CliffordAlgebra Q\n[PROOFSTEP]\nhaveI f_apply : \u2200 x y, f x y = (\u03b9 Q x.1 + algebraMap R _ x.2) * (\u03b9 Q y.1 - algebraMap R _ y.2) := fun x y => by rfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nx y : M \u00d7 R\n\u22a2 \u2191(\u2191f x) y =\n    (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n      (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\n\u22a2 { x // x \u2208 even (Q' Q) } \u2192\u2090[R] CliffordAlgebra Q\n[PROOFSTEP]\nhaveI hc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (algebraMap _ _ r) x := Algebra.commutes\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\n\u22a2 { x // x \u2208 even (Q' Q) } \u2192\u2090[R] CliffordAlgebra Q\n[PROOFSTEP]\nhaveI hm : \u2200 m : M \u00d7 R, \u03b9 Q m.1 * \u03b9 Q m.1 - algebraMap R _ m.2 * algebraMap R _ m.2 = algebraMap R _ (Q' Q m) :=\n  by\n  intro m\n  rw [\u03b9_sq_scalar, \u2190 RingHom.map_mul, \u2190 RingHom.map_sub, sub_eq_add_neg, Q'_apply, sub_eq_add_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\n\u22a2 \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n[PROOFSTEP]\nintro m\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nm : M \u00d7 R\n\u22a2 \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n    \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n[PROOFSTEP]\nrw [\u03b9_sq_scalar, \u2190 RingHom.map_mul, \u2190 RingHom.map_sub, sub_eq_add_neg, Q'_apply, sub_eq_add_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n\u22a2 { x // x \u2208 even (Q' Q) } \u2192\u2090[R] CliffordAlgebra Q\n[PROOFSTEP]\nrefine' even.lift (Q' Q) \u27e8f, _, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n\u22a2 \u2200 (m : M \u00d7 R), \u2191(\u2191f m) m = \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n[PROOFSTEP]\nsimp_rw [f_apply]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n\u22a2 \u2200 (m\u2081 m\u2082 m\u2083 : M \u00d7 R), \u2191(\u2191f m\u2081) m\u2082 * \u2191(\u2191f m\u2082) m\u2083 = \u2191(Q' Q) m\u2082 \u2022 \u2191(\u2191f m\u2081) m\u2083\n[PROOFSTEP]\nsimp_rw [f_apply]\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n\u22a2 \u2200 (m : M \u00d7 R),\n    (\u2191(\u03b9 Q) m.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m.snd) *\n        (\u2191(\u03b9 Q) m.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m.snd) =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n[PROOFSTEP]\nintro m\n[GOAL]\ncase refine'_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\nm : M \u00d7 R\n\u22a2 (\u2191(\u03b9 Q) m.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m.snd) *\n      (\u2191(\u03b9 Q) m.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m.snd) =\n    \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n[PROOFSTEP]\nrw [\u2190 (hc _ _).symm.mul_self_sub_mul_self_eq, hm]\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\n\u22a2 \u2200 (m\u2081 m\u2082 m\u2083 : M \u00d7 R),\n    (\u2191(\u03b9 Q) m\u2081.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m\u2081.snd) *\n          (\u2191(\u03b9 Q) m\u2082.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m\u2082.snd) *\n        ((\u2191(\u03b9 Q) m\u2082.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m\u2082.snd) *\n          (\u2191(\u03b9 Q) m\u2083.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m\u2083.snd)) =\n      \u2191(Q' Q) m\u2082 \u2022\n        ((\u2191(\u03b9 Q) m\u2081.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m\u2081.snd) *\n          (\u2191(\u03b9 Q) m\u2083.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m\u2083.snd))\n[PROOFSTEP]\nintro m\u2081 m\u2082 m\u2083\n[GOAL]\ncase refine'_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\nf_apply :\n  \u2200 (x y : M \u00d7 R),\n    \u2191(\u2191f x) y =\n      (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n        (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\nhc : \u2200 (r : R) (x : CliffordAlgebra Q), Commute (\u2191(algebraMap R (CliffordAlgebra Q)) r) x\nhm :\n  \u2200 (m : M \u00d7 R),\n    \u2191(\u03b9 Q) m.fst * \u2191(\u03b9 Q) m.fst -\n        \u2191(algebraMap R (CliffordAlgebra Q)) m.snd * \u2191(algebraMap R (CliffordAlgebra Q)) m.snd =\n      \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)\nm\u2081 m\u2082 m\u2083 : M \u00d7 R\n\u22a2 (\u2191(\u03b9 Q) m\u2081.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m\u2081.snd) *\n        (\u2191(\u03b9 Q) m\u2082.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m\u2082.snd) *\n      ((\u2191(\u03b9 Q) m\u2082.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m\u2082.snd) *\n        (\u2191(\u03b9 Q) m\u2083.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m\u2083.snd)) =\n    \u2191(Q' Q) m\u2082 \u2022\n      ((\u2191(\u03b9 Q) m\u2081.fst + \u2191(algebraMap R (CliffordAlgebra Q)) m\u2081.snd) *\n        (\u2191(\u03b9 Q) m\u2083.fst - \u2191(algebraMap R (CliffordAlgebra Q)) m\u2083.snd))\n[PROOFSTEP]\nrw [\u2190 mul_smul_comm, \u2190 mul_assoc, mul_assoc (_ + _), \u2190 (hc _ _).symm.mul_self_sub_mul_self_eq', Algebra.smul_def, \u2190\n  mul_assoc, hm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : M \u00d7 R\n\u22a2 \u2191(ofEven Q) (\u2191(\u2191(even.\u03b9 (Q' Q)).bilin x) y) =\n    (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n      (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\n[PROOFSTEP]\nunfold ofEven\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : M \u00d7 R\n\u22a2 \u2191(let f :=\n          LinearMap.compl\u2082\n            (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n              (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n            (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n              LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R));\n        \u2191(even.lift (Q' Q))\n          { bilin := f,\n            contract :=\n              (_ :\n                \u2200 (m : M \u00d7 R),\n                  \u2191(\u2191(LinearMap.compl\u2082\n                              (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                              (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                          m)\n                      m =\n                    \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)),\n            contract_mid :=\n              (_ :\n                \u2200 (m\u2081 m\u2082 m\u2083 : M \u00d7 R),\n                  \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2081)\n                        m\u2082 *\n                      \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2082)\n                        m\u2083 =\n                    \u2191(Q' Q) m\u2082 \u2022\n                      \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2081)\n                        m\u2083) })\n      (\u2191(\u2191(even.\u03b9 (Q' Q)).bilin x) y) =\n    (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n      (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\n[PROOFSTEP]\nlift_lets\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : M \u00d7 R\n\u22a2 let f :=\n    LinearMap.compl\u2082\n      (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n        (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n          LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R));\n  \u2191(\u2191(even.lift (Q' Q))\n          { bilin := f,\n            contract :=\n              (_ :\n                \u2200 (m : M \u00d7 R),\n                  \u2191(\u2191(LinearMap.compl\u2082\n                              (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                              (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                          m)\n                      m =\n                    \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)),\n            contract_mid :=\n              (_ :\n                \u2200 (m\u2081 m\u2082 m\u2083 : M \u00d7 R),\n                  \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2081)\n                        m\u2082 *\n                      \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2082)\n                        m\u2083 =\n                    \u2191(Q' Q) m\u2082 \u2022\n                      \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2081)\n                        m\u2083) })\n      (\u2191(\u2191(even.\u03b9 (Q' Q)).bilin x) y) =\n    (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n      (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\n[PROOFSTEP]\nintro f\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : M \u00d7 R\nf : M \u00d7 R \u2192\u2097[R] M \u00d7 R \u2192\u2097[R] CliffordAlgebra Q :=\n  LinearMap.compl\u2082\n    (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n      (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n        LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n    (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n      LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R))\n\u22a2 \u2191(\u2191(even.lift (Q' Q))\n          { bilin := f,\n            contract :=\n              (_ :\n                \u2200 (m : M \u00d7 R),\n                  \u2191(\u2191(LinearMap.compl\u2082\n                              (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                              (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                          m)\n                      m =\n                    \u2191(algebraMap R (CliffordAlgebra Q)) (\u2191(Q' Q) m)),\n            contract_mid :=\n              (_ :\n                \u2200 (m\u2081 m\u2082 m\u2083 : M \u00d7 R),\n                  \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2081)\n                        m\u2082 *\n                      \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2082)\n                        m\u2083 =\n                    \u2191(Q' Q) m\u2082 \u2022\n                      \u2191(\u2191(LinearMap.compl\u2082\n                                (LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))\n                                  (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) +\n                                    LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                                (LinearMap.comp (\u03b9 Q) (LinearMap.fst R M R) -\n                                  LinearMap.comp (Algebra.linearMap R (CliffordAlgebra Q)) (LinearMap.snd R M R)))\n                            m\u2081)\n                        m\u2083) })\n      (\u2191(\u2191(even.\u03b9 (Q' Q)).bilin x) y) =\n    (\u2191(\u03b9 Q) x.fst + \u2191(algebraMap R (CliffordAlgebra Q)) x.snd) *\n      (\u2191(\u03b9 Q) y.fst - \u2191(algebraMap R (CliffordAlgebra Q)) y.snd)\n[PROOFSTEP]\nrefine @even.lift_\u03b9 R (M \u00d7 R) _ _ _ (Q' Q) _ _ _ \u27e8f, ?_, ?_\u27e9 x y\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\n\u22a2 \u2191(\u2191(toEven Q) (\u2191(ofEven Q) (\u2191(\u2191(even.\u03b9 (Q' Q)).bilin (m\u2081, r\u2081)) (m\u2082, r\u2082)))) =\n    (e0 Q * \u2191(v Q) m\u2081 + \u2191(algebraMap R (CliffordAlgebra (Q' Q))) r\u2081) *\n      (e0 Q * \u2191(v Q) m\u2082 - \u2191(algebraMap R (CliffordAlgebra (Q' Q))) r\u2082)\n[PROOFSTEP]\nrw [ofEven_\u03b9, AlgHom.map_mul, AlgHom.map_add, AlgHom.map_sub, AlgHom.commutes, AlgHom.commutes, Subalgebra.coe_mul,\n  Subalgebra.coe_add, Subalgebra.coe_sub, toEven_\u03b9, toEven_\u03b9, Subalgebra.coe_algebraMap, Subalgebra.coe_algebraMap]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\n\u22a2 (e0 Q * \u2191(v Q) m\u2081 + \u2191(algebraMap R (CliffordAlgebra (Q' Q))) r\u2081) *\n      (e0 Q * \u2191(v Q) m\u2082 - \u2191(algebraMap R (CliffordAlgebra (Q' Q))) r\u2082) =\n    e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 - r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081 -\n      \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082)\n[PROOFSTEP]\nrw [mul_sub, add_mul, add_mul, \u2190 Algebra.commutes, \u2190 Algebra.smul_def, \u2190 map_mul, \u2190 Algebra.smul_def,\n  sub_add_eq_sub_sub, smul_mul_assoc, smul_mul_assoc]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\n\u22a2 e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 - r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081 -\n      \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082) =\n    \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082 + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 + \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q + r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q\n[PROOFSTEP]\nhave h1 : e0 Q * v Q m\u2081 * (e0 Q * v Q m\u2082) = v Q m\u2081 * v Q m\u2082 := by rw [\u2190 mul_assoc, e0_mul_v_mul_e0]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\n\u22a2 e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) = \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082\n[PROOFSTEP]\nrw [\u2190 mul_assoc, e0_mul_v_mul_e0]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\nh1 : e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) = \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082\n\u22a2 e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 - r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081 -\n      \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082) =\n    \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082 + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 + \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q + r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q\n[PROOFSTEP]\nhave h2 : -(r\u2082 \u2022 e0 Q * v Q m\u2081) = v Q m\u2081 * r\u2082 \u2022 e0 Q := by rw [mul_smul_comm, smul_mul_assoc, \u2190 smul_neg, neg_e0_mul_v]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\nh1 : e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) = \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082\n\u22a2 -(r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081) = \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q\n[PROOFSTEP]\nrw [mul_smul_comm, smul_mul_assoc, \u2190 smul_neg, neg_e0_mul_v]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\nh1 : e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) = \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082\nh2 : -(r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081) = \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q\n\u22a2 e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 - r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081 -\n      \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082) =\n    \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082 + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 + \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q + r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q\n[PROOFSTEP]\nhave h3 : -algebraMap R _ (r\u2081 * r\u2082) = r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q := by\n  rw [Algebra.algebraMap_eq_smul_one, smul_mul_smul, e0_mul_e0, smul_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\nh1 : e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) = \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082\nh2 : -(r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081) = \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q\n\u22a2 -\u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082) = r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one, smul_mul_smul, e0_mul_e0, smul_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\nh1 : e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) = \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082\nh2 : -(r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081) = \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q\nh3 : -\u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082) = r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q\n\u22a2 e0 Q * \u2191(v Q) m\u2081 * (e0 Q * \u2191(v Q) m\u2082) + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 - r\u2082 \u2022 e0 Q * \u2191(v Q) m\u2081 -\n      \u2191(algebraMap R (CliffordAlgebra (Q' Q))) (r\u2081 * r\u2082) =\n    \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082 + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 + \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q + r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg, h1, h2, h3]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm\u2081\u271d m\u2082\u271d : M \u00d7 R\nm\u2081 : M\nr\u2081 : R\nm\u2082 : M\nr\u2082 : R\n\u22a2 \u2191(v Q) m\u2081 * \u2191(v Q) m\u2082 + r\u2081 \u2022 e0 Q * \u2191(v Q) m\u2082 + \u2191(v Q) m\u2081 * r\u2082 \u2022 e0 Q + r\u2081 \u2022 e0 Q * r\u2082 \u2022 e0 Q =\n    \u2191(\u03b9 (Q' Q)) (m\u2081, r\u2081) * \u2191(\u03b9 (Q' Q)) (m\u2082, r\u2082)\n[PROOFSTEP]\nrw [\u03b9_eq_v_add_smul_e0, \u03b9_eq_v_add_smul_e0, mul_add, add_mul, add_mul, add_assoc]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(ofEven Q) (\u2191(toEven Q) (\u2191(\u03b9 Q) m)) =\n    \u2191(ofEven Q) { val := \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m)), property := (_ : \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m)) \u2208 even (Q' Q)) }\n[PROOFSTEP]\nrw [Subtype.coe_eta]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(ofEven Q) { val := \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m)), property := (_ : \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m)) \u2208 even (Q' Q)) } =\n    (\u2191(\u03b9 Q) 0 + \u2191(algebraMap R (CliffordAlgebra Q)) 1) * (\u2191(\u03b9 Q) m - \u2191(algebraMap R (CliffordAlgebra Q)) 0)\n[PROOFSTEP]\nsimp_rw [toEven_\u03b9]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(ofEven Q) { val := e0 Q * \u2191(v Q) m, property := (_ : (fun x => x \u2208 even (Q' Q)) (e0 Q * \u2191(v Q) m)) } =\n    (\u2191(\u03b9 Q) 0 + \u2191(algebraMap R (CliffordAlgebra Q)) 1) * (\u2191(\u03b9 Q) m - \u2191(algebraMap R (CliffordAlgebra Q)) 0)\n[PROOFSTEP]\nexact ofEven_\u03b9 Q _ _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 (\u2191(\u03b9 Q) 0 + \u2191(algebraMap R (CliffordAlgebra Q)) 1) * (\u2191(\u03b9 Q) m - \u2191(algebraMap R (CliffordAlgebra Q)) 0) = \u2191(\u03b9 Q) m\n[PROOFSTEP]\nrw [map_one, map_zero, map_zero, sub_zero, zero_add, one_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : CliffordAlgebra Q\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute x))) = \u2191reverse \u2191(\u2191(toEven Q) x)\n[PROOFSTEP]\ninduction x using CliffordAlgebra.induction\n[GOAL]\ncase h_grade0\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nr\u271d : R\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(algebraMap R (CliffordAlgebra Q)) r\u271d)))) =\n    \u2191reverse \u2191(\u2191(toEven Q) (\u2191(algebraMap R (CliffordAlgebra Q)) r\u271d))\ncase h_grade1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx\u271d : M\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(\u03b9 Q) x\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) x\u271d))\ncase h_mul\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 * b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 * b\u271d))\ncase h_add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 + b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 + b\u271d))\n[PROOFSTEP]\ncase h_grade0 r => simp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nr : R\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(algebraMap R (CliffordAlgebra Q)) r)))) =\n    \u2191reverse \u2191(\u2191(toEven Q) (\u2191(algebraMap R (CliffordAlgebra Q)) r))\n[PROOFSTEP]\ncase h_grade0 r => simp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nr : R\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(algebraMap R (CliffordAlgebra Q)) r)))) =\n    \u2191reverse \u2191(\u2191(toEven Q) (\u2191(algebraMap R (CliffordAlgebra Q)) r))\n[PROOFSTEP]\nsimp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes]\n[GOAL]\ncase h_grade1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx\u271d : M\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(\u03b9 Q) x\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) x\u271d))\ncase h_mul\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 * b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 * b\u271d))\ncase h_add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 + b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 + b\u271d))\n[PROOFSTEP]\ncase h_grade1 m =>\n  -- porting note: added `letI`\n  letI : SubtractionMonoid (even (Q' Q)) := AddGroup.toSubtractionMonoid\n  simp only [involute_\u03b9, Subalgebra.coe_neg, toEven_\u03b9, reverse.map_mul, reverse_v, reverse_e0, reverse_\u03b9, neg_e0_mul_v,\n    map_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(\u03b9 Q) m)))) = \u2191reverse \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m))\n[PROOFSTEP]\ncase h_grade1 m =>\n  -- porting note: added `letI`\n  letI : SubtractionMonoid (even (Q' Q)) := AddGroup.toSubtractionMonoid\n  simp only [involute_\u03b9, Subalgebra.coe_neg, toEven_\u03b9, reverse.map_mul, reverse_v, reverse_e0, reverse_\u03b9, neg_e0_mul_v,\n    map_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(\u03b9 Q) m)))) = \u2191reverse \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m))\n[PROOFSTEP]\nletI : SubtractionMonoid (even (Q' Q)) := AddGroup.toSubtractionMonoid\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nm : M\nthis : SubtractionMonoid { x // x \u2208 even (Q' Q) } := AddGroup.toSubtractionMonoid\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (\u2191(\u03b9 Q) m)))) = \u2191reverse \u2191(\u2191(toEven Q) (\u2191(\u03b9 Q) m))\n[PROOFSTEP]\nsimp only [involute_\u03b9, Subalgebra.coe_neg, toEven_\u03b9, reverse.map_mul, reverse_v, reverse_e0, reverse_\u03b9, neg_e0_mul_v,\n  map_neg]\n[GOAL]\ncase h_mul\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 * b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 * b\u271d))\ncase h_add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 + b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 + b\u271d))\n[PROOFSTEP]\ncase h_mul x y hx hy => simp only [map_mul, Subalgebra.coe_mul, reverse.map_mul, hx, hy]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : CliffordAlgebra Q\nhx : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute x))) = \u2191reverse \u2191(\u2191(toEven Q) x)\nhy : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute y))) = \u2191reverse \u2191(\u2191(toEven Q) y)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (x * y)))) = \u2191reverse \u2191(\u2191(toEven Q) (x * y))\n[PROOFSTEP]\ncase h_mul x y hx hy => simp only [map_mul, Subalgebra.coe_mul, reverse.map_mul, hx, hy]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : CliffordAlgebra Q\nhx : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute x))) = \u2191reverse \u2191(\u2191(toEven Q) x)\nhy : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute y))) = \u2191reverse \u2191(\u2191(toEven Q) y)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (x * y)))) = \u2191reverse \u2191(\u2191(toEven Q) (x * y))\n[PROOFSTEP]\nsimp only [map_mul, Subalgebra.coe_mul, reverse.map_mul, hx, hy]\n[GOAL]\ncase h_add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\na\u271d\u00b2 b\u271d : CliffordAlgebra Q\na\u271d\u00b9 : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute a\u271d\u00b2))) = \u2191reverse \u2191(\u2191(toEven Q) a\u271d\u00b2)\na\u271d : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute b\u271d))) = \u2191reverse \u2191(\u2191(toEven Q) b\u271d)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (a\u271d\u00b2 + b\u271d)))) = \u2191reverse \u2191(\u2191(toEven Q) (a\u271d\u00b2 + b\u271d))\n[PROOFSTEP]\ncase h_add x y hx hy => simp only [map_add, Subalgebra.coe_add, hx, hy]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : CliffordAlgebra Q\nhx : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute x))) = \u2191reverse \u2191(\u2191(toEven Q) x)\nhy : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute y))) = \u2191reverse \u2191(\u2191(toEven Q) y)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (x + y)))) = \u2191reverse \u2191(\u2191(toEven Q) (x + y))\n[PROOFSTEP]\ncase h_add x y hx hy => simp only [map_add, Subalgebra.coe_add, hx, hy]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y : CliffordAlgebra Q\nhx : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute x))) = \u2191reverse \u2191(\u2191(toEven Q) x)\nhy : \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute y))) = \u2191reverse \u2191(\u2191(toEven Q) y)\n\u22a2 \u2191(\u2191(toEven Q) (\u2191reverse (\u2191involute (x + y)))) = \u2191reverse \u2191(\u2191(toEven Q) (x + y))\n[PROOFSTEP]\nsimp only [map_add, Subalgebra.coe_add, hx, hy]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' : QuadraticForm R M\nh : Q' = -Q\nthis\u271d : AddCommGroup { x // x \u2208 even Q' } := AddSubgroupClass.toAddCommGroup (even Q')\nthis : HasDistribNeg { x // x \u2208 even Q' } := NonUnitalNonAssocRing.toHasDistribNeg\nm : M\n\u22a2 \u2191(\u2191(-(even.\u03b9 Q').bilin) m) m = \u2191(algebraMap R { x // x \u2208 even Q' }) (\u2191Q m)\n[PROOFSTEP]\nsimp_rw [LinearMap.neg_apply, EvenHom.contract, h, QuadraticForm.neg_apply, map_neg, neg_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' : QuadraticForm R M\nh : Q' = -Q\nthis\u271d : AddCommGroup { x // x \u2208 even Q' } := AddSubgroupClass.toAddCommGroup (even Q')\nthis : HasDistribNeg { x // x \u2208 even Q' } := NonUnitalNonAssocRing.toHasDistribNeg\nm\u2081 m\u2082 m\u2083 : M\n\u22a2 \u2191(\u2191(-(even.\u03b9 Q').bilin) m\u2081) m\u2082 * \u2191(\u2191(-(even.\u03b9 Q').bilin) m\u2082) m\u2083 = \u2191Q m\u2082 \u2022 \u2191(\u2191(-(even.\u03b9 Q').bilin) m\u2081) m\u2083\n[PROOFSTEP]\nsimp_rw [LinearMap.neg_apply, neg_mul_neg, EvenHom.contract_mid, h, QuadraticForm.neg_apply, smul_neg, neg_smul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' : QuadraticForm R M\nh : Q' = -Q\nh' : Q = -Q'\n\u22a2 AlgHom.comp (evenToNeg Q' Q h') (evenToNeg Q Q' h) = AlgHom.id R { x // x \u2208 even Q }\n[PROOFSTEP]\next m\u2081 m\u2082 : 4\n[GOAL]\ncase h.bilin.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' : QuadraticForm R M\nh : Q' = -Q\nh' : Q = -Q'\nm\u2081 m\u2082 : M\n\u22a2 \u2191(\u2191(EvenHom.compr\u2082 (even.\u03b9 Q) (AlgHom.comp (evenToNeg Q' Q h') (evenToNeg Q Q' h))).bilin m\u2081) m\u2082 =\n    \u2191(\u2191(EvenHom.compr\u2082 (even.\u03b9 Q) (AlgHom.id R { x // x \u2208 even Q })).bilin m\u2081) m\u2082\n[PROOFSTEP]\ndsimp only [EvenHom.compr\u2082_bilin, LinearMap.compr\u2082_apply, AlgHom.toLinearMap_apply, AlgHom.comp_apply, AlgHom.id_apply]\n[GOAL]\ncase h.bilin.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ Q' : QuadraticForm R M\nh : Q' = -Q\nh' : Q = -Q'\nm\u2081 m\u2082 : M\n\u22a2 \u2191(evenToNeg Q' Q h') (\u2191(evenToNeg Q Q' h) (\u2191(\u2191(even.\u03b9 Q).bilin m\u2081) m\u2082)) = \u2191(\u2191(even.\u03b9 Q).bilin m\u2081) m\u2082\n[PROOFSTEP]\nrw [evenToNeg_\u03b9, map_neg, evenToNeg_\u03b9, neg_neg]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv", "llama_tokens": 23647, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744673038222, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.5662864055808184}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Mul M\ninst\u271d : ContinuousMul M\na b : M\n\u22a2 \ud835\udcdd a * \ud835\udcdd b \u2264 \ud835\udcdd (a * b)\n[PROOFSTEP]\nrw [\u2190 map\u2082_mul, \u2190 map_uncurry_prod, \u2190 nhds_prod_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Mul M\ninst\u271d : ContinuousMul M\na b : M\n\u22a2 map (Function.uncurry fun x x_1 => x * x_1) (\ud835\udcdd (a, b)) \u2264 \ud835\udcdd (a * b)\n[PROOFSTEP]\nexact continuous_mul.tendsto _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Mul M\ninst\u271d\u2075 : ContinuousMul M\ninst\u271d\u2074 : TopologicalSpace N\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : ContinuousMul N\ninst\u271d\u00b9 : T2Space N\nf : \u03b9 \u2192 N\u02e3\nr\u2081 r\u2082 : N\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u2081 : Tendsto (fun x => \u2191(f x)) l (\ud835\udcdd r\u2081)\nh\u2082 : Tendsto (fun x => \u2191(f x)\u207b\u00b9) l (\ud835\udcdd r\u2082)\n\u22a2 r\u2081 * r\u2082 = 1\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Mul M\ninst\u271d\u2075 : ContinuousMul M\ninst\u271d\u2074 : TopologicalSpace N\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : ContinuousMul N\ninst\u271d\u00b9 : T2Space N\nf : \u03b9 \u2192 N\u02e3\nr\u2081 r\u2082 : N\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u2081 : Tendsto (fun x => \u2191(f x)) l (\ud835\udcdd r\u2081)\nh\u2082 : Tendsto (fun x => \u2191(f x)\u207b\u00b9) l (\ud835\udcdd r\u2082)\n\u22a2 1 = r\u2081 * r\u2082\n[PROOFSTEP]\nsimpa using h\u2081.mul h\u2082\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Mul M\ninst\u271d\u2075 : ContinuousMul M\ninst\u271d\u2074 : TopologicalSpace N\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : ContinuousMul N\ninst\u271d\u00b9 : T2Space N\nf : \u03b9 \u2192 N\u02e3\nr\u2081 r\u2082 : N\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u2081 : Tendsto (fun x => \u2191(f x)) l (\ud835\udcdd r\u2081)\nh\u2082 : Tendsto (fun x => \u2191(f x)\u207b\u00b9) l (\ud835\udcdd r\u2082)\n\u22a2 r\u2082 * r\u2081 = 1\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Mul M\ninst\u271d\u2075 : ContinuousMul M\ninst\u271d\u2074 : TopologicalSpace N\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : ContinuousMul N\ninst\u271d\u00b9 : T2Space N\nf : \u03b9 \u2192 N\u02e3\nr\u2081 r\u2082 : N\nl : Filter \u03b9\ninst\u271d : NeBot l\nh\u2081 : Tendsto (fun x => \u2191(f x)) l (\ud835\udcdd r\u2081)\nh\u2082 : Tendsto (fun x => \u2191(f x)\u207b\u00b9) l (\ud835\udcdd r\u2082)\n\u22a2 1 = r\u2082 * r\u2081\n[PROOFSTEP]\nsimpa using h\u2082.mul h\u2081\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\n\u22a2 Continuous fun p => p.fst * p.snd\n[PROOFSTEP]\nrw [continuous_iff_continuousAt]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\n\u22a2 \u2200 (x : M \u00d7 M), ContinuousAt (fun p => p.fst * p.snd) x\n[PROOFSTEP]\nrintro \u27e8x\u2080, y\u2080\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\n\u22a2 ContinuousAt (fun p => p.fst * p.snd) (x\u2080, y\u2080)\n[PROOFSTEP]\nhave key : (fun p : M \u00d7 M => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry (\u00b7 * \u00b7) :=\n  by\n  ext p\n  simp [uncurry, mul_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\n\u22a2 (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\n[PROOFSTEP]\next p\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\np : M \u00d7 M\n\u22a2 x\u2080 * p.fst * (p.snd * y\u2080) = (((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1) p\n[PROOFSTEP]\nsimp [uncurry, mul_assoc]\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\n\u22a2 ContinuousAt (fun p => p.fst * p.snd) (x\u2080, y\u2080)\n[PROOFSTEP]\nhave key\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x :=\n  by\n  ext x\n  simp [mul_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\n\u22a2 ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\nx : M\n\u22a2 ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) x = x\u2080 * y\u2080 * x\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\nkey\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x\n\u22a2 ContinuousAt (fun p => p.fst * p.snd) (x\u2080, y\u2080)\n[PROOFSTEP]\ncalc\n  map (uncurry (\u00b7 * \u00b7)) (\ud835\udcdd (x\u2080, y\u2080)) = map (uncurry (\u00b7 * \u00b7)) (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080) := by rw [nhds_prod_eq]\n  _ = map (fun p : M \u00d7 M => x\u2080 * p.1 * (p.2 * y\u2080)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) :=\n    -- Porting note: `rw` was able to prove this\n            -- Now it fails with `failed to rewrite using equation theorems for 'Function.uncurry'`\n            -- and `failed to rewrite using equation theorems for 'Function.comp'`.\n            -- Removing those two lemmas, the `rw` would succeed, but then needs a `rfl`.by\n    simp_rw [uncurry, hleft x\u2080, hright y\u2080, prod_map_map_eq, Filter.map_map, Function.comp]\n  _ = map ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) (map (uncurry (\u00b7 * \u00b7)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)) := by rw [key, \u2190 Filter.map_map]\n  _ \u2264 map ((fun x : M => x\u2080 * x) \u2218 fun x => x * y\u2080) (\ud835\udcdd 1) := (map_mono hmul)\n  _ = \ud835\udcdd (x\u2080 * y\u2080) := by rw [\u2190 Filter.map_map, \u2190 hright, hleft y\u2080, Filter.map_map, key\u2082, \u2190 hleft]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\nkey\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x\n\u22a2 map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd (x\u2080, y\u2080)) = map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080)\n[PROOFSTEP]\nrw [nhds_prod_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\nkey\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x\n\u22a2 map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080) = map (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)\n[PROOFSTEP]\nsimp_rw [uncurry, hleft x\u2080, hright y\u2080, prod_map_map_eq, Filter.map_map, Function.comp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\nkey\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x\n\u22a2 map (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) =\n    map ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) (map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1))\n[PROOFSTEP]\nrw [key, \u2190 Filter.map_map]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 y\u2080 : M\nkey : (fun p => x\u2080 * p.fst * (p.snd * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1\nkey\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x\n\u22a2 map ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) (\ud835\udcdd 1) = \ud835\udcdd (x\u2080 * y\u2080)\n[PROOFSTEP]\nrw [\u2190 Filter.map_map, \u2190 hright, hleft y\u2080, Filter.map_map, key\u2082, \u2190 hleft]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\n\u22a2 ContinuousMul M\n[PROOFSTEP]\napply ContinuousMul.of_nhds_one hmul hleft\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\n\u22a2 \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\n[PROOFSTEP]\nintro x\u2080\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : Mul M\u271d\ninst\u271d\u00b2 : ContinuousMul M\u271d\nM : Type u\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : TopologicalSpace M\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nx\u2080 : M\n\u22a2 \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp_rw [mul_comm, hleft x\u2080]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u2074 : TopologicalSpace M\u2082\ninst\u271d\u00b3 : T2Space M\u2082\ninst\u271d\u00b2 : Mul M\u2081\ninst\u271d\u00b9 : Mul M\u2082\ninst\u271d : ContinuousMul M\u2082\n\u22a2 IsClosed {f | \u2200 (x y : M\u2081), f (x * y) = f x * f y}\n[PROOFSTEP]\nsimp only [setOf_forall]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u2074 : TopologicalSpace M\u2082\ninst\u271d\u00b3 : T2Space M\u2082\ninst\u271d\u00b2 : Mul M\u2081\ninst\u271d\u00b9 : Mul M\u2082\ninst\u271d : ContinuousMul M\u2082\n\u22a2 IsClosed (\u22c2 (i : M\u2081) (i_1 : M\u2081), {x | x (i * i_1) = x i * x i_1})\n[PROOFSTEP]\nexact\n  isClosed_iInter fun x =>\n    isClosed_iInter fun y =>\n      isClosed_eq\n        (continuous_apply _)\n          -- Porting note: proof was:\n                    -- `((continuous_apply _).mul (continuous_apply _))`\n        (by continuity)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : TopologicalSpace X\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u2074 : TopologicalSpace M\u2082\ninst\u271d\u00b3 : T2Space M\u2082\ninst\u271d\u00b2 : Mul M\u2081\ninst\u271d\u00b9 : Mul M\u2082\ninst\u271d : ContinuousMul M\u2082\nx y : M\u2081\n\u22a2 Continuous fun x_1 => x_1 x * x_1 y\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN\u271d : Type u_5\ninst\u271d\u2076 : TopologicalSpace X\nM : Type u_6\nN : Type u_7\nF : Type u_8\ninst\u271d\u2075 : Mul M\ninst\u271d\u2074 : Mul N\ninst\u271d\u00b3 : MulHomClass F M N\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : TopologicalSpace N\ninst\u271d : ContinuousMul N\nf : F\nhf : Inducing \u2191f\n\u22a2 Continuous (\u2191f \u2218 fun p => p.fst * p.snd)\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), map_mul f] using hf.continuous.fst'.mul hf.continuous.snd'\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 s\n[PROOFSTEP]\nhave : (fun a : M \u00d7 M => a.1 * a.2) \u207b\u00b9' s \u2208 \ud835\udcdd ((1, 1) : M \u00d7 M) := tendsto_mul (by simpa only [one_mul] using hs)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\n\u22a2 s \u2208 \ud835\udcdd (1 * 1)\n[PROOFSTEP]\nsimpa only [one_mul] using hs\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\nthis : (fun a => a.fst * a.snd) \u207b\u00b9' s \u2208 \ud835\udcdd (1, 1)\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 s\n[PROOFSTEP]\nsimpa only [prod_subset_iff] using exists_nhds_square this\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nu : Set M\nhu : u \u2208 \ud835\udcdd 1\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd 1 \u2227 \u2200 {v w s t : M}, v \u2208 V \u2192 w \u2208 V \u2192 s \u2208 V \u2192 t \u2208 V \u2192 v * w * s * t \u2208 u\n[PROOFSTEP]\nrcases exists_nhds_one_split hu with \u27e8W, W1, h\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nu : Set M\nhu : u \u2208 \ud835\udcdd 1\nW : Set M\nW1 : W \u2208 \ud835\udcdd 1\nh : \u2200 (v : M), v \u2208 W \u2192 \u2200 (w : M), w \u2208 W \u2192 v * w \u2208 u\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd 1 \u2227 \u2200 {v w s t : M}, v \u2208 V \u2192 w \u2208 V \u2192 s \u2208 V \u2192 t \u2208 V \u2192 v * w * s * t \u2208 u\n[PROOFSTEP]\nrcases exists_nhds_one_split W1 with \u27e8V, V1, h'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nu : Set M\nhu : u \u2208 \ud835\udcdd 1\nW : Set M\nW1 : W \u2208 \ud835\udcdd 1\nh : \u2200 (v : M), v \u2208 W \u2192 \u2200 (w : M), w \u2208 W \u2192 v * w \u2208 u\nV : Set M\nV1 : V \u2208 \ud835\udcdd 1\nh' : \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 W\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd 1 \u2227 \u2200 {v w s t : M}, v \u2208 V \u2192 w \u2208 V \u2192 s \u2208 V \u2192 t \u2208 V \u2192 v * w * s * t \u2208 u\n[PROOFSTEP]\nuse V, V1\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nu : Set M\nhu : u \u2208 \ud835\udcdd 1\nW : Set M\nW1 : W \u2208 \ud835\udcdd 1\nh : \u2200 (v : M), v \u2208 W \u2192 \u2200 (w : M), w \u2208 W \u2192 v * w \u2208 u\nV : Set M\nV1 : V \u2208 \ud835\udcdd 1\nh' : \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 W\n\u22a2 \u2200 {v w s t : M}, v \u2208 V \u2192 w \u2208 V \u2192 s \u2208 V \u2192 t \u2208 V \u2192 v * w * s * t \u2208 u\n[PROOFSTEP]\nintro v w s t v_in w_in s_in t_in\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nu : Set M\nhu : u \u2208 \ud835\udcdd 1\nW : Set M\nW1 : W \u2208 \ud835\udcdd 1\nh : \u2200 (v : M), v \u2208 W \u2192 \u2200 (w : M), w \u2208 W \u2192 v * w \u2208 u\nV : Set M\nV1 : V \u2208 \ud835\udcdd 1\nh' : \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 W\nv w s t : M\nv_in : v \u2208 V\nw_in : w \u2208 V\ns_in : s \u2208 V\nt_in : t \u2208 V\n\u22a2 v * w * s * t \u2208 u\n[PROOFSTEP]\nsimpa only [mul_assoc] using h _ (h' v v_in w w_in) _ (h' s s_in t t_in)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nU : Set M\nhU : U \u2208 \ud835\udcdd 1\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 V * V \u2286 U\n[PROOFSTEP]\nrcases exists_open_nhds_one_split hU with \u27e8V, Vo, V1, hV\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nU : Set M\nhU : U \u2208 \ud835\udcdd 1\nV : Set M\nVo : IsOpen V\nV1 : 1 \u2208 V\nhV : \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 U\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 V * V \u2286 U\n[PROOFSTEP]\nuse V, Vo, V1\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nU : Set M\nhU : U \u2208 \ud835\udcdd 1\nV : Set M\nVo : IsOpen V\nV1 : 1 \u2208 V\nhV : \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 U\n\u22a2 V * V \u2286 U\n[PROOFSTEP]\nrintro _ \u27e8x, y, hx, hy, rfl\u27e9\n[GOAL]\ncase right.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nU : Set M\nhU : U \u2208 \ud835\udcdd 1\nV : Set M\nVo : IsOpen V\nV1 : 1 \u2208 V\nhV : \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 U\nx y : M\nhx : x \u2208 V\nhy : y \u2208 V\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 U\n[PROOFSTEP]\nexact hV _ hx _ hy\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns t : Set M\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact (s * t)\n[PROOFSTEP]\nrw [\u2190 image_mul_prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns t : Set M\nhs : IsCompact s\nht : IsCompact t\n\u22a2 IsCompact ((fun x => x.fst * x.snd) '' s \u00d7\u02e2 t)\n[PROOFSTEP]\nexact (hs.prod ht).image continuous_mul\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 \u03b1 \u2192 M\nx : Filter \u03b1\na : \u03b9 \u2192 M\nx\u271d : \u2200 (i : \u03b9), i \u2208 [] \u2192 Tendsto (f i) x (\ud835\udcdd (a i))\n\u22a2 Tendsto (fun b => List.prod (List.map (fun c => f c b) [])) x (\ud835\udcdd (List.prod (List.map a [])))\n[PROOFSTEP]\nsimp [tendsto_const_nhds]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 M\nx : Filter \u03b1\na : \u03b9 \u2192 M\nf : \u03b9\nl : List \u03b9\nh : \u2200 (i : \u03b9), i \u2208 f :: l \u2192 Tendsto (f\u271d i) x (\ud835\udcdd (a i))\n\u22a2 Tendsto (fun b => List.prod (List.map (fun c => f\u271d c b) (f :: l))) x (\ud835\udcdd (List.prod (List.map a (f :: l))))\n[PROOFSTEP]\nsimp only [List.map_cons, List.prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 M\nx : Filter \u03b1\na : \u03b9 \u2192 M\nf : \u03b9\nl : List \u03b9\nh : \u2200 (i : \u03b9), i \u2208 f :: l \u2192 Tendsto (f\u271d i) x (\ud835\udcdd (a i))\n\u22a2 Tendsto (fun b => f\u271d f b * List.prod (List.map (fun c => f\u271d c b) l)) x (\ud835\udcdd (a f * List.prod (List.map a l)))\n[PROOFSTEP]\nexact (h f (List.mem_cons_self _ _)).mul (tendsto_list_prod l fun c hc => h c (List.mem_cons_of_mem _ hc))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nl : List \u03b9\nt : Set X\nh : \u2200 (i : \u03b9), i \u2208 l \u2192 ContinuousOn (f i) t\n\u22a2 ContinuousOn (fun a => List.prod (List.map (fun i => f i a) l)) t\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nl : List \u03b9\nt : Set X\nh : \u2200 (i : \u03b9), i \u2208 l \u2192 ContinuousOn (f i) t\nx : X\nhx : x \u2208 t\n\u22a2 ContinuousWithinAt (fun a => List.prod (List.map (fun i => f i a) l)) t x\n[PROOFSTEP]\nrw [continuousWithinAt_iff_continuousAt_restrict _ hx]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nl : List \u03b9\nt : Set X\nh : \u2200 (i : \u03b9), i \u2208 l \u2192 ContinuousOn (f i) t\nx : X\nhx : x \u2208 t\n\u22a2 ContinuousAt (restrict t fun a => List.prod (List.map (fun i => f i a) l)) { val := x, property := hx }\n[PROOFSTEP]\nrefine' tendsto_list_prod _ fun i hi => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nl : List \u03b9\nt : Set X\nh : \u2200 (i : \u03b9), i \u2208 l \u2192 ContinuousOn (f i) t\nx : X\nhx : x \u2208 t\ni : \u03b9\nhi : i \u2208 l\n\u22a2 Tendsto (fun b => f i \u2191b) (\ud835\udcdd { val := x, property := hx }) (\ud835\udcdd (f i \u2191{ val := x, property := hx }))\n[PROOFSTEP]\nspecialize h i hi x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nl : List \u03b9\nt : Set X\nx : X\nhx : x \u2208 t\ni : \u03b9\nhi : i \u2208 l\nh : ContinuousWithinAt (f i) t x\n\u22a2 Tendsto (fun b => f i \u2191b) (\ud835\udcdd { val := x, property := hx }) (\ud835\udcdd (f i \u2191{ val := x, property := hx }))\n[PROOFSTEP]\nrw [continuousWithinAt_iff_continuousAt_restrict _ hx] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nl : List \u03b9\nt : Set X\nx : X\nhx : x \u2208 t\ni : \u03b9\nhi : i \u2208 l\nh : ContinuousAt (restrict t (f i)) { val := x, property := hx }\n\u22a2 Tendsto (fun b => f i \u2191b) (\ud835\udcdd { val := x, property := hx }) (\ud835\udcdd (f i \u2191{ val := x, property := hx }))\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\n\u22a2 Continuous fun a => a ^ 0\n[PROOFSTEP]\nsimpa using continuous_const\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nk : \u2115\n\u22a2 Continuous fun a => a ^ (k + 1)\n[PROOFSTEP]\nsimp only [pow_succ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\nk : \u2115\n\u22a2 Continuous fun a => a * a ^ k\n[PROOFSTEP]\nexact continuous_id.mul (continuous_pow _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\na b : M\nha : b * a = 1\n\u22a2 Tendsto (fun x => a * x) (cocompact M) (cocompact M)\n[PROOFSTEP]\nrefine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_left b))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\na b : M\nha : b * a = 1\n\u22a2 Tendsto ((fun b_1 => b * b_1) \u2218 fun x => a * x) (cocompact M) (cocompact M)\n[PROOFSTEP]\nsimp only [comp_mul_left, ha, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\na b : M\nha : b * a = 1\n\u22a2 Tendsto (fun x => x) (cocompact M) (cocompact M)\n[PROOFSTEP]\nexact Filter.tendsto_id\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\na b : M\nha : a * b = 1\n\u22a2 Tendsto (fun x => x * a) (cocompact M) (cocompact M)\n[PROOFSTEP]\nrefine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_right b))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\na b : M\nha : a * b = 1\n\u22a2 Tendsto ((fun b_1 => b_1 * b) \u2218 fun x => x * a) (cocompact M) (cocompact M)\n[PROOFSTEP]\nsimp only [comp_mul_right, ha, mul_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\na b : M\nha : a * b = 1\n\u22a2 Tendsto (fun x => x) (cocompact M) (cocompact M)\n[PROOFSTEP]\nexact Filter.tendsto_id\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : ContinuousMul M\nR : Type u_6\nA : Type u_7\ninst\u271d\u2074 : Monoid A\ninst\u271d\u00b3 : SMul R A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : ContinuousMul A\nq : R\n\u22a2 Continuous fun x => q \u2022 x\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 smul_one_mul q (_ : A)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : ContinuousMul M\nR : Type u_6\nA : Type u_7\ninst\u271d\u2074 : Monoid A\ninst\u271d\u00b3 : SMul R A\ninst\u271d\u00b2 : IsScalarTower R A A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : ContinuousMul A\nq : R\n\u22a2 Continuous fun x => q \u2022 1 * x\n[PROOFSTEP]\nexact continuous_const.mul continuous_id\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : ContinuousMul M\nR : Type u_6\nA : Type u_7\ninst\u271d\u2074 : Monoid A\ninst\u271d\u00b3 : SMul R A\ninst\u271d\u00b2 : SMulCommClass R A A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : ContinuousMul A\nq : R\n\u22a2 Continuous fun x => q \u2022 x\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 mul_smul_one q (_ : A)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : TopologicalSpace X\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : Monoid M\ninst\u271d\u2075 : ContinuousMul M\nR : Type u_6\nA : Type u_7\ninst\u271d\u2074 : Monoid A\ninst\u271d\u00b3 : SMul R A\ninst\u271d\u00b2 : SMulCommClass R A A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : ContinuousMul A\nq : R\n\u22a2 Continuous fun x => x * q \u2022 1\n[PROOFSTEP]\nexact continuous_id.mul continuous_const\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 \u03b1 \u2192 M\nx : Filter \u03b1\na : \u03b9 \u2192 M\ns : Multiset \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Tendsto (f i) x (\ud835\udcdd (a i))) \u2192\n    Tendsto (fun b => Multiset.prod (Multiset.map (fun c => f c b) s)) x (\ud835\udcdd (Multiset.prod (Multiset.map a s)))\n[PROOFSTEP]\nrcases s with \u27e8l\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 \u03b1 \u2192 M\nx : Filter \u03b1\na : \u03b9 \u2192 M\ns : Multiset \u03b9\nl : List \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 Quot.mk Setoid.r l \u2192 Tendsto (f i) x (\ud835\udcdd (a i))) \u2192\n    Tendsto (fun b => Multiset.prod (Multiset.map (fun c => f c b) (Quot.mk Setoid.r l))) x\n      (\ud835\udcdd (Multiset.prod (Multiset.map a (Quot.mk Setoid.r l))))\n[PROOFSTEP]\nsimpa using tendsto_list_prod l\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 Continuous (f i)) \u2192 Continuous fun a => Multiset.prod (Multiset.map (fun i => f i a) s)\n[PROOFSTEP]\nrcases s with \u27e8l\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\nl : List \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 Quot.mk Setoid.r l \u2192 Continuous (f i)) \u2192\n    Continuous fun a => Multiset.prod (Multiset.map (fun i => f i a) (Quot.mk Setoid.r l))\n[PROOFSTEP]\nsimpa using continuous_list_prod l\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\nt : Set X\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 ContinuousOn (f i) t) \u2192 ContinuousOn (fun a => Multiset.prod (Multiset.map (fun i => f i a) s)) t\n[PROOFSTEP]\nrcases s with \u27e8l\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\nt : Set X\nl : List \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 Quot.mk Setoid.r l \u2192 ContinuousOn (f i) t) \u2192\n    ContinuousOn (fun a => Multiset.prod (Multiset.map (fun i => f i a) (Quot.mk Setoid.r l))) t\n[PROOFSTEP]\nsimpa using continuousOn_list_prod l\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\u271d\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nX : Type u_6\nM : Type u_7\ninst\u271d : CommMonoid M\ns : Finset \u03b9\nl : Filter X\nf g : \u03b9 \u2192 X \u2192 M\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 f i =\u1da0[l] g i\n\u22a2 \u220f i in s, f i =\u1da0[l] \u220f i in s, g i\n[PROOFSTEP]\nreplace hs : \u2200\u1da0 x in l, \u2200 i \u2208 s, f i x = g i x\n[GOAL]\ncase hs\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\u271d\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nX : Type u_6\nM : Type u_7\ninst\u271d : CommMonoid M\ns : Finset \u03b9\nl : Filter X\nf g : \u03b9 \u2192 X \u2192 M\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 f i =\u1da0[l] g i\n\u22a2 \u2200\u1da0 (x : X) in l, \u2200 (i : \u03b9), i \u2208 s \u2192 f i x = g i x\n[PROOFSTEP]\nrwa [eventually_all_finset]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\u271d\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nX : Type u_6\nM : Type u_7\ninst\u271d : CommMonoid M\ns : Finset \u03b9\nl : Filter X\nf g : \u03b9 \u2192 X \u2192 M\nhs : \u2200\u1da0 (x : X) in l, \u2200 (i : \u03b9), i \u2208 s \u2192 f i x = g i x\n\u22a2 \u220f i in s, f i =\u1da0[l] \u220f i in s, g i\n[PROOFSTEP]\nfilter_upwards [hs] with x hx\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX\u271d : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\u271d\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nX : Type u_6\nM : Type u_7\ninst\u271d : CommMonoid M\ns : Finset \u03b9\nl : Filter X\nf g : \u03b9 \u2192 X \u2192 M\nhs : \u2200\u1da0 (x : X) in l, \u2200 (i : \u03b9), i \u2208 s \u2192 f i x = g i x\nx : X\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 f i x = g i x\n\u22a2 Finset.prod s (fun i => f i) x = Finset.prod s (fun i => g i) x\n[PROOFSTEP]\nsimp only [Finset.prod_apply, Finset.prod_congr rfl hx]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nM : Type u_6\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 X \u2192 M\nhf : LocallyFinite fun i => mulSupport (f i)\nx\u2080 : X\n\u22a2 \u2203 I, \u2200\u1da0 (x : X) in \ud835\udcdd x\u2080, (mulSupport fun i => f i x) \u2286 \u2191I\n[PROOFSTEP]\nrcases hf x\u2080 with \u27e8U, hxU, hUf\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nM : Type u_6\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 X \u2192 M\nhf : LocallyFinite fun i => mulSupport (f i)\nx\u2080 : X\nU : Set X\nhxU : U \u2208 \ud835\udcdd x\u2080\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i \u2229 U)}\n\u22a2 \u2203 I, \u2200\u1da0 (x : X) in \ud835\udcdd x\u2080, (mulSupport fun i => f i x) \u2286 \u2191I\n[PROOFSTEP]\nrefine' \u27e8hUf.toFinset, mem_of_superset hxU fun y hy i hi => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nM : Type u_6\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 X \u2192 M\nhf : LocallyFinite fun i => mulSupport (f i)\nx\u2080 : X\nU : Set X\nhxU : U \u2208 \ud835\udcdd x\u2080\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i \u2229 U)}\ny : X\nhy : y \u2208 U\ni : \u03b9\nhi : i \u2208 mulSupport fun i => f i y\n\u22a2 i \u2208 \u2191(Finite.toFinset hUf)\n[PROOFSTEP]\nrw [hUf.coe_toFinset]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM\u271d : Type u_4\nN : Type u_5\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace M\u271d\ninst\u271d\u00b2 : CommMonoid M\u271d\ninst\u271d\u00b9 : ContinuousMul M\u271d\nM : Type u_6\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 X \u2192 M\nhf : LocallyFinite fun i => mulSupport (f i)\nx\u2080 : X\nU : Set X\nhxU : U \u2208 \ud835\udcdd x\u2080\nhUf : Set.Finite {i | Set.Nonempty ((fun i => mulSupport (f i)) i \u2229 U)}\ny : X\nhy : y \u2208 U\ni : \u03b9\nhi : i \u2208 mulSupport fun i => f i y\n\u22a2 i \u2208 {i | Set.Nonempty ((fun i => mulSupport (f i)) i \u2229 U)}\n[PROOFSTEP]\nexact \u27e8y, hi, hy\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n\u22a2 Continuous fun x => \u220f\u1da0 (i : \u03b9), f i x\n[PROOFSTEP]\nrefine' continuous_iff_continuousAt.2 fun x => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\n\u22a2 ContinuousAt (fun x => \u220f\u1da0 (i : \u03b9), f i x) x\n[PROOFSTEP]\nrcases finprod_eventually_eq_prod hf x with \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset \u03b9\nhs : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u220f\u1da0 (i : \u03b9), f i y = \u220f i in s, f i y\n\u22a2 ContinuousAt (fun x => \u220f\u1da0 (i : \u03b9), f i x) x\n[PROOFSTEP]\nrefine' ContinuousAt.congr _ (EventuallyEq.symm hs)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset \u03b9\nhs : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u220f\u1da0 (i : \u03b9), f i y = \u220f i in s, f i y\n\u22a2 ContinuousAt (fun x => \u220f i in s, f i x) x\n[PROOFSTEP]\nexact tendsto_finset_prod _ fun i _ => (hc i).continuousAt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\np : \u03b9 \u2192 Prop\nhc : \u2200 (i : \u03b9), p i \u2192 Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n\u22a2 Continuous fun x => \u220f\u1da0 (i : \u03b9) (_ : p i), f i x\n[PROOFSTEP]\nsimp only [\u2190 finprod_subtype_eq_finprod_cond]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\np : \u03b9 \u2192 Prop\nhc : \u2200 (i : \u03b9), p i \u2192 Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n\u22a2 Continuous fun x => \u220f\u1da0 (j : { i // p i }), f (\u2191j) x\n[PROOFSTEP]\nexact continuous_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nts : \u03b9' \u2192 TopologicalSpace M\nh' : \u2200 (i : \u03b9'), ContinuousMul M\n\u22a2 ContinuousMul M\n[PROOFSTEP]\nrw [\u2190 sInf_range]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nts : \u03b9' \u2192 TopologicalSpace M\nh' : \u2200 (i : \u03b9'), ContinuousMul M\n\u22a2 ContinuousMul M\n[PROOFSTEP]\nexact continuousMul_sInf (Set.forall_range_iff.mpr h')\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nt\u2081 t\u2082 : TopologicalSpace M\nh\u2081 : ContinuousMul M\nh\u2082 : ContinuousMul M\n\u22a2 ContinuousMul M\n[PROOFSTEP]\nrw [inf_eq_iInf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nt\u2081 t\u2082 : TopologicalSpace M\nh\u2081 : ContinuousMul M\nh\u2082 : ContinuousMul M\n\u22a2 ContinuousMul M\n[PROOFSTEP]\nrefine' continuousMul_iInf fun b => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nt\u2081 t\u2082 : TopologicalSpace M\nh\u2081 : ContinuousMul M\nh\u2082 : ContinuousMul M\nb : Bool\n\u22a2 ContinuousMul M\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nt\u2081 t\u2082 : TopologicalSpace M\nh\u2081 : ContinuousMul M\nh\u2082 : ContinuousMul M\n\u22a2 ContinuousMul M\n[PROOFSTEP]\nassumption\n[GOAL]\ncase true\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X\n\u03b9' : Sort u_6\ninst\u271d : Mul M\nt\u2081 t\u2082 : TopologicalSpace M\nh\u2081 : ContinuousMul M\nh\u2082 : ContinuousMul M\n\u22a2 ContinuousMul M\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Monoid", "llama_tokens": 19925, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199714402812, "lm_q2_score": 0.6757646140788307, "lm_q1q2_score": 0.5660339367450629}}
{"text": "[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 a * a = 0 \u2194\n    \u2200 (x y : R),\n      \u2191(\u2191(LinearEquiv.toEquiv (LinearEquiv.symm (LinearMap.ringLmapEquivSelf R \u2115 A))) a) x *\n          \u2191(\u2191(LinearEquiv.toEquiv (LinearEquiv.symm (LinearMap.ringLmapEquivSelf R \u2115 A))) a) y =\n        0\n[PROOFSTEP]\ndsimp\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 a * a = 0 \u2194 \u2200 (x y : R), x \u2022 a * y \u2022 a = 0\n[PROOFSTEP]\nsimp_rw [smul_mul_smul]\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 a * a = 0 \u2194 \u2200 (x y : R), (x * y) \u2022 (a * a) = 0\n[PROOFSTEP]\nrefine' \u27e8fun h x y => h.symm \u25b8 smul_zero _, fun h => by simpa using h 1 1\u27e9\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\nh : \u2200 (x y : R), (x * y) \u2022 (a * a) = 0\n\u22a2 a * a = 0\n[PROOFSTEP]\nsimpa using h 1 1\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\ne : { e // e * e = 0 }\n\u22a2 \u2191(\u2191lift e) \u03b5 = \u2191e\n[PROOFSTEP]\nsimp only [lift_apply_apply, fst_eps, map_zero, snd_eps, one_smul, zero_add]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.DualNumber", "llama_tokens": 617, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256472515684, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.5659860623023389}}
{"text": "[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 count p 0 = 0\n[PROOFSTEP]\nrw [count, List.range_zero, List.countp, List.countp.go]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 Fintype { i // i < n \u2227 p i }\n[PROOFSTEP]\napply Fintype.ofFinset ((Finset.range n).filter p)\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 filter p (range n) \u2194 x \u2208 fun x => x < n \u2227 p x\n[PROOFSTEP]\nintro x\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn x : \u2115\n\u22a2 x \u2208 filter p (range n) \u2194 x \u2208 fun x => x < n \u2227 p x\n[PROOFSTEP]\nrw [mem_filter, mem_range]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn x : \u2115\n\u22a2 x < n \u2227 p x \u2194 x \u2208 fun x => x < n \u2227 p x\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p n = card (filter p (range n))\n[PROOFSTEP]\nrw [count, List.countp_eq_length_filter]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 List.length (List.filter (fun b => decide (p b)) (List.range n)) = card (filter p (range n))\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p n = Fintype.card { k // k < n \u2227 p k }\n[PROOFSTEP]\nrw [count_eq_card_filter_range, \u2190 Fintype.card_ofFinset, \u2190 CountSet.fintype]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 (Fintype.card \u2191fun x => x < n \u2227 p x) = Fintype.card { k // k < n \u2227 p k }\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p (n + 1) = count p n + if p n then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : p n\n\u22a2 count p (n + 1) = count p n + 1\n[PROOFSTEP]\nsimp [count, List.range_succ, h]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : \u00acp n\n\u22a2 count p (n + 1) = count p n + 0\n[PROOFSTEP]\nsimp [count, List.range_succ, h]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p n \u2264 count p (n + 1)\n[PROOFSTEP]\nby_cases h : p n\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : p n\n\u22a2 count p n \u2264 count p (n + 1)\n[PROOFSTEP]\nsimp [count_succ, h]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : \u00acp n\n\u22a2 count p n \u2264 count p (n + 1)\n[PROOFSTEP]\nsimp [count_succ, h]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 count p (a + b) = count p a + count (fun k => p (a + k)) b\n[PROOFSTEP]\nhave : Disjoint ((range a).filter p) (((range b).map <| addLeftEmbedding a).filter p) :=\n  by\n  apply disjoint_filter_filter\n  rw [Finset.disjoint_left]\n  simp_rw [mem_map, mem_range, addLeftEmbedding_apply]\n  rintro x hx \u27e8c, _, rfl\u27e9\n  exact (self_le_add_right _ _).not_lt hx\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 Disjoint (filter p (range a)) (filter p (map (addLeftEmbedding a) (range b)))\n[PROOFSTEP]\napply disjoint_filter_filter\n[GOAL]\ncase a\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 Disjoint (range a) (map (addLeftEmbedding a) (range b))\n[PROOFSTEP]\nrw [Finset.disjoint_left]\n[GOAL]\ncase a\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 \u2200 \u2983a_1 : \u2115\u2984, a_1 \u2208 range a \u2192 \u00aca_1 \u2208 map (addLeftEmbedding a) (range b)\n[PROOFSTEP]\nsimp_rw [mem_map, mem_range, addLeftEmbedding_apply]\n[GOAL]\ncase a\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 \u2200 \u2983a_1 : \u2115\u2984, a_1 < a \u2192 \u00ac\u2203 a_3, a_3 < b \u2227 a + a_3 = a_1\n[PROOFSTEP]\nrintro x hx \u27e8c, _, rfl\u27e9\n[GOAL]\ncase a.intro.intro\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b c : \u2115\nleft\u271d : c < b\nhx : a + c < a\n\u22a2 False\n[PROOFSTEP]\nexact (self_le_add_right _ _).not_lt hx\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\nthis : Disjoint (filter p (range a)) (filter p (map (addLeftEmbedding a) (range b)))\n\u22a2 count p (a + b) = count p a + count (fun k => p (a + k)) b\n[PROOFSTEP]\nsimp_rw [count_eq_card_filter_range, range_add, filter_union, card_disjoint_union this, filter_map, addLeftEmbedding,\n  card_map]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\nthis : Disjoint (filter p (range a)) (filter p (map (addLeftEmbedding a) (range b)))\n\u22a2 card (filter p (range a)) +\n      card\n        (filter (p \u2218 \u2191{ toFun := fun h => a + h, inj' := (_ : Function.Injective ((fun x x_1 => x + x_1) a)) })\n          (range b)) =\n    card (filter p (range a)) + card (filter (fun k => p (a + k)) (range b))\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 count p (a + b) = count (fun k => p (k + b)) a + count p b\n[PROOFSTEP]\nrw [add_comm, count_add, add_comm]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\na b : \u2115\n\u22a2 count (fun k => p (b + k)) a + count p b = count (fun k => p (k + b)) a + count p b\n[PROOFSTEP]\nsimp_rw [add_comm b]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 count p 1 = if p 0 then 1 else 0\n[PROOFSTEP]\nsimp [count_succ]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p (n + 1) = count (fun k => p (k + 1)) n + if p 0 then 1 else 0\n[PROOFSTEP]\nrw [count_add', count_one]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p n < count p (n + 1) \u2194 p n\n[PROOFSTEP]\nby_cases h : p n\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : p n\n\u22a2 count p n < count p (n + 1) \u2194 p n\n[PROOFSTEP]\nsimp [count_succ, h]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : \u00acp n\n\u22a2 count p n < count p (n + 1) \u2194 p n\n[PROOFSTEP]\nsimp [count_succ, h]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p (n + 1) = count p n + 1 \u2194 p n\n[PROOFSTEP]\nby_cases h : p n\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : p n\n\u22a2 count p (n + 1) = count p n + 1 \u2194 p n\n[PROOFSTEP]\nsimp [h, count_succ]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : \u00acp n\n\u22a2 count p (n + 1) = count p n + 1 \u2194 p n\n[PROOFSTEP]\nsimp [h, count_succ]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p (n + 1) = count p n \u2194 \u00acp n\n[PROOFSTEP]\nby_cases h : p n\n[GOAL]\ncase pos\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : p n\n\u22a2 count p (n + 1) = count p n \u2194 \u00acp n\n[PROOFSTEP]\nsimp [h, count_succ]\n[GOAL]\ncase neg\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\nh : \u00acp n\n\u22a2 count p (n + 1) = count p n \u2194 \u00acp n\n[PROOFSTEP]\nsimp [h, count_succ]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 \u2191(count p n) \u2264 Cardinal.mk \u2191{k | p k}\n[PROOFSTEP]\nrw [count_eq_card_fintype, \u2190 Cardinal.mk_fintype]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 Cardinal.mk { k // k < n \u2227 p k } \u2264 Cardinal.mk \u2191{k | p k}\n[PROOFSTEP]\nexact Cardinal.mk_subtype_mono fun x hx \u21a6 hx.2\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nm n : \u2115\nhm : p m\nhn : p n\nheq : count p m = count p n\n\u22a2 m = n\n[PROOFSTEP]\nby_contra' h : m \u2260 n\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nm n : \u2115\nhm : p m\nhn : p n\nheq : count p m = count p n\nh : m \u2260 n\n\u22a2 False\n[PROOFSTEP]\nwlog hmn : m < n\n[GOAL]\ncase inr\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nm n : \u2115\nhm : p m\nhn : p n\nheq : count p m = count p n\nh : m \u2260 n\nthis : \u2200 {p : \u2115 \u2192 Prop} [inst : DecidablePred p] {m n : \u2115}, p m \u2192 p n \u2192 count p m = count p n \u2192 m \u2260 n \u2192 m < n \u2192 False\nhmn : \u00acm < n\n\u22a2 False\n[PROOFSTEP]\nexact this hn hm heq.symm h.symm (h.lt_or_lt.resolve_left hmn)\n[GOAL]\np\u271d : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\u271d\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nm n : \u2115\nhm : p m\nhn : p n\nheq : count p m = count p n\nh : m \u2260 n\nhmn : m < n\n\u22a2 False\n[PROOFSTEP]\nsimpa [heq] using count_strict_mono hm hmn\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : Set.Finite (setOf p)\nn : \u2115\n\u22a2 count p n \u2264 card (Set.Finite.toFinset hp)\n[PROOFSTEP]\nrw [count_eq_card_filter_range]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : Set.Finite (setOf p)\nn : \u2115\n\u22a2 card (filter p (range n)) \u2264 card (Set.Finite.toFinset hp)\n[PROOFSTEP]\nexact Finset.card_mono fun x hx \u21a6 hp.mem_toFinset.2 (mem_filter.1 hx).2\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u2115 \u2192 Prop\ninst\u271d : DecidablePred q\nn : \u2115\nhpq : \u2200 (k : \u2115), p k \u2192 q k\n\u22a2 count p n \u2264 count q n\n[PROOFSTEP]\nsimp only [count_eq_card_filter_range]\n[GOAL]\np : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u2115 \u2192 Prop\ninst\u271d : DecidablePred q\nn : \u2115\nhpq : \u2200 (k : \u2115), p k \u2192 q k\n\u22a2 card (filter p (range n)) \u2264 card (filter q (range n))\n[PROOFSTEP]\nexact card_le_of_subset ((range n).monotone_filter_right hpq)\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Count", "llama_tokens": 3893, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396212, "lm_q2_score": 0.7090191276365462, "lm_q1q2_score": 0.565929692843972}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : PartialOrder \u03b2\nf g : C(\u03b1, \u03b2)\nx\u271d : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : PartialOrder \u03b2\ng : C(\u03b1, \u03b2)\ntoFun\u271d : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d : Continuous toFun\u271d\nx\u271d : (fun f => f.toFun) (mk toFun\u271d) = (fun f => f.toFun) g\n\u22a2 mk toFun\u271d = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : PartialOrder \u03b2\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d\u00b9 : Continuous toFun\u271d\u00b9\ntoFun\u271d : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d : Continuous toFun\u271d\nx\u271d : (fun f => f.toFun) (mk toFun\u271d\u00b9) = (fun f => f.toFun) (mk toFun\u271d)\n\u22a2 mk toFun\u271d\u00b9 = mk toFun\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : OrderClosedTopology \u03b2\nsrc\u271d\u00b9 : PartialOrder C(\u03b1, \u03b2) := partialOrder\nsrc\u271d : Sup C(\u03b1, \u03b2) := sup\nf g : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : \u03b1), \u2191f a \u2264 \u2191(f \u2294 g) a\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : OrderClosedTopology \u03b2\nsrc\u271d\u00b9 : PartialOrder C(\u03b1, \u03b2) := partialOrder\nsrc\u271d : Sup C(\u03b1, \u03b2) := sup\nf g : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : \u03b1), \u2191g a \u2264 \u2191(f \u2294 g) a\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : OrderClosedTopology \u03b2\nsrc\u271d\u00b9 : PartialOrder C(\u03b1, \u03b2) := partialOrder\nsrc\u271d : Sup C(\u03b1, \u03b2) := sup\nf\u2081 f\u2082 g : C(\u03b1, \u03b2)\nw\u2081 : f\u2081 \u2264 g\nw\u2082 : f\u2082 \u2264 g\na : \u03b1\n\u22a2 \u2191(f\u2081 \u2294 f\u2082) a \u2264 \u2191g a\n[PROOFSTEP]\nsimp [le_def.mp w\u2081 a, le_def.mp w\u2082 a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : OrderClosedTopology \u03b2\nsrc\u271d\u00b9 : PartialOrder C(\u03b1, \u03b2) := partialOrder\nsrc\u271d : Inf C(\u03b1, \u03b2) := inf\nf g : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : \u03b1), \u2191(f \u2293 g) a \u2264 \u2191f a\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : OrderClosedTopology \u03b2\nsrc\u271d\u00b9 : PartialOrder C(\u03b1, \u03b2) := partialOrder\nsrc\u271d : Inf C(\u03b1, \u03b2) := inf\nf g : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : \u03b1), \u2191(f \u2293 g) a \u2264 \u2191g a\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : OrderClosedTopology \u03b2\nsrc\u271d\u00b9 : PartialOrder C(\u03b1, \u03b2) := partialOrder\nsrc\u271d : Inf C(\u03b1, \u03b2) := inf\nf\u2081 f\u2082 g : C(\u03b1, \u03b2)\nw\u2081 : f\u2081 \u2264 f\u2082\nw\u2082 : f\u2081 \u2264 g\na : \u03b1\n\u22a2 \u2191f\u2081 a \u2264 \u2191(f\u2082 \u2293 g) a\n[PROOFSTEP]\nsimp [le_def.mp w\u2081 a, le_def.mp w\u2082 a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b3\ninst\u271d : OrderClosedTopology \u03b3\n\u03b9 : Type u_4\ns : Finset \u03b9\nH : Finset.Nonempty s\nf : \u03b9 \u2192 C(\u03b2, \u03b3)\n\u22a2 \u2191(Finset.sup' s H f) = Finset.sup' s H fun a => \u2191(f a)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LinearOrder \u03b3\ninst\u271d : OrderClosedTopology \u03b3\n\u03b9 : Type u_4\ns : Finset \u03b9\nH : Finset.Nonempty s\nf : \u03b9 \u2192 C(\u03b2, \u03b3)\nx\u271d : \u03b2\n\u22a2 \u2191(Finset.sup' s H f) x\u271d = Finset.sup' s H (fun a => \u2191(f a)) x\u271d\n[PROOFSTEP]\nsimp [sup'_apply]\n", "meta": {"mathlib_filename": "Mathlib.Topology.ContinuousFunction.Ordered", "llama_tokens": 1951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267728417087, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.565914550922654}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b1\nx : \u03b2\nh : IsLUB (\u2191f '' s) x\n\u22a2 \u2200 {x y : \u03b1}, \u2191f x \u2264 \u2191f y \u2194 x \u2264 y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b1\nx : \u03b2\nh : IsLUB s (\u2191(symm f) x)\n\u22a2 \u2200 {x y : \u03b2}, \u2191(symm f) x \u2264 \u2191(symm f) y \u2194 x \u2264 y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b1\nx : \u03b1\n\u22a2 IsLUB (\u2191f '' s) (\u2191f x) \u2194 IsLUB s x\n[PROOFSTEP]\nrw [isLUB_image, f.symm_apply_apply]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b2\nx : \u03b1\n\u22a2 IsLUB (\u2191f \u207b\u00b9' s) x \u2194 IsLUB s (\u2191f x)\n[PROOFSTEP]\nrw [\u2190 f.symm_symm, \u2190 image_eq_preimage, isLUB_image]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b2\nx : \u03b2\n\u22a2 IsLUB (\u2191f \u207b\u00b9' s) (\u2191(symm f) x) \u2194 IsLUB s x\n[PROOFSTEP]\nrw [isLUB_preimage, f.apply_symm_apply]\n", "meta": {"mathlib_filename": "Mathlib.Order.Bounds.OrderIso", "llama_tokens": 562, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.800691997339971, "lm_q2_score": 0.7057850340255386, "lm_q1q2_score": 0.5651164285865679}}
{"text": "[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\n\u22a2 centerMass \u2205 w z = 0\n[PROOFSTEP]\nsimp only [centerMass, sum_empty, smul_zero]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhne : i \u2260 j\n\u22a2 centerMass {i, j} w z = (w i / (w i + w j)) \u2022 z i + (w j / (w i + w j)) \u2022 z j\n[PROOFSTEP]\nsimp only [centerMass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nha : \u00aci \u2208 t\nhw : \u2211 j in t, w j \u2260 0\n\u22a2 centerMass (insert i t) w z =\n    (w i / (w i + \u2211 j in t, w j)) \u2022 z i + ((\u2211 j in t, w j) / (w i + \u2211 j in t, w j)) \u2022 centerMass t w z\n[PROOFSTEP]\nsimp only [centerMass, sum_insert ha, smul_add, (mul_smul _ _ _).symm, \u2190 div_eq_inv_mul]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nha : \u00aci \u2208 t\nhw : \u2211 j in t, w j \u2260 0\n\u22a2 (w i / (w i + \u2211 i in t, w i)) \u2022 z i + (w i + \u2211 i in t, w i)\u207b\u00b9 \u2022 \u2211 i in t, w i \u2022 z i =\n    (w i / (w i + \u2211 i in t, w i)) \u2022 z i +\n      ((\u2211 i in t, w i) / (w i + \u2211 i in t, w i) * (\u2211 i in t, w i)\u207b\u00b9) \u2022 \u2211 i in t, w i \u2022 z i\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nha : \u00aci \u2208 t\nhw : \u2211 j in t, w j \u2260 0\n\u22a2 (w i + \u2211 i in t, w i)\u207b\u00b9 = (\u2211 i in t, w i) / (w i + \u2211 i in t, w i) * (\u2211 i in t, w i)\u207b\u00b9\n[PROOFSTEP]\nrw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw : w i \u2260 0\n\u22a2 centerMass {i} w z = z i\n[PROOFSTEP]\nrw [centerMass, sum_singleton, sum_singleton, \u2190 mul_smul, inv_mul_cancel hw, one_smul]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw : \u2211 i in t, w i = 1\n\u22a2 centerMass t w z = \u2211 i in t, w i \u2022 z i\n[PROOFSTEP]\nsimp only [Finset.centerMass, hw, inv_one, one_smul]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\n\u22a2 (centerMass t w fun i => c \u2022 z i) = c \u2022 centerMass t w z\n[PROOFSTEP]\nsimp only [Finset.centerMass, Finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 a \u2022 centerMass s ws zs + b \u2022 centerMass t wt zt =\n    centerMass (disjSum s t) (Sum.elim (fun i => a * ws i) fun j => b * wt j) (Sum.elim zs zt)\n[PROOFSTEP]\nrw [s.centerMass_eq_of_sum_1 _ hws, t.centerMass_eq_of_sum_1 _ hwt, smul_sum, smul_sum, \u2190 Finset.sum_sum_elim,\n  Finset.centerMass_eq_of_sum_1]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 x in disjSum s t, Sum.elim (fun x => a \u2022 ws x \u2022 zs x) (fun x => b \u2022 wt x \u2022 zt x) x =\n    \u2211 i in disjSum s t, Sum.elim (fun i => a * ws i) (fun j => b * wt j) i \u2022 Sum.elim zs zt i\n[PROOFSTEP]\ncongr with \u27e8\u27e9\n[GOAL]\ncase e_f.h.inl\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\nval\u271d : \u03b9\n\u22a2 Sum.elim (fun x => a \u2022 ws x \u2022 zs x) (fun x => b \u2022 wt x \u2022 zt x) (Sum.inl val\u271d) =\n    Sum.elim (fun i => a * ws i) (fun j => b * wt j) (Sum.inl val\u271d) \u2022 Sum.elim zs zt (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Sum.elim_inr, mul_smul]\n[GOAL]\ncase e_f.h.inr\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\nval\u271d : \u03b9'\n\u22a2 Sum.elim (fun x => a \u2022 ws x \u2022 zs x) (fun x => b \u2022 wt x \u2022 zt x) (Sum.inr val\u271d) =\n    Sum.elim (fun i => a * ws i) (fun j => b * wt j) (Sum.inr val\u271d) \u2022 Sum.elim zs zt (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Sum.elim_inr, mul_smul]\n[GOAL]\ncase hw\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 i in disjSum s t, Sum.elim (fun i => a * ws i) (fun j => b * wt j) i = 1\n[PROOFSTEP]\nrw [sum_sum_elim, \u2190 mul_sum, \u2190 mul_sum, hws, hwt, mul_one, mul_one, hab]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz\u271d : \u03b9 \u2192 E\ns : Finset \u03b9\nw\u2081 w\u2082 : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\na b : R\nhab : a + b = 1\n\u22a2 a \u2022 centerMass s w\u2081 z + b \u2022 centerMass s w\u2082 z = centerMass s (fun i => a * w\u2081 i + b * w\u2082 i) z\n[PROOFSTEP]\nhave hw : (\u2211 i in s, (a * w\u2081 i + b * w\u2082 i)) = 1 := by simp only [mul_sum.symm, sum_add_distrib, mul_one, *]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz\u271d : \u03b9 \u2192 E\ns : Finset \u03b9\nw\u2081 w\u2082 : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 i in s, (a * w\u2081 i + b * w\u2082 i) = 1\n[PROOFSTEP]\nsimp only [mul_sum.symm, sum_add_distrib, mul_one, *]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz\u271d : \u03b9 \u2192 E\ns : Finset \u03b9\nw\u2081 w\u2082 : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2081 : \u2211 i in s, w\u2081 i = 1\nhw\u2082 : \u2211 i in s, w\u2082 i = 1\na b : R\nhab : a + b = 1\nhw : \u2211 i in s, (a * w\u2081 i + b * w\u2082 i) = 1\n\u22a2 a \u2022 centerMass s w\u2081 z + b \u2022 centerMass s w\u2082 z = centerMass s (fun i => a * w\u2081 i + b * w\u2082 i) z\n[PROOFSTEP]\nsimp only [Finset.centerMass_eq_of_sum_1, Finset.centerMass_eq_of_sum_1 _ _ hw, smul_sum, sum_add_distrib, add_smul,\n  mul_smul, *]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i \u2208 t\n\u22a2 centerMass t (fun j => if i = j then 1 else 0) z = z i\n[PROOFSTEP]\nrw [Finset.centerMass_eq_of_sum_1]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i \u2208 t\n\u22a2 \u2211 i_1 in t, (if i = i_1 then 1 else 0) \u2022 z i_1 = z i\ncase hw\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i \u2208 t\n\u22a2 (\u2211 i_1 in t, if i = i_1 then 1 else 0) = 1\n[PROOFSTEP]\ntrans \u2211 j in t, if i = j then z i else 0\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i \u2208 t\n\u22a2 \u2211 i_1 in t, (if i = i_1 then 1 else 0) \u2022 z i_1 = \u2211 j in t, if i = j then z i else 0\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i\u271d \u2208 t\ni : \u03b9\n\u22a2 (if i\u271d = i then 1 else 0) \u2022 z i = if i\u271d = i then z i\u271d else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i\u271d \u2208 t\ni : \u03b9\nh : i\u271d = i\n\u22a2 1 \u2022 z i = z i\u271d\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i\u271d \u2208 t\ni : \u03b9\nh : \u00aci\u271d = i\n\u22a2 0 \u2022 z i = 0\n[PROOFSTEP]\nexacts [h \u25b8 one_smul _ _, zero_smul _ _]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i \u2208 t\n\u22a2 (\u2211 j in t, if i = j then z i else 0) = z i\n[PROOFSTEP]\nrw [sum_ite_eq, if_pos hi]\n[GOAL]\ncase hw\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhi : i \u2208 t\n\u22a2 (\u2211 i_1 in t, if i = i_1 then 1 else 0) = 1\n[PROOFSTEP]\nrw [sum_ite_eq, if_pos hi]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nt' : Finset \u03b9\nht : t \u2286 t'\nh : \u2200 (i : \u03b9), i \u2208 t' \u2192 \u00aci \u2208 t \u2192 w i = 0\n\u22a2 centerMass t w z = centerMass t' w z\n[PROOFSTEP]\nrw [centerMass, sum_subset ht h, smul_sum, centerMass, smul_sum]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nt' : Finset \u03b9\nht : t \u2286 t'\nh : \u2200 (i : \u03b9), i \u2208 t' \u2192 \u00aci \u2208 t \u2192 w i = 0\n\u22a2 \u2211 x in t, (\u2211 x in t', w x)\u207b\u00b9 \u2022 w x \u2022 z x = \u2211 x in t', (\u2211 i in t', w i)\u207b\u00b9 \u2022 w x \u2022 z x\n[PROOFSTEP]\napply sum_subset ht\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nt' : Finset \u03b9\nht : t \u2286 t'\nh : \u2200 (i : \u03b9), i \u2208 t' \u2192 \u00aci \u2208 t \u2192 w i = 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 t' \u2192 \u00acx \u2208 t \u2192 (\u2211 x in t', w x)\u207b\u00b9 \u2022 w x \u2022 z x = 0\n[PROOFSTEP]\nintro i hit' hit\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nt' : Finset \u03b9\nht : t \u2286 t'\nh : \u2200 (i : \u03b9), i \u2208 t' \u2192 \u00aci \u2208 t \u2192 w i = 0\ni : \u03b9\nhit' : i \u2208 t'\nhit : \u00aci \u2208 t\n\u22a2 (\u2211 x in t', w x)\u207b\u00b9 \u2022 w i \u2022 z i = 0\n[PROOFSTEP]\nrw [h i hit' hit, zero_smul, smul_zero]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ni : \u03b9\nhit : i \u2208 t\nhit' : \u00aci \u2208 filter (fun i => w i \u2260 0) t\n\u22a2 w i = 0\n[PROOFSTEP]\nsimpa only [hit, mem_filter, true_and_iff, Ne.def, Classical.not_not] using hit'\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : 0 < \u2211 i in s, w i\n\u22a2 s \u2260 \u2205\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nf : \u03b9 \u2192 \u03b1\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 0 \u2264 w i\nhw\u2081 : 0 < \u2211 i in \u2205, w i\n\u22a2 False\n[PROOFSTEP]\nsimp at hw\u2081 \n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : 0 < \u2211 i in s, w i\n\u22a2 centerMass s w f \u2264 sup' s (_ : Finset.Nonempty s) f\n[PROOFSTEP]\nrw [centerMass, inv_smul_le_iff hw\u2081, sum_smul]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : 0 < \u2211 i in s, w i\n\u22a2 \u2211 i in s, w i \u2022 f i \u2264 \u2211 i in s, w i \u2022 sup' s (_ : Finset.Nonempty s) f\n[PROOFSTEP]\nexact sum_le_sum fun i hi => smul_le_smul_of_nonneg (le_sup' _ hi) <| hw\u2080 i hi\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : 0 < \u2211 i in s, w i\n\u22a2 s \u2260 \u2205\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nf : \u03b9 \u2192 \u03b1\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 0 \u2264 w i\nhw\u2081 : 0 < \u2211 i in \u2205, w i\n\u22a2 False\n[PROOFSTEP]\nsimp at hw\u2081 \n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\n\u22a2 (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\n[PROOFSTEP]\ninduction' t using Finset.induction with i t hi ht\n[GOAL]\ncase empty\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in \u2205, w i \u2192 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 z i \u2208 s) \u2192 centerMass \u2205 w z \u2208 s\n[PROOFSTEP]\nsimp [lt_irrefl]\n[GOAL]\ncase insert\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\n\u22a2 (\u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1) \u2192\n    0 < \u2211 i in insert i t, w i \u2192 (\u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s) \u2192 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nintro h\u2080 hpos hmem\n[GOAL]\ncase insert\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < \u2211 i in insert i t, w i\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nhave zi : z i \u2208 s := hmem _ (mem_insert_self _ _)\n[GOAL]\ncase insert\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < \u2211 i in insert i t, w i\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nhave hs\u2080 : \u2200 j \u2208 t, 0 \u2264 w j := fun j hj => h\u2080 j <| mem_insert_of_mem hj\n[GOAL]\ncase insert\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < \u2211 i in insert i t, w i\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nrw [sum_insert hi] at hpos \n[GOAL]\ncase insert\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nby_cases hsum_t : \u2211 j in t, w j = 0\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nhave ws : \u2200 j \u2208 t, w j = 0 := (sum_eq_zero_iff_of_nonneg hs\u2080).1 hsum_t\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\nws : \u2200 (j : \u03b9), j \u2208 t \u2192 w j = 0\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nhave wz : \u2211 j in t, w j \u2022 z j = 0 := sum_eq_zero fun i hi => by simp [ws i hi]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d\u00b9 j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni\u271d : \u03b9\nt : Finset \u03b9\nhi\u271d : \u00aci\u271d \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i : \u03b9), i \u2208 insert i\u271d t \u2192 0 \u2264 w i\nhpos : 0 < w i\u271d + \u2211 x in t, w x\nhmem : \u2200 (i : \u03b9), i \u2208 insert i\u271d t \u2192 z i \u2208 s\nzi : z i\u271d \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\nws : \u2200 (j : \u03b9), j \u2208 t \u2192 w j = 0\ni : \u03b9\nhi : i \u2208 t\n\u22a2 w i \u2022 z i = 0\n[PROOFSTEP]\nsimp [ws i hi]\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\nws : \u2200 (j : \u03b9), j \u2208 t \u2192 w j = 0\nwz : \u2211 j in t, w j \u2022 z j = 0\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nsimp only [centerMass, sum_insert hi, wz, hsum_t, add_zero]\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\nws : \u2200 (j : \u03b9), j \u2208 t \u2192 w j = 0\nwz : \u2211 j in t, w j \u2022 z j = 0\n\u22a2 (w i)\u207b\u00b9 \u2022 w i \u2022 z i \u2208 s\n[PROOFSTEP]\nsimp only [hsum_t, add_zero] at hpos \n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\nws : \u2200 (j : \u03b9), j \u2208 t \u2192 w j = 0\nwz : \u2211 j in t, w j \u2022 z j = 0\nhpos : 0 < w i\n\u22a2 (w i)\u207b\u00b9 \u2022 w i \u2022 z i \u2208 s\n[PROOFSTEP]\nrw [\u2190 mul_smul, inv_mul_cancel (ne_of_gt hpos), one_smul]\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u2211 j in t, w j = 0\nws : \u2200 (j : \u03b9), j \u2208 t \u2192 w j = 0\nwz : \u2211 j in t, w j \u2022 z j = 0\nhpos : 0 < w i\n\u22a2 z i \u2208 s\n[PROOFSTEP]\nexact zi\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u00ac\u2211 j in t, w j = 0\n\u22a2 centerMass (insert i t) w z \u2208 s\n[PROOFSTEP]\nrw [Finset.centerMass_insert _ _ _ hi hsum_t]\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u00ac\u2211 j in t, w j = 0\n\u22a2 (w i / (w i + \u2211 j in t, w j)) \u2022 z i + ((\u2211 j in t, w j) / (w i + \u2211 j in t, w j)) \u2022 centerMass t (fun j => w j) z \u2208 s\n[PROOFSTEP]\nrefine' convex_iff_div.1 hs zi (ht hs\u2080 _ _) _ (sum_nonneg hs\u2080) hpos\n[GOAL]\ncase neg.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u00ac\u2211 j in t, w j = 0\n\u22a2 0 < \u2211 i in t, w i\n[PROOFSTEP]\nexact lt_of_le_of_ne (sum_nonneg hs\u2080) (Ne.symm hsum_t)\n[GOAL]\ncase neg.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u00ac\u2211 j in t, w j = 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase neg.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j\u271d : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u00ac\u2211 j in t, w j = 0\nj : \u03b9\nhj : j \u2208 t\n\u22a2 z j \u2208 s\n[PROOFSTEP]\nexact hmem j (mem_insert_of_mem hj)\n[GOAL]\ncase neg.refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\ni : \u03b9\nt : Finset \u03b9\nhi : \u00aci \u2208 t\nht : (\u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i) \u2192 0 < \u2211 i in t, w i \u2192 (\u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s) \u2192 centerMass t w z \u2208 s\nh\u2080 : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 0 \u2264 w i_1\nhpos : 0 < w i + \u2211 x in t, w x\nhmem : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 z i_1 \u2208 s\nzi : z i \u2208 s\nhs\u2080 : \u2200 (j : \u03b9), j \u2208 t \u2192 0 \u2264 w j\nhsum_t : \u00ac\u2211 j in t, w j = 0\n\u22a2 0 \u2264 w i\n[PROOFSTEP]\nexact h\u2080 _ (mem_insert_self _ _)\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nh\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 \u2211 i in t, w i \u2022 z i \u2208 s\n[PROOFSTEP]\nsimpa only [h\u2081, centerMass, inv_one, one_smul] using hs.centerMass_mem h\u2080 (h\u2081.symm \u25b8 zero_lt_one) hz\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\n\u22a2 \u2211\u1da0 (i : \u03b9), w i \u2022 z i \u2208 s\n[PROOFSTEP]\nhave hfin_w : (support (w \u2218 PLift.down)).Finite := by\n  by_contra H\n  rw [finsum, dif_neg H] at h\u2081 \n  exact zero_ne_one h\u2081\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\n\u22a2 Set.Finite (support (w \u2218 PLift.down))\n[PROOFSTEP]\nby_contra H\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nH : \u00acSet.Finite (support (w \u2218 PLift.down))\n\u22a2 False\n[PROOFSTEP]\nrw [finsum, dif_neg H] at h\u2081 \n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : 0 = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nH : \u00acSet.Finite (support (w \u2218 PLift.down))\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one h\u2081\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nhfin_w : Set.Finite (support (w \u2218 PLift.down))\n\u22a2 \u2211\u1da0 (i : \u03b9), w i \u2022 z i \u2208 s\n[PROOFSTEP]\nhave hsub : support ((fun i => w i \u2022 z i) \u2218 PLift.down) \u2286 hfin_w.toFinset :=\n  (support_smul_subset_left _ _).trans hfin_w.coe_toFinset.ge\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nhfin_w : Set.Finite (support (w \u2218 PLift.down))\nhsub : support ((fun i => w i \u2022 z i) \u2218 PLift.down) \u2286 \u2191(Finite.toFinset hfin_w)\n\u22a2 \u2211\u1da0 (i : \u03b9), w i \u2022 z i \u2208 s\n[PROOFSTEP]\nrw [finsum_eq_sum_pLift_of_support_subset hsub]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nhfin_w : Set.Finite (support (w \u2218 PLift.down))\nhsub : support ((fun i => w i \u2022 z i) \u2218 PLift.down) \u2286 \u2191(Finite.toFinset hfin_w)\n\u22a2 \u2211 i in Finite.toFinset hfin_w, w i.down \u2022 z i.down \u2208 s\n[PROOFSTEP]\nrefine' hs.sum_mem (fun _ _ => h\u2080 _) _ fun i hi => hz _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nhfin_w : Set.Finite (support (w \u2218 PLift.down))\nhsub : support ((fun i => w i \u2022 z i) \u2218 PLift.down) \u2286 \u2191(Finite.toFinset hfin_w)\n\u22a2 \u2211 i in Finite.toFinset hfin_w, w i.down = 1\n[PROOFSTEP]\nrwa [finsum, dif_pos hfin_w] at h\u2081 \n[GOAL]\ncase refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\n\u03b9 : Sort u_7\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Convex R s\nh\u2080 : \u2200 (i : \u03b9), 0 \u2264 w i\nh\u2081 : \u2211\u1da0 (i : \u03b9), w i = 1\nhz : \u2200 (i : \u03b9), w i \u2260 0 \u2192 z i \u2208 s\nhfin_w : Set.Finite (support (w \u2218 PLift.down))\nhsub : support ((fun i => w i \u2022 z i) \u2218 PLift.down) \u2286 \u2191(Finite.toFinset hfin_w)\ni : PLift \u03b9\nhi : i \u2208 Finite.toFinset hfin_w\n\u22a2 w i.down \u2260 0\n[PROOFSTEP]\nrwa [hfin_w.mem_toFinset] at hi \n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\n\u22a2 Convex R s \u2194\n    \u2200 (t : Finset E) (w : E \u2192 R),\n      (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\n[PROOFSTEP]\nrefine' \u27e8fun hs t w hw\u2080 hw\u2081 hts => hs.sum_mem hw\u2080 hw\u2081 hts, _\u27e9\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\n\u22a2 (\u2200 (t : Finset E) (w : E \u2192 R),\n      (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s) \u2192\n    Convex R s\n[PROOFSTEP]\nintro h x hx y hy a b ha hb hab\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 s\n[PROOFSTEP]\nby_cases h_cases : x = y\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : x = y\n\u22a2 a \u2022 x + b \u2022 y \u2208 s\n[PROOFSTEP]\nrw [h_cases, \u2190 add_smul, hab, one_smul]\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : x = y\n\u22a2 y \u2208 s\n[PROOFSTEP]\nexact hy\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 a \u2022 x + b \u2022 y \u2208 s\n[PROOFSTEP]\nconvert\n  h { x, y } (fun z => if z = y then b else a) _ _\n    _\n      -- Porting note: Original proof had 2 `simp_intro i hi`\n[GOAL]\ncase h.e'_4\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 a \u2022 x + b \u2022 y = \u2211 x in {x, y}, (if x = y then b else a) \u2022 x\n[PROOFSTEP]\nsimp only [sum_pair h_cases, if_neg h_cases, if_pos trivial]\n[GOAL]\ncase neg.convert_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 \u2200 (i : E), i \u2208 {x, y} \u2192 0 \u2264 (fun z => if z = y then b else a) i\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase neg.convert_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\na\u271d : i \u2208 {x, y}\n\u22a2 0 \u2264 (fun z => if z = y then b else a) i\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase neg.convert_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\na\u271d : i \u2208 {x, y}\n\u22a2 0 \u2264 if i = y then b else a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\na\u271d : i \u2208 {x, y}\nh\u271d : i = y\n\u22a2 0 \u2264 b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\na\u271d : i \u2208 {x, y}\nh\u271d : \u00aci = y\n\u22a2 0 \u2264 a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.convert_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 \u2211 i in {x, y}, (fun z => if z = y then b else a) i = 1\n[PROOFSTEP]\nsimp only [sum_pair h_cases, if_neg h_cases, if_pos trivial, hab]\n[GOAL]\ncase neg.convert_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 \u2200 (x_1 : E), x_1 \u2208 {x, y} \u2192 x_1 \u2208 s\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase neg.convert_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\nhi : i \u2208 {x, y}\n\u22a2 i \u2208 s\n[PROOFSTEP]\nsimp only [Finset.mem_singleton, Finset.mem_insert] at hi \n[GOAL]\ncase neg.convert_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\nhi : i = x \u2228 i = y\n\u22a2 i \u2208 s\n[PROOFSTEP]\ncases hi\n[GOAL]\ncase neg.convert_3.inl\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\nh\u271d : i = x\n\u22a2 i \u2208 s\n[PROOFSTEP]\nsubst i\n[GOAL]\ncase neg.convert_3.inr\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\ni : E\nh\u271d : i = y\n\u22a2 i \u2208 s\n[PROOFSTEP]\nsubst i\n[GOAL]\ncase neg.convert_3.inl\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 x \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.convert_3.inr\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nh :\n  \u2200 (t : Finset E) (w : E \u2192 R),\n    (\u2200 (i : E), i \u2208 t \u2192 0 \u2264 w i) \u2192 \u2211 i in t, w i = 1 \u2192 (\u2200 (x : E), x \u2208 t \u2192 x \u2208 s) \u2192 \u2211 x in t, w x \u2022 x \u2208 s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nh_cases : \u00acx = y\n\u22a2 y \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\n\u03b9 : Type u_7\nt : Finset \u03b9\np : \u03b9 \u2192 E\nw : \u03b9 \u2192 R\nhw\u2082 : \u2211 i in t, w i = 1\n\u22a2 \u2191(affineCombination R t p) w = centerMass t w p\n[PROOFSTEP]\nrw [affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ w _ hw\u2082 (0 : E), Finset.weightedVSubOfPoint_apply,\n  vadd_eq_add, add_zero, t.centerMass_eq_of_sum_1 _ hw\u2082]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\n\u03b9 : Type u_7\nt : Finset \u03b9\np : \u03b9 \u2192 E\nw : \u03b9 \u2192 R\nhw\u2082 : \u2211 i in t, w i = 1\n\u22a2 \u2211 i in t, w i \u2022 (p i -\u1d65 0) = \u2211 i in t, w i \u2022 p i\n[PROOFSTEP]\nsimp_rw [vsub_eq_sub, sub_zero]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nv : \u03b9 \u2192 E\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\n\u22a2 \u2191(affineCombination R s v) w \u2208 \u2191(convexHull R) (Set.range v)\n[PROOFSTEP]\nrw [affineCombination_eq_centerMass hw\u2081]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nv : \u03b9 \u2192 E\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\n\u22a2 centerMass s (fun i => w i) v \u2208 \u2191(convexHull R) (Set.range v)\n[PROOFSTEP]\napply s.centerMass_mem_convexHull hw\u2080\n[GOAL]\ncase hws\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nv : \u03b9 \u2192 E\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\n\u22a2 0 < \u2211 i in s, w i\n[PROOFSTEP]\nsimp [hw\u2081]\n[GOAL]\ncase hz\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nv : \u03b9 \u2192 E\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 v i \u2208 Set.range v\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nhs : Finset.Nonempty s\n\u22a2 centroid R s id \u2208 \u2191(convexHull R) \u2191s\n[PROOFSTEP]\nrw [s.centroid_eq_centerMass hs]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nhs : Finset.Nonempty s\n\u22a2 centerMass s (centroidWeights R s) id \u2208 \u2191(convexHull R) \u2191s\n[PROOFSTEP]\napply s.centerMass_id_mem_convexHull\n[GOAL]\ncase hw\u2080\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nhs : Finset.Nonempty s\n\u22a2 \u2200 (i : E), i \u2208 s \u2192 0 \u2264 centroidWeights R s i\n[PROOFSTEP]\nsimp only [inv_nonneg, imp_true_iff, Nat.cast_nonneg, Finset.centroidWeights_apply]\n[GOAL]\ncase hws\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nhs : Finset.Nonempty s\n\u22a2 0 < \u2211 i in s, centroidWeights R s i\n[PROOFSTEP]\nhave hs_card : (s.card : R) \u2260 0 := by simp [Finset.nonempty_iff_ne_empty.mp hs]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nhs : Finset.Nonempty s\n\u22a2 \u2191(card s) \u2260 0\n[PROOFSTEP]\nsimp [Finset.nonempty_iff_ne_empty.mp hs]\n[GOAL]\ncase hws\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nhs : Finset.Nonempty s\nhs_card : \u2191(card s) \u2260 0\n\u22a2 0 < \u2211 i in s, centroidWeights R s i\n[PROOFSTEP]\nsimp only [hs_card, Finset.sum_const, nsmul_eq_mul, mul_inv_cancel, Ne.def, not_false_iff, Finset.centroidWeights_apply,\n  zero_lt_one]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\n\u22a2 \u2191(convexHull R) (Set.range v) = {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nrefine' Subset.antisymm (convexHull_min _ _) _\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\n\u22a2 Set.range v \u2286 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\nx : E\nhx : x \u2208 Set.range v\n\u22a2 x \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Set.mem_range.mp hx\n[GOAL]\ncase refine'_1.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\nx : E\nhx : x \u2208 Set.range v\ni : \u03b9\nhi : v i = x\n\u22a2 x \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nrefine' \u27e8{ i }, Function.const \u03b9 (1 : R), by simp, by simp, by simp [hi]\u27e9\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\nx : E\nhx : x \u2208 Set.range v\ni : \u03b9\nhi : v i = x\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 {i} \u2192 0 \u2264 const \u03b9 1 i_1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\nx : E\nhx : x \u2208 Set.range v\ni : \u03b9\nhi : v i = x\n\u22a2 Finset.sum {i} (const \u03b9 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\nx : E\nhx : x \u2208 Set.range v\ni : \u03b9\nhi : v i = x\n\u22a2 \u2191(affineCombination R {i} v) (const \u03b9 1) = x\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\n\u22a2 Convex R {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nrintro x \u27e8s, w, hw\u2080, hw\u2081, rfl\u27e9 y \u27e8s', w', hw\u2080', hw\u2081', rfl\u27e9 a b ha hb hab\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191(affineCombination R s v) w + b \u2022 \u2191(affineCombination R s' v) w' \u2208\n    {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nlet W : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\n\u22a2 a \u2022 \u2191(affineCombination R s v) w + b \u2022 \u2191(affineCombination R s' v) w' \u2208\n    {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nhave hW\u2081 : (s \u222a s').sum W = 1 := by\n  rw [sum_add_distrib, \u2190 sum_subset (subset_union_left s s'), \u2190 sum_subset (subset_union_right s s'),\n        sum_ite_of_true _ _ fun i hi => hi, sum_ite_of_true _ _ fun i hi => hi, \u2190 mul_sum, \u2190 mul_sum, hw\u2081, hw\u2081', \u2190\n        add_mul, hab, mul_one] <;>\n      intro i _ hi' <;>\n    simp [hi']\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\n\u22a2 Finset.sum (s \u222a s') W = 1\n[PROOFSTEP]\nrw [sum_add_distrib, \u2190 sum_subset (subset_union_left s s'), \u2190 sum_subset (subset_union_right s s'),\n  sum_ite_of_true _ _ fun i hi => hi, sum_ite_of_true _ _ fun i hi => hi, \u2190 mul_sum, \u2190 mul_sum, hw\u2081, hw\u2081', \u2190 add_mul,\n  hab, mul_one]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u222a s' \u2192 \u00acx \u2208 s' \u2192 (if x \u2208 s' then b * w' x else 0) = 0\n[PROOFSTEP]\nintro i _ hi'\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u222a s' \u2192 \u00acx \u2208 s \u2192 (if x \u2208 s then a * w x else 0) = 0\n[PROOFSTEP]\nintro i _ hi'\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\ni : \u03b9\na\u271d : i \u2208 s \u222a s'\nhi' : \u00aci \u2208 s'\n\u22a2 (if i \u2208 s' then b * w' i else 0) = 0\n[PROOFSTEP]\nsimp [hi']\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\ni : \u03b9\na\u271d : i \u2208 s \u222a s'\nhi' : \u00aci \u2208 s\n\u22a2 (if i \u2208 s then a * w i else 0) = 0\n[PROOFSTEP]\nsimp [hi']\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 a \u2022 \u2191(affineCombination R s v) w + b \u2022 \u2191(affineCombination R s' v) w' \u2208\n    {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x}\n[PROOFSTEP]\nrefine' \u27e8s \u222a s', W, _, hW\u2081, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u222a s' \u2192 0 \u2264 W i\n[PROOFSTEP]\nrintro i -\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nby_cases hi : i \u2208 s\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nby_cases hi' : i \u2208 s'\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nby_cases hi' : i \u2208 s'\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\nhi : i \u2208 s\nhi' : i \u2208 s'\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nsimp [hi, hi', add_nonneg, mul_nonneg ha (hw\u2080 i _), mul_nonneg hb (hw\u2080' i _)]\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\nhi : i \u2208 s\nhi' : \u00aci \u2208 s'\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nsimp [hi, hi', add_nonneg, mul_nonneg ha (hw\u2080 i _), mul_nonneg hb (hw\u2080' i _)]\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\nhi : \u00aci \u2208 s\nhi' : i \u2208 s'\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nsimp [hi, hi', add_nonneg, mul_nonneg ha (hw\u2080 i _), mul_nonneg hb (hw\u2080' i _)]\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\nhi : \u00aci \u2208 s\nhi' : \u00aci \u2208 s'\n\u22a2 0 \u2264 W i\n[PROOFSTEP]\nsimp [hi, hi', add_nonneg, mul_nonneg ha (hw\u2080 i _), mul_nonneg hb (hw\u2080' i _)]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 \u2191(affineCombination R (s \u222a s') v) W = a \u2022 \u2191(affineCombination R s v) w + b \u2022 \u2191(affineCombination R s' v) w'\n[PROOFSTEP]\nsimp_rw [affineCombination_eq_linear_combination (s \u222a s') v _ hW\u2081, affineCombination_eq_linear_combination s v w hw\u2081,\n  affineCombination_eq_linear_combination s' v w' hw\u2081', add_smul, sum_add_distrib]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 \u2211 x in s \u222a s', (if x \u2208 s then a * w x else 0) \u2022 v x + \u2211 x in s \u222a s', (if x \u2208 s' then b * w' x else 0) \u2022 v x =\n    a \u2022 \u2211 i in s, w i \u2022 v i + b \u2022 \u2211 i in s', w' i \u2022 v i\n[PROOFSTEP]\nrw [\u2190 sum_subset (subset_union_left s s'), \u2190 sum_subset (subset_union_right s s')]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 \u2211 x in s, (if x \u2208 s then a * w x else 0) \u2022 v x + \u2211 x in s', (if x \u2208 s' then b * w' x else 0) \u2022 v x =\n    a \u2022 \u2211 i in s, w i \u2022 v i + b \u2022 \u2211 i in s', w' i \u2022 v i\n[PROOFSTEP]\nsimp only [ite_smul, sum_ite_of_true _ _ fun _ hi => hi, mul_smul, \u2190 smul_sum]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u222a s' \u2192 \u00acx \u2208 s' \u2192 (if x \u2208 s' then b * w' x else 0) \u2022 v x = 0\n[PROOFSTEP]\nintro i _ hi'\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\na\u271d : i \u2208 s \u222a s'\nhi' : \u00aci \u2208 s'\n\u22a2 (if i \u2208 s' then b * w' i else 0) \u2022 v i = 0\n[PROOFSTEP]\nsimp [hi']\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u222a s' \u2192 \u00acx \u2208 s \u2192 (if x \u2208 s then a * w x else 0) \u2022 v x = 0\n[PROOFSTEP]\nintro i _ hi'\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ns' : Finset \u03b9\nw' : \u03b9 \u2192 R\nhw\u2080' : \u2200 (i : \u03b9), i \u2208 s' \u2192 0 \u2264 w' i\nhw\u2081' : Finset.sum s' w' = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nW : \u03b9 \u2192 R := fun i => (if i \u2208 s then a * w i else 0) + if i \u2208 s' then b * w' i else 0\nhW\u2081 : Finset.sum (s \u222a s') W = 1\ni : \u03b9\na\u271d : i \u2208 s \u222a s'\nhi' : \u00aci \u2208 s\n\u22a2 (if i \u2208 s then a * w i else 0) \u2022 v i = 0\n[PROOFSTEP]\nsimp [hi']\n[GOAL]\ncase refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\n\u22a2 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s v) w = x} \u2286 \u2191(convexHull R) (Set.range v)\n[PROOFSTEP]\nrintro x \u27e8s, w, hw\u2080, hw\u2081, rfl\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz v : \u03b9 \u2192 E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\n\u22a2 \u2191(affineCombination R s v) w \u2208 \u2191(convexHull R) (Set.range v)\n[PROOFSTEP]\nexact affineCombination_mem_convexHull hw\u2080 hw\u2081\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 \u2191(convexHull R) s = {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\n[PROOFSTEP]\nrefine' Subset.antisymm (convexHull_min _ _) _\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 s \u2286 {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nx : E\nhx : x \u2208 s\n\u22a2 x \u2208 {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\n[PROOFSTEP]\nuse PUnit, { PUnit.unit }, fun _ => 1, fun _ => x, fun _ _ => zero_le_one, Finset.sum_singleton, fun _ _ => hx\n[GOAL]\ncase h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nx : E\nhx : x \u2208 s\n\u22a2 (centerMass {PUnit.unit} (fun x => 1) fun x_1 => x) = x\n[PROOFSTEP]\nsimp only [Finset.centerMass, Finset.sum_singleton, inv_one, one_smul]\n[GOAL]\ncase refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 Convex R {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\n[PROOFSTEP]\nrintro x \u27e8\u03b9, sx, wx, zx, hwx\u2080, hwx\u2081, hzx, rfl\u27e9 y \u27e8\u03b9', sy, wy, zy, hwy\u2080, hwy\u2081, hzy, rfl\u27e9 a b ha hb hab\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 centerMass sx wx zx + b \u2022 centerMass sy wy zy \u2208 {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\n[PROOFSTEP]\nrw [Finset.centerMass_segment' _ _ _ _ _ _ hwx\u2081 hwy\u2081 _ _ hab]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 centerMass (disjSum sx sy) (Sum.elim (fun i => a * wx i) fun j => b * wy j) (Sum.elim zx zy) \u2208\n    {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\n[PROOFSTEP]\nrefine' \u27e8_, _, _, _, _, _, _, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2200 (i : \u03b9 \u2295 \u03b9'), i \u2208 disjSum sx sy \u2192 0 \u2264 Sum.elim (fun i => a * wx i) (fun j => b * wy j) i\n[PROOFSTEP]\nrintro i hi\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 disjSum sx sy\n\u22a2 0 \u2264 Sum.elim (fun i => a * wx i) (fun j => b * wy j) i\n[PROOFSTEP]\nrw [Finset.mem_disjSum] at hi \n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9 \u2295 \u03b9'\nhi : (\u2203 a, a \u2208 sx \u2227 Sum.inl a = i) \u2228 \u2203 b, b \u2208 sy \u2227 Sum.inr b = i\n\u22a2 0 \u2264 Sum.elim (fun i => a * wx i) (fun j => b * wy j) i\n[PROOFSTEP]\nrcases hi with (\u27e8j, hj, rfl\u27e9 | \u27e8j, hj, rfl\u27e9)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.inl.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nj : \u03b9\nhj : j \u2208 sx\n\u22a2 0 \u2264 Sum.elim (fun i => a * wx i) (fun j => b * wy j) (Sum.inl j)\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Sum.elim_inr]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.inr.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nj : \u03b9'\nhj : j \u2208 sy\n\u22a2 0 \u2264 Sum.elim (fun i => a * wx i) (fun j => b * wy j) (Sum.inr j)\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Sum.elim_inr]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.inl.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nj : \u03b9\nhj : j \u2208 sx\n\u22a2 0 \u2264 a * wx j\n[PROOFSTEP]\napply_rules [mul_nonneg, hwx\u2080, hwy\u2080]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.inr.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nj : \u03b9'\nhj : j \u2208 sy\n\u22a2 0 \u2264 b * wy j\n[PROOFSTEP]\napply_rules [mul_nonneg, hwx\u2080, hwy\u2080]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2211 i in disjSum sx sy, Sum.elim (fun i => a * wx i) (fun j => b * wy j) i = 1\n[PROOFSTEP]\nsimp [Finset.sum_sum_elim, Finset.mul_sum.symm, *]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2200 (i : \u03b9 \u2295 \u03b9'), i \u2208 disjSum sx sy \u2192 Sum.elim zx zy i \u2208 s\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 disjSum sx sy\n\u22a2 Sum.elim zx zy i \u2208 s\n[PROOFSTEP]\nrw [Finset.mem_disjSum] at hi \n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9 \u2295 \u03b9'\nhi : (\u2203 a, a \u2208 sx \u2227 Sum.inl a = i) \u2228 \u2203 b, b \u2208 sy \u2227 Sum.inr b = i\n\u22a2 Sum.elim zx zy i \u2208 s\n[PROOFSTEP]\nrcases hi with (\u27e8j, hj, rfl\u27e9 | \u27e8j, hj, rfl\u27e9)\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inl.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nj : \u03b9\nhj : j \u2208 sx\n\u22a2 Sum.elim zx zy (Sum.inl j) \u2208 s\n[PROOFSTEP]\napply_rules [hzx, hzy]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.inr.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9'\u271d : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nsx : Finset \u03b9\nwx : \u03b9 \u2192 R\nzx : \u03b9 \u2192 E\nhwx\u2080 : \u2200 (i : \u03b9), i \u2208 sx \u2192 0 \u2264 wx i\nhwx\u2081 : \u2211 i in sx, wx i = 1\nhzx : \u2200 (i : \u03b9), i \u2208 sx \u2192 zx i \u2208 s\n\u03b9' : Type\nsy : Finset \u03b9'\nwy : \u03b9' \u2192 R\nzy : \u03b9' \u2192 E\nhwy\u2080 : \u2200 (i : \u03b9'), i \u2208 sy \u2192 0 \u2264 wy i\nhwy\u2081 : \u2211 i in sy, wy i = 1\nhzy : \u2200 (i : \u03b9'), i \u2208 sy \u2192 zy i \u2208 s\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nj : \u03b9'\nhj : j \u2208 sy\n\u22a2 Sum.elim zx zy (Sum.inr j) \u2208 s\n[PROOFSTEP]\napply_rules [hzx, hzy]\n[GOAL]\ncase refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x} \u2286 \u2191(convexHull R) s\n[PROOFSTEP]\nrintro _ \u27e8\u03b9, t, w, z, hw\u2080, hw\u2081, hz, rfl\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 centerMass t w z \u2208 \u2191(convexHull R) s\n[PROOFSTEP]\nexact t.centerMass_mem_convexHull hw\u2080 (hw\u2081.symm \u25b8 zero_lt_one) hz\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\n\u22a2 \u2191(convexHull R) \u2191s = {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nrefine' Set.Subset.antisymm (convexHull_min _ _) _\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\n\u22a2 \u2191s \u2286 {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nrw [Finset.mem_coe] at hx \n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 s\n\u22a2 x \u2208 {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nrefine' \u27e8_, _, _, Finset.centerMass_ite_eq _ _ _ hx\u27e9\n[GOAL]\ncase refine'_1.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 s\n\u22a2 \u2200 (y : E), y \u2208 s \u2192 0 \u2264 if x = y then 1 else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 s\ny\u271d : E\na\u271d : y\u271d \u2208 s\n\u22a2 0 \u2264 if x = y\u271d then 1 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 s\ny\u271d : E\na\u271d : y\u271d \u2208 s\nh\u271d : x = y\u271d\n\u22a2 0 \u2264 1\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 s\ny\u271d : E\na\u271d : y\u271d \u2208 s\nh\u271d : \u00acx = y\u271d\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nexacts [zero_le_one, le_refl 0]\n[GOAL]\ncase refine'_1.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\nhx : x \u2208 s\n\u22a2 (\u2211 y in s, if x = y then 1 else 0) = 1\n[PROOFSTEP]\nrw [Finset.sum_ite_eq, if_pos hx]\n[GOAL]\ncase refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\n\u22a2 Convex R {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nrintro x \u27e8wx, hwx\u2080, hwx\u2081, rfl\u27e9 y \u27e8wy, hwy\u2080, hwy\u2081, rfl\u27e9 a b ha hb hab\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nwx : E \u2192 R\nhwx\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wx y\nhwx\u2081 : \u2211 y in s, wx y = 1\nwy : E \u2192 R\nhwy\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wy y\nhwy\u2081 : \u2211 y in s, wy y = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 centerMass s wx id + b \u2022 centerMass s wy id \u2208 {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nrw [Finset.centerMass_segment _ _ _ _ hwx\u2081 hwy\u2081 _ _ hab]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nwx : E \u2192 R\nhwx\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wx y\nhwx\u2081 : \u2211 y in s, wx y = 1\nwy : E \u2192 R\nhwy\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wy y\nhwy\u2081 : \u2211 y in s, wy y = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 centerMass s (fun i => a * wx i + b * wy i) id \u2208 {x | \u2203 w x_1 x_2, centerMass s w id = x}\n[PROOFSTEP]\nrefine' \u27e8_, _, _, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nwx : E \u2192 R\nhwx\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wx y\nhwx\u2081 : \u2211 y in s, wx y = 1\nwy : E \u2192 R\nhwy\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wy y\nhwy\u2081 : \u2211 y in s, wy y = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2200 (y : E), y \u2208 s \u2192 0 \u2264 a * wx y + b * wy y\n[PROOFSTEP]\nrintro i hi\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nwx : E \u2192 R\nhwx\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wx y\nhwx\u2081 : \u2211 y in s, wx y = 1\nwy : E \u2192 R\nhwy\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wy y\nhwy\u2081 : \u2211 y in s, wy y = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : E\nhi : i \u2208 s\n\u22a2 0 \u2264 a * wx i + b * wy i\n[PROOFSTEP]\napply_rules [add_nonneg, mul_nonneg, hwx\u2080, hwy\u2080]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nwx : E \u2192 R\nhwx\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wx y\nhwx\u2081 : \u2211 y in s, wx y = 1\nwy : E \u2192 R\nhwy\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 wy y\nhwy\u2081 : \u2211 y in s, wy y = 1\na b : R\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2211 y in s, (a * wx y + b * wy y) = 1\n[PROOFSTEP]\nsimp only [Finset.sum_add_distrib, Finset.mul_sum.symm, mul_one, *]\n[GOAL]\ncase refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\n\u22a2 {x | \u2203 w x_1 x_2, centerMass s w id = x} \u2286 \u2191(convexHull R) \u2191s\n[PROOFSTEP]\nrintro _ \u27e8w, hw\u2080, hw\u2081, rfl\u27e9\n[GOAL]\ncase refine'_3.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nw : E \u2192 R\nhw\u2080 : \u2200 (y : E), y \u2208 s \u2192 0 \u2264 w y\nhw\u2081 : \u2211 y in s, w y = 1\n\u22a2 centerMass s w id \u2208 \u2191(convexHull R) \u2191s\n[PROOFSTEP]\nexact s.centerMass_mem_convexHull (fun x hx => hw\u2080 _ hx) (hw\u2081.symm \u25b8 zero_lt_one) fun x hx => hx\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\n\u22a2 x \u2208 \u2191(convexHull R) \u2191s \u2194 \u2203 w x_1 x_2, centerMass s w id = x\n[PROOFSTEP]\nrw [Finset.convexHull_eq, Set.mem_setOf_eq]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\nhs : Set.Finite s\n\u22a2 \u2191(convexHull R) s = {x | \u2203 w x_1 x_2, centerMass (Finite.toFinset hs) w id = x}\n[PROOFSTEP]\nsimpa only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, exists_prop] using hs.toFinset.convexHull_eq\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 \u2191(convexHull R) s = \u22c3 (t : Finset E) (_ : \u2191t \u2286 s), \u2191(convexHull R) \u2191t\n[PROOFSTEP]\nrefine' Subset.antisymm _ _\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 \u2191(convexHull R) s \u2286 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s), \u2191(convexHull R) \u2191t\n[PROOFSTEP]\nrw [_root_.convexHull_eq]\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x} \u2286 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s), \u2191(convexHull R) \u2191t\n[PROOFSTEP]\nrintro x \u27e8\u03b9, t, w, z, hw\u2080, hw\u2081, hz, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 centerMass t w z \u2208 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s), \u2191(convexHull R) \u2191t\n[PROOFSTEP]\nsimp only [mem_iUnion]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 \u2203 i i_1, centerMass t w z \u2208 \u2191(convexHull R) \u2191i\n[PROOFSTEP]\nrefine' \u27e8t.image z, _, _\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 \u2191(Finset.image z t) \u2286 s\n[PROOFSTEP]\nrw [coe_image, Set.image_subset_iff]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 \u2191t \u2286 z \u207b\u00b9' s\n[PROOFSTEP]\nexact hz\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 centerMass t w z \u2208 \u2191(convexHull R) \u2191(Finset.image z t)\n[PROOFSTEP]\napply t.centerMass_mem_convexHull hw\u2080\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2.hws\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 0 < \u2211 i in t, w i\n[PROOFSTEP]\nsimp only [hw\u2081, zero_lt_one]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.refine'_2.hz\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz\u271d : \u03b9\u271d \u2192 E\ns : Set E\n\u03b9 : Type\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 t \u2192 0 \u2264 w i\nhw\u2081 : \u2211 i in t, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 s\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \u2192 z i \u2208 \u2191(Finset.image z t)\n[PROOFSTEP]\nexact fun i hi => Finset.mem_coe.2 (Finset.mem_image_of_mem _ hi)\n[GOAL]\ncase refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Set E\n\u22a2 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s), \u2191(convexHull R) \u2191t \u2286 \u2191(convexHull R) s\n[PROOFSTEP]\nexact iUnion_subset fun i => iUnion_subset convexHull_mono\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nt : Set F\nx : E\ny : F\nhx : x \u2208 \u2191(convexHull R) s\nhy : y \u2208 \u2191(convexHull R) t\n\u22a2 (x, y) \u2208 \u2191(convexHull R) (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [_root_.convexHull_eq] at hx hy \u22a2\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nt : Set F\nx : E\ny : F\nhx : x \u2208 {x | \u2203 \u03b9 t w z x_1 x_2 x_3, centerMass t w z = x}\nhy : y \u2208 {x | \u2203 \u03b9 t_1 w z x_1 x_2 x_3, centerMass t_1 w z = x}\n\u22a2 (x, y) \u2208 {x | \u2203 \u03b9 t_1 w z x_1 x_2 x_3, centerMass t_1 w z = x}\n[PROOFSTEP]\nobtain \u27e8\u03b9, a, w, S, hw, hw', hS, hSp\u27e9 := hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\nhy : y \u2208 {x | \u2203 \u03b9 t_1 w z x_1 x_2 x_3, centerMass t_1 w z = x}\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u22a2 (x, y) \u2208 {x | \u2203 \u03b9 t_1 w z x_1 x_2 x_3, centerMass t_1 w z = x}\n[PROOFSTEP]\nobtain \u27e8\u03ba, b, v, T, hv, hv', hT, hTp\u27e9 := hy\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\n\u22a2 (x, y) \u2208 {x | \u2203 \u03b9 t_1 w z x_1 x_2 x_3, centerMass t_1 w z = x}\n[PROOFSTEP]\nhave h_sum : \u2211 i : \u03b9 \u00d7 \u03ba in a \u00d7\u02e2 b, w i.fst * v i.snd = 1 :=\n  by\n  rw [Finset.sum_product, \u2190 hw']\n  congr\n  ext i\n  have : \u2211 y : \u03ba in b, w i * v y = \u2211 y : \u03ba in b, v y * w i :=\n    by\n    congr\n    ext\n    simp [mul_comm]\n  rw [this, \u2190 Finset.sum_mul, hv']\n  simp\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\n\u22a2 \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n[PROOFSTEP]\nrw [Finset.sum_product, \u2190 hw']\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\n\u22a2 \u2211 x in a, \u2211 y in b, w (x, y).fst * v (x, y).snd = \u2211 i in a, w i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\n\u22a2 (fun x => \u2211 y in b, w (x, y).fst * v (x, y).snd) = fun i => w i\n[PROOFSTEP]\next i\n[GOAL]\ncase e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\ni : \u03b9\n\u22a2 \u2211 y in b, w (i, y).fst * v (i, y).snd = w i\n[PROOFSTEP]\nhave : \u2211 y : \u03ba in b, w i * v y = \u2211 y : \u03ba in b, v y * w i :=\n  by\n  congr\n  ext\n  simp [mul_comm]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\ni : \u03b9\n\u22a2 \u2211 y in b, w i * v y = \u2211 y in b, v y * w i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\ni : \u03b9\n\u22a2 (fun y => w i * v y) = fun y => v y * w i\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\ni : \u03b9\nx\u271d : \u03ba\n\u22a2 w i * v x\u271d = v x\u271d * w i\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\ncase e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\ni : \u03b9\nthis : \u2211 y in b, w i * v y = \u2211 y in b, v y * w i\n\u22a2 \u2211 y in b, w (i, y).fst * v (i, y).snd = w i\n[PROOFSTEP]\nrw [this, \u2190 Finset.sum_mul, hv']\n[GOAL]\ncase e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\ni : \u03b9\nthis : \u2211 y in b, w i * v y = \u2211 y in b, v y * w i\n\u22a2 1 * w i = w i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (x, y) \u2208 {x | \u2203 \u03b9 t_1 w z x_1 x_2 x_3, centerMass t_1 w z = x}\n[PROOFSTEP]\nrefine' \u27e8\u03b9 \u00d7 \u03ba, a \u00d7\u02e2 b, fun p => w p.1 * v p.2, fun p => (S p.1, T p.2), fun p hp => _, h_sum, fun p hp => _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\np : \u03b9 \u00d7 \u03ba\nhp : p \u2208 a \u00d7\u02e2 b\n\u22a2 0 \u2264 (fun p => w p.fst * v p.snd) p\n[PROOFSTEP]\nrw [mem_product] at hp \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\np : \u03b9 \u00d7 \u03ba\nhp : p.fst \u2208 a \u2227 p.snd \u2208 b\n\u22a2 0 \u2264 (fun p => w p.fst * v p.snd) p\n[PROOFSTEP]\nexact mul_nonneg (hw p.1 hp.1) (hv p.2 hp.2)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\np : \u03b9 \u00d7 \u03ba\nhp : p \u2208 a \u00d7\u02e2 b\n\u22a2 (fun p => (S p.fst, T p.snd)) p \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nrw [mem_product] at hp \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\np : \u03b9 \u00d7 \u03ba\nhp : p.fst \u2208 a \u2227 p.snd \u2208 b\n\u22a2 (fun p => (S p.fst, T p.snd)) p \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nexact \u27e8hS p.1 hp.1, hT p.2 hp.2\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (centerMass (a \u00d7\u02e2 b) (fun p => w p.fst * v p.snd) fun p => (S p.fst, T p.snd)) = (x, y)\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (centerMass (a \u00d7\u02e2 b) (fun p => w p.fst * v p.snd) fun p => (S p.fst, T p.snd)).fst = (x, y).fst\n[PROOFSTEP]\nrw [\u2190 hSp, Finset.centerMass_eq_of_sum_1 _ _ hw', Finset.centerMass_eq_of_sum_1 _ _ h_sum]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (\u2211 i in a \u00d7\u02e2 b, (w i.fst * v i.snd) \u2022 (S i.fst, T i.snd)).fst = (\u2211 i in a, w i \u2022 S i, y).fst\n[PROOFSTEP]\nsimp_rw [Prod.fst_sum, Prod.smul_mk]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 \u2211 x in a \u00d7\u02e2 b, (w x.fst * v x.snd) \u2022 S x.fst = \u2211 i in a, w i \u2022 S i\n[PROOFSTEP]\nrw [Finset.sum_product]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 \u2211 x in a, \u2211 y in b, (w (x, y).fst * v (x, y).snd) \u2022 S (x, y).fst = \u2211 i in a, w i \u2022 S i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081.e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (fun x => \u2211 y in b, (w (x, y).fst * v (x, y).snd) \u2022 S (x, y).fst) = fun i => w i \u2022 S i\n[PROOFSTEP]\next i\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081.e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\ni : \u03b9\n\u22a2 \u2211 y in b, (w (i, y).fst * v (i, y).snd) \u2022 S (i, y).fst = w i \u2022 S i\n[PROOFSTEP]\nhave : (\u2211 j : \u03ba in b, (w i * v j) \u2022 S i) = \u2211 j : \u03ba in b, v j \u2022 w i \u2022 S i :=\n  by\n  congr\n  ext\n  rw [mul_smul, smul_comm]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\ni : \u03b9\n\u22a2 \u2211 j in b, (w i * v j) \u2022 S i = \u2211 j in b, v j \u2022 w i \u2022 S i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\ni : \u03b9\n\u22a2 (fun j => (w i * v j) \u2022 S i) = fun j => v j \u2022 w i \u2022 S i\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\ni : \u03b9\nx\u271d : \u03ba\n\u22a2 (w i * v x\u271d) \u2022 S i = v x\u271d \u2022 w i \u2022 S i\n[PROOFSTEP]\nrw [mul_smul, smul_comm]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2081.e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\ni : \u03b9\nthis : \u2211 j in b, (w i * v j) \u2022 S i = \u2211 j in b, v j \u2022 w i \u2022 S i\n\u22a2 \u2211 y in b, (w (i, y).fst * v (i, y).snd) \u2022 S (i, y).fst = w i \u2022 S i\n[PROOFSTEP]\nrw [this, \u2190 Finset.sum_smul, hv', one_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (centerMass (a \u00d7\u02e2 b) (fun p => w p.fst * v p.snd) fun p => (S p.fst, T p.snd)).snd = (x, y).snd\n[PROOFSTEP]\nrw [\u2190 hTp, Finset.centerMass_eq_of_sum_1 _ _ hv', Finset.centerMass_eq_of_sum_1 _ _ h_sum]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (\u2211 i in a \u00d7\u02e2 b, (w i.fst * v i.snd) \u2022 (S i.fst, T i.snd)).snd = (x, \u2211 i in b, v i \u2022 T i).snd\n[PROOFSTEP]\nsimp_rw [Prod.snd_sum, Prod.smul_mk]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 \u2211 x in a \u00d7\u02e2 b, (w x.fst * v x.snd) \u2022 T x.snd = \u2211 i in b, v i \u2022 T i\n[PROOFSTEP]\nrw [Finset.sum_product, Finset.sum_comm]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 \u2211 y in b, \u2211 x in a, (w (x, y).fst * v (x, y).snd) \u2022 T (x, y).snd = \u2211 i in b, v i \u2022 T i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082.e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\n\u22a2 (fun y => \u2211 x in a, (w (x, y).fst * v (x, y).snd) \u2022 T (x, y).snd) = fun i => v i \u2022 T i\n[PROOFSTEP]\next j\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082.e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\nj : \u03ba\n\u22a2 \u2211 x in a, (w (x, j).fst * v (x, j).snd) \u2022 T (x, j).snd = v j \u2022 T j\n[PROOFSTEP]\nsimp_rw [mul_smul]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.h\u2082.e_f.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j\u271d : \u03b9\u271d\nc : R\nt\u271d : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\nt : Set F\nx : E\ny : F\n\u03b9 : Type\na : Finset \u03b9\nw : \u03b9 \u2192 R\nS : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), i \u2208 a \u2192 0 \u2264 w i\nhw' : \u2211 i in a, w i = 1\nhS : \u2200 (i : \u03b9), i \u2208 a \u2192 S i \u2208 s\nhSp : centerMass a w S = x\n\u03ba : Type\nb : Finset \u03ba\nv : \u03ba \u2192 R\nT : \u03ba \u2192 F\nhv : \u2200 (i : \u03ba), i \u2208 b \u2192 0 \u2264 v i\nhv' : \u2211 i in b, v i = 1\nhT : \u2200 (i : \u03ba), i \u2208 b \u2192 T i \u2208 t\nhTp : centerMass b v T = y\nh_sum : \u2211 i in a \u00d7\u02e2 b, w i.fst * v i.snd = 1\nj : \u03ba\n\u22a2 \u2211 x in a, w x \u2022 v j \u2022 T j = v j \u2022 T j\n[PROOFSTEP]\nrw [\u2190 Finset.sum_smul, hw', one_smul]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns t : Set E\n\u22a2 \u2191(convexHull R) (s + t) = \u2191(convexHull R) s + \u2191(convexHull R) t\n[PROOFSTEP]\nsimp_rw [\u2190 image2_add, \u2190 image_prod, IsLinearMap.isLinearMap_add.convexHull_image, convexHull_prod]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns t : Set E\n\u22a2 \u2191(convexHull R) (s - t) = \u2191(convexHull R) s - \u2191(convexHull R) t\n[PROOFSTEP]\nsimp_rw [sub_eq_add_neg, convexHull_add, convexHull_neg]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\n\u22a2 \u2191(convexHull R) (Set.range fun i j => if i = j then 1 else 0) = stdSimplex R \u03b9\n[PROOFSTEP]\nrefine' Subset.antisymm (convexHull_min _ (convex_stdSimplex R \u03b9)) _\n[GOAL]\ncase refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\n\u22a2 (Set.range fun i j => if i = j then 1 else 0) \u2286 stdSimplex R \u03b9\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni\u271d j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\ni : \u03b9\n\u22a2 (fun i j => if i = j then 1 else 0) i \u2208 stdSimplex R \u03b9\n[PROOFSTEP]\nexact ite_eq_mem_stdSimplex R i\n[GOAL]\ncase refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\n\u22a2 stdSimplex R \u03b9 \u2286 \u2191(convexHull R) (Set.range fun i j => if i = j then 1 else 0)\n[PROOFSTEP]\nrintro w \u27e8hw\u2080, hw\u2081\u27e9\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf w : \u03b9 \u2192 R\nhw\u2080 : \u2200 (x : \u03b9), 0 \u2264 w x\nhw\u2081 : \u2211 x : \u03b9, w x = 1\n\u22a2 w \u2208 \u2191(convexHull R) (Set.range fun i j => if i = j then 1 else 0)\n[PROOFSTEP]\nrw [pi_eq_sum_univ w, \u2190 Finset.univ.centerMass_eq_of_sum_1 _ hw\u2081]\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw\u271d : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf w : \u03b9 \u2192 R\nhw\u2080 : \u2200 (x : \u03b9), 0 \u2264 w x\nhw\u2081 : \u2211 x : \u03b9, w x = 1\n\u22a2 (centerMass Finset.univ (fun i => w i) fun i j => if i = j then 1 else 0) \u2208\n    \u2191(convexHull R) (Set.range fun i j => if i = j then 1 else 0)\n[PROOFSTEP]\nexact Finset.univ.centerMass_mem_convexHull (fun i _ => hw\u2080 i) (hw\u2081.symm \u25b8 zero_lt_one) fun i _ => mem_range_self i\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\ns : Set E\nhs : Set.Finite s\n\u22a2 \u2191(convexHull R) s = \u2191(\u2211 x : \u2191s, LinearMap.smulRight (LinearMap.proj x) \u2191x) '' stdSimplex R \u2191s\n[PROOFSTEP]\nrw [\u2190 @convexHull_basis_eq_stdSimplex _ _ _ hs.fintype, \u2190 LinearMap.convexHull_image, \u2190 Set.range_comp]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\ns : Set E\nhs : Set.Finite s\n\u22a2 \u2191(convexHull R) s =\n    \u2191(convexHull R) (range (\u2191(\u2211 x : \u2191s, LinearMap.smulRight (LinearMap.proj x) \u2191x) \u2218 fun i j => if i = j then 1 else 0))\n[PROOFSTEP]\nsimp_rw [Function.comp]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\ns : Set E\nhs : Set.Finite s\n\u22a2 \u2191(convexHull R) s =\n    \u2191(convexHull R)\n      (range fun x => \u2191(\u2211 x : \u2191s, LinearMap.smulRight (LinearMap.proj x) \u2191x) fun j => if x = j then 1 else 0)\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\ns : Set E\nhs : Set.Finite s\n\u22a2 s = range fun x => \u2191(\u2211 x : \u2191s, LinearMap.smulRight (LinearMap.proj x) \u2191x) fun j => if x = j then 1 else 0\n[PROOFSTEP]\nconvert Subtype.range_coe.symm\n[GOAL]\ncase h.e'_3.h.e'_3.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192 R\ns : Set E\nhs : Set.Finite s\nx\u271d : \u2191s\n\u22a2 (\u2191(\u2211 x : \u2191s, LinearMap.smulRight (LinearMap.proj x) \u2191x) fun j => if x\u271d = j then 1 else 0) = \u2191x\u271d\n[PROOFSTEP]\nsimp [LinearMap.sum_apply, ite_smul _ (1 : R), Finset.filter_eq, @Finset.mem_univ _ hs.fintype _]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\n\u22a2 \u2191(convexHull R) (Set.range \u2191b) = {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n[PROOFSTEP]\nrw [convexHull_range_eq_exists_affineCombination]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\n\u22a2 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s \u2191b) w = x} = {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\nx : E\n\u22a2 x \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s \u2191b) w = x} \u2194 x \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n[PROOFSTEP]\nrefine' \u27e8_, fun hx => _\u27e9\n[GOAL]\ncase h.refine'_1\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\nx : E\n\u22a2 x \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s \u2191b) w = x} \u2192 x \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n[PROOFSTEP]\nrintro \u27e8s, w, hw\u2080, hw\u2081, rfl\u27e9 i\n[GOAL]\ncase h.refine'_1.intro.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ni : \u03b9\n\u22a2 0 \u2264 \u2191(coord b i) (\u2191(affineCombination R s \u2191b) w)\n[PROOFSTEP]\nby_cases hi : i \u2208 s\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 \u2191(coord b i) (\u2191(affineCombination R s \u2191b) w)\n[PROOFSTEP]\nrw [b.coord_apply_combination_of_mem hi hw\u2081]\n[GOAL]\ncase pos\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 w i\n[PROOFSTEP]\nexact hw\u2080 i hi\n[GOAL]\ncase neg\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2080 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw\u2081 : Finset.sum s w = 1\ni : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 0 \u2264 \u2191(coord b i) (\u2191(affineCombination R s \u2191b) w)\n[PROOFSTEP]\nrw [b.coord_apply_combination_of_not_mem hi hw\u2081]\n[GOAL]\ncase h.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\nx : E\nhx : x \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n\u22a2 x \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s \u2191b) w = x}\n[PROOFSTEP]\nhave hx' : x \u2208 affineSpan R (range b) := by\n  rw [b.tot]\n  exact AffineSubspace.mem_top R E x\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\nx : E\nhx : x \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n\u22a2 x \u2208 affineSpan R (Set.range \u2191b)\n[PROOFSTEP]\nrw [b.tot]\n[GOAL]\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\nx : E\nhx : x \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\nexact AffineSubspace.mem_top R E x\n[GOAL]\ncase h.refine'_2\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\nx : E\nhx : x \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\nhx' : x \u2208 affineSpan R (Set.range \u2191b)\n\u22a2 x \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s \u2191b) w = x}\n[PROOFSTEP]\nobtain \u27e8s, w, hw\u2081, rfl\u27e9 := (mem_affineSpan_iff_eq_affineCombination R E).mp hx'\n[GOAL]\ncase h.refine'_2.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2081 : \u2211 i in s, w i = 1\nhx : \u2191(affineCombination R s \u2191b) w \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\nhx' : \u2191(affineCombination R s \u2191b) w \u2208 affineSpan R (Set.range \u2191b)\n\u22a2 \u2191(affineCombination R s \u2191b) w \u2208 {x | \u2203 s w x_1 x_2, \u2191(affineCombination R s \u2191b) w = x}\n[PROOFSTEP]\nrefine' \u27e8s, w, _, hw\u2081, rfl\u27e9\n[GOAL]\ncase h.refine'_2.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2081 : \u2211 i in s, w i = 1\nhx : \u2191(affineCombination R s \u2191b) w \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\nhx' : \u2191(affineCombination R s \u2191b) w \u2208 affineSpan R (Set.range \u2191b)\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.refine'_2.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2081 : \u2211 i in s, w i = 1\nhx : \u2191(affineCombination R s \u2191b) w \u2208 {x | \u2200 (i : \u03b9), 0 \u2264 \u2191(coord b i) x}\nhx' : \u2191(affineCombination R s \u2191b) w \u2208 affineSpan R (Set.range \u2191b)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 w i\n[PROOFSTEP]\nspecialize hx i\n[GOAL]\ncase h.refine'_2.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2081 : \u2211 i in s, w i = 1\nhx' : \u2191(affineCombination R s \u2191b) w \u2208 affineSpan R (Set.range \u2191b)\ni : \u03b9\nhi : i \u2208 s\nhx : 0 \u2264 \u2191(coord b i) (\u2191(affineCombination R s \u2191b) w)\n\u22a2 0 \u2264 w i\n[PROOFSTEP]\nrw [b.coord_apply_combination_of_mem hi hw\u2081] at hx \n[GOAL]\ncase h.refine'_2.intro.intro.intro\nR : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9\u271d : Type u_4\n\u03b9' : Type u_5\n\u03b1 : Type u_6\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : AddCommGroup F\ninst\u271d\u2075 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : Module R \u03b1\ninst\u271d\u00b9 : OrderedSMul R \u03b1\ns\u271d : Set E\ni\u271d j : \u03b9\u271d\nc : R\nt : Finset \u03b9\u271d\nw\u271d : \u03b9\u271d \u2192 R\nz : \u03b9\u271d \u2192 E\ninst\u271d : Fintype \u03b9\u271d\nf : \u03b9\u271d \u2192 R\n\u03b9 : Type u_7\nb : AffineBasis \u03b9 R E\ns : Finset \u03b9\nw : \u03b9 \u2192 R\nhw\u2081 : \u2211 i in s, w i = 1\nhx' : \u2191(affineCombination R s \u2191b) w \u2208 affineSpan R (Set.range \u2191b)\ni : \u03b9\nhi : i \u2208 s\nhx : 0 \u2264 w i\n\u22a2 0 \u2264 w i\n[PROOFSTEP]\nexact hx\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Combination", "llama_tokens": 93746, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936324115011, "lm_q2_score": 0.7371581684030623, "lm_q1q2_score": 0.5648796105273916}}
{"text": "[GOAL]\n\u22a2 ConcreteCategory MeasCat\n[PROOFSTEP]\nunfold MeasCat\n[GOAL]\n\u22a2 ConcreteCategory (Bundled MeasurableSpace)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03bc : MeasureTheory.Measure (MeasureTheory.Measure \u211d\u22650\u221e)\n\u22a2 \u222b\u207b (x : \u211d\u22650\u221e), x \u2202Measure.join \u03bc = \u222b\u207b (x : \u211d\u22650\u221e), x \u2202Measure.map (fun m => \u222b\u207b (x : \u211d\u22650\u221e), x \u2202m) \u03bc\n[PROOFSTEP]\nrw [Measure.lintegral_join, lintegral_map]\n[GOAL]\ncase hf\n\u03bc : MeasureTheory.Measure (MeasureTheory.Measure \u211d\u22650\u221e)\n\u22a2 Measurable fun x => x\n[PROOFSTEP]\napply_rules [measurable_id, Measure.measurable_lintegral]\n[GOAL]\ncase hg\n\u03bc : MeasureTheory.Measure (MeasureTheory.Measure \u211d\u22650\u221e)\n\u22a2 Measurable fun m => \u222b\u207b (x : \u211d\u22650\u221e), x \u2202m\n[PROOFSTEP]\napply_rules [measurable_id, Measure.measurable_lintegral]\n[GOAL]\n\u03bc : MeasureTheory.Measure (MeasureTheory.Measure \u211d\u22650\u221e)\n\u22a2 Measurable fun x => x\n[PROOFSTEP]\napply_rules [measurable_id, Measure.measurable_lintegral]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Category.MeasCat", "llama_tokens": 401, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8670357460591569, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5647479569029702}}
{"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\nv i\u2081 : \u03b1\nvi\u2081 : v * i\u2081 = 1\niv\u2081 : i\u2081 * v = 1\nv' i\u2082 : \u03b1\nvi\u2082 : v' * i\u2082 = 1\niv\u2082 : i\u2082 * v' = 1\ne : \u2191{ val := v, inv := i\u2081, val_inv := vi\u2081, inv_val := iv\u2081 } = \u2191{ val := v', inv := i\u2082, val_inv := vi\u2082, inv_val := iv\u2082 }\n\u22a2 { val := v, inv := i\u2081, val_inv := vi\u2081, inv_val := iv\u2081 } = { val := v', inv := i\u2082, val_inv := vi\u2082, inv_val := iv\u2082 }\n[PROOFSTEP]\nsimp only at e \n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\nv i\u2081 : \u03b1\nvi\u2081 : v * i\u2081 = 1\niv\u2081 : i\u2081 * v = 1\nv' i\u2082 : \u03b1\nvi\u2082 : v' * i\u2082 = 1\niv\u2082 : i\u2082 * v' = 1\ne : v = v'\n\u22a2 { val := v, inv := i\u2081, val_inv := vi\u2081, inv_val := iv\u2081 } = { val := v', inv := i\u2082, val_inv := vi\u2082, inv_val := iv\u2082 }\n[PROOFSTEP]\nsubst v'\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\nv i\u2081 : \u03b1\nvi\u2081 : v * i\u2081 = 1\niv\u2081 : i\u2081 * v = 1\ni\u2082 : \u03b1\nvi\u2082 : v * i\u2082 = 1\niv\u2082 : i\u2082 * v = 1\n\u22a2 { val := v, inv := i\u2081, val_inv := vi\u2081, inv_val := iv\u2081 } = { val := v, inv := i\u2082, val_inv := vi\u2082, inv_val := iv\u2082 }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_inv\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\nv i\u2081 : \u03b1\nvi\u2081 : v * i\u2081 = 1\niv\u2081 : i\u2081 * v = 1\ni\u2082 : \u03b1\nvi\u2082 : v * i\u2082 = 1\niv\u2082 : i\u2082 * v = 1\n\u22a2 i\u2081 = i\u2082\n[PROOFSTEP]\nsimpa only [iv\u2082, vi\u2081, one_mul, mul_one] using mul_assoc i\u2082 v i\u2081\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 \u2191u\u2081 * \u2191u\u2082 * (u\u2082.inv * u\u2081.inv) = 1\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_assoc u\u2082.val, val_inv, one_mul, val_inv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 u\u2082.inv * u\u2081.inv * (\u2191u\u2081 * \u2191u\u2082) = 1\n[PROOFSTEP]\nrw [mul_assoc, \u2190 mul_assoc u\u2081.inv, inv_val, one_mul, inv_val]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u a : \u03b1\u02e3\n\u22a2 \u2191a = 1 \u2194 a = 1\n[PROOFSTEP]\nrw [\u2190 Units.val_one, eq_iff]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : \u2191u = a\n\u22a2 \u2191u\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrw [\u2190 h, u.inv_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : \u2191u = a\n\u22a2 a * \u2191u\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 h, u.mul_inv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u a : \u03b1\u02e3\nb : \u03b1\n\u22a2 \u2191a * (\u2191a\u207b\u00b9 * b) = b\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_inv, one_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u a : \u03b1\u02e3\nb : \u03b1\n\u22a2 \u2191a\u207b\u00b9 * (\u2191a * b) = b\n[PROOFSTEP]\nrw [\u2190 mul_assoc, inv_mul, one_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u : \u03b1\u02e3\na : \u03b1\nb : \u03b1\u02e3\n\u22a2 a * \u2191b * \u2191b\u207b\u00b9 = a\n[PROOFSTEP]\nrw [mul_assoc, mul_inv, mul_one]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u : \u03b1\u02e3\na : \u03b1\nb : \u03b1\u02e3\n\u22a2 a * \u2191b\u207b\u00b9 * \u2191b = a\n[PROOFSTEP]\nrw [mul_assoc, inv_mul, mul_one]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c\u271d u a : \u03b1\u02e3\nb c : \u03b1\nh : \u2191a * b = \u2191a * c\n\u22a2 b = c\n[PROOFSTEP]\nsimpa only [inv_mul_cancel_left] using congr_arg (fun x : \u03b1 => \u2191(a\u207b\u00b9 : \u03b1\u02e3) * x) h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c\u271d u a : \u03b1\u02e3\nb c : \u03b1\nh : b * \u2191a = c * \u2191a\n\u22a2 b = c\n[PROOFSTEP]\nsimpa only [mul_inv_cancel_right] using congr_arg (fun x : \u03b1 => x * \u2191(a\u207b\u00b9 : \u03b1\u02e3)) h\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u : \u03b1\u02e3\na b : \u03b1\nh : a = b * \u2191c\u207b\u00b9\n\u22a2 a * \u2191c = b\n[PROOFSTEP]\nrw [h, inv_mul_cancel_right]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u : \u03b1\u02e3\na b : \u03b1\nh : a * \u2191c = b\n\u22a2 a = b * \u2191c\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 h, mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c\u271d u : \u03b1\u02e3\na c : \u03b1\nh : a = \u2191b\u207b\u00b9 * c\n\u22a2 \u2191b * a = c\n[PROOFSTEP]\nrw [h, mul_inv_cancel_left]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c\u271d u : \u03b1\u02e3\na c : \u03b1\nh : \u2191b * a = c\n\u22a2 a = \u2191b\u207b\u00b9 * c\n[PROOFSTEP]\nrw [\u2190 h, inv_mul_cancel_left]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b\u271d c\u271d u : \u03b1\u02e3\nb c : \u03b1\nh : \u2191a\u207b\u00b9 * b = c\n\u22a2 b = \u2191a * c\n[PROOFSTEP]\nrw [\u2190 h, mul_inv_cancel_left]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b\u271d c\u271d u : \u03b1\u02e3\nb c : \u03b1\nh : b = \u2191a * c\n\u22a2 \u2191a\u207b\u00b9 * b = c\n[PROOFSTEP]\nrw [h, inv_mul_cancel_left]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c\u271d u : \u03b1\u02e3\na c : \u03b1\nh : a * \u2191b\u207b\u00b9 = c\n\u22a2 a = c * \u2191b\n[PROOFSTEP]\nrw [\u2190 h, inv_mul_cancel_right]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c\u271d u : \u03b1\u02e3\na c : \u03b1\nh : a = c * \u2191b\n\u22a2 a * \u2191b\u207b\u00b9 = c\n[PROOFSTEP]\nrw [h, mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : a * \u2191u = 1\n\u22a2 \u2191u\u207b\u00b9 = 1 * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : a * \u2191u = 1\n\u22a2 1 * \u2191u\u207b\u00b9 = a\n[PROOFSTEP]\nrw [\u2190 h, mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : \u2191u * a = 1\n\u22a2 \u2191u\u207b\u00b9 = \u2191u\u207b\u00b9 * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : \u2191u * a = 1\n\u22a2 \u2191u\u207b\u00b9 * 1 = a\n[PROOFSTEP]\nrw [\u2190 h, inv_mul_cancel_left]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\n\u22a2 a * \u2191u = 1 \u2194 a = \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_inv_eq_one, inv_inv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\n\u22a2 \u2191u * a = 1 \u2194 \u2191u\u207b\u00b9 = a\n[PROOFSTEP]\nrw [\u2190 inv_mul_eq_one, inv_inv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c u u\u2081 u\u2082 : \u03b1\u02e3\nh : \u2191u\u2081 = \u2191u\u2082\n\u22a2 \u2191u\u2081 * \u2191u\u2082\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [h, u\u2082.mul_inv]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c a : \u03b1\nu : \u03b1\u02e3\n\u22a2 a * (\u2191u\u207b\u00b9 * \u2191u) = a\n[PROOFSTEP]\nrw [Units.inv_mul, mul_one]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c a : \u03b1\nu : \u03b1\u02e3\n\u22a2 a * (\u2191u * \u2191u\u207b\u00b9) = a\n[PROOFSTEP]\nrw [Units.mul_inv, mul_one]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c x : \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 x /\u209a u\u2081 /\u209a u\u2082 = x /\u209a (u\u2082 * u\u2081)\n[PROOFSTEP]\nsimp only [divp, mul_inv_rev, Units.val_mul, mul_assoc]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c x : \u03b1\nu : \u03b1\u02e3\ny : \u03b1\n\u22a2 x /\u209a u * \u2191u = y * \u2191u \u2194 y * \u2191u = x\n[PROOFSTEP]\nrw [divp_mul_cancel]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c x : \u03b1\nu : \u03b1\u02e3\ny : \u03b1\n\u22a2 x = y * \u2191u \u2194 y * \u2191u = x\n[PROOFSTEP]\nexact \u27e8Eq.symm, Eq.symm\u27e9\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c x : \u03b1\nu : \u03b1\u02e3\ny : \u03b1\n\u22a2 x = y /\u209a u \u2194 x * \u2191u = y\n[PROOFSTEP]\nrw [eq_comm, divp_eq_iff_mul_eq]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c a : \u03b1\nu : \u03b1\u02e3\n\u22a2 a /\u209a u * \u2191u = 1 * \u2191u \u2194 a = \u2191u\n[PROOFSTEP]\nrw [divp_mul_cancel, one_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c : \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191u\u207b\u00b9 = 1 /\u209a u\n[PROOFSTEP]\nrw [one_divp]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c : \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191(1 / u) = 1 /\u209a u\n[PROOFSTEP]\nrw [one_div, one_divp]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c : \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 \u2191(u\u2081 / u\u2082) = \u2191u\u2081 /\u209a u\u2082\n[PROOFSTEP]\nrw [divp, division_def, Units.val_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CommMonoid \u03b1\nx y : \u03b1\nu : \u03b1\u02e3\n\u22a2 x /\u209a u * y = x * y /\u209a u\n[PROOFSTEP]\nrw [divp, divp, mul_right_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CommMonoid \u03b1\nx y : \u03b1\nux uy : \u03b1\u02e3\n\u22a2 x /\u209a ux = y /\u209a uy \u2194 x * \u2191uy = y * \u2191ux\n[PROOFSTEP]\nrw [divp_eq_iff_mul_eq, divp_mul_eq_mul_divp, divp_eq_iff_mul_eq]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : CommMonoid \u03b1\nx y : \u03b1\nux uy : \u03b1\u02e3\n\u22a2 x /\u209a ux * (y /\u209a uy) = x * y /\u209a (ux * uy)\n[PROOFSTEP]\nrw [divp_mul_eq_mul_divp, divp_assoc', divp_divp_eq_divp_mul]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na b : \u03b1\nh : a * b = 1\n\u22a2 b * a = 1\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na b : \u03b1\nh : a * b = 1\n\u22a2 b * a = 1\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\n\u03b1 : Type u\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na b : \u03b1\n\u22a2 a = 1 \u2227 b = 1 \u2192 a * b = 1\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nexact mul_one _\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na : M\nh : IsUnit a\n\u22a2 \u2203 b, a * b = 1\n[PROOFSTEP]\nrcases h with \u27e8\u27e8a, b, hab, _\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na b : M\nhab : a * b = 1\ninv_val\u271d : b * a = 1\n\u22a2 \u2203 b_1, \u2191{ val := a, inv := b, val_inv := hab, inv_val := inv_val\u271d } * b_1 = 1\n[PROOFSTEP]\nexact \u27e8b, hab\u27e9\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na : M\nh : IsUnit a\n\u22a2 \u2203 b, b * a = 1\n[PROOFSTEP]\nrcases h with \u27e8\u27e8a, b, _, hba\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na b : M\nval_inv\u271d : a * b = 1\nhba : b * a = 1\n\u22a2 \u2203 b_1, b_1 * \u2191{ val := a, inv := b, val_inv := val_inv\u271d, inv_val := hba } = 1\n[PROOFSTEP]\nexact \u27e8b, hba\u27e9\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : CommMonoid M\na : M\n\u22a2 IsUnit a \u2194 \u2203 b, b * a = 1\n[PROOFSTEP]\nsimp [isUnit_iff_exists_inv, mul_comm]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\nx y : M\n\u22a2 IsUnit x \u2192 IsUnit y \u2192 IsUnit (x * y)\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9 \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\nx y : M\u02e3\n\u22a2 IsUnit (\u2191x * \u2191y)\n[PROOFSTEP]\nexact \u27e8x * y, Units.val_mul _ _\u27e9\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na : M\nu : M\u02e3\nx\u271d : IsUnit (a * \u2191u)\nv : M\u02e3\nhv : \u2191v = a * \u2191u\n\u22a2 IsUnit a\n[PROOFSTEP]\nhave : IsUnit (a * \u2191u * \u2191u\u207b\u00b9) := by exists v * u\u207b\u00b9; rw [\u2190 hv, Units.val_mul]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na : M\nu : M\u02e3\nx\u271d : IsUnit (a * \u2191u)\nv : M\u02e3\nhv : \u2191v = a * \u2191u\n\u22a2 IsUnit (a * \u2191u * \u2191u\u207b\u00b9)\n[PROOFSTEP]\nexists v * u\u207b\u00b9\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na : M\nu : M\u02e3\nx\u271d : IsUnit (a * \u2191u)\nv : M\u02e3\nhv : \u2191v = a * \u2191u\n\u22a2 \u2191(v * u\u207b\u00b9) = a * \u2191u * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 hv, Units.val_mul]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na : M\nu : M\u02e3\nx\u271d : IsUnit (a * \u2191u)\nv : M\u02e3\nhv : \u2191v = a * \u2191u\nthis : IsUnit (a * \u2191u * \u2191u\u207b\u00b9)\n\u22a2 IsUnit a\n[PROOFSTEP]\nrwa [mul_assoc, Units.mul_inv, mul_one] at this \n[GOAL]\n\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\n\u22a2 IsUnit a\n[PROOFSTEP]\nhave : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a)) := by exists u\u207b\u00b9 * v; rw [\u2190 hv, Units.val_mul]\n[GOAL]\n\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\n\u22a2 IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a))\n[PROOFSTEP]\nexists u\u207b\u00b9 * v\n[GOAL]\n\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\n\u22a2 \u2191(u\u207b\u00b9 * v) = \u2191u\u207b\u00b9 * (\u2191u * a)\n[PROOFSTEP]\nrw [\u2190 hv, Units.val_mul]\n[GOAL]\n\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\nthis : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a))\n\u22a2 IsUnit a\n[PROOFSTEP]\nrwa [\u2190 mul_assoc, Units.inv_mul, one_mul] at this \n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : CommMonoid M\nx y : M\nhu : IsUnit (x * y)\nz : M\nhz : x * y * z = 1\n\u22a2 x * (y * z) = 1\n[PROOFSTEP]\nrwa [\u2190 mul_assoc]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : CommMonoid M\nx y : M\nhu : IsUnit (x * y)\n\u22a2 IsUnit (y * x)\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na b c : M\nh : IsUnit a\n\u22a2 a * \u2191(IsUnit.unit h)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 h.unit.mul_inv]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : Monoid M\na b c : M\nh : IsUnit a\n\u22a2 a * \u2191(IsUnit.unit h)\u207b\u00b9 = \u2191(IsUnit.unit h) * \u2191(IsUnit.unit h)\u207b\u00b9\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : DivisionMonoid M\na : M\n\u22a2 IsUnit a \u2192 a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrintro \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : DivisionMonoid M\nu : M\u02e3\n\u22a2 (\u2191u)\u207b\u00b9 * \u2191u = 1\n[PROOFSTEP]\nrw [\u2190 Units.val_inv_eq_inv_val, Units.inv_mul]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : DivisionMonoid M\na : M\n\u22a2 IsUnit a \u2192 a * a\u207b\u00b9 = 1\n[PROOFSTEP]\nrintro \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\nM : Type u_1\nN : Type u_2\ninst\u271d : DivisionMonoid M\nu : M\u02e3\n\u22a2 \u2191u * (\u2191u)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 Units.val_inv_eq_inv_val, Units.mul_inv]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nhM : Monoid M\nh : \u2200 (a : M), IsUnit a\na : M\n\u22a2 a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nchange \u2191(h a).unit\u207b\u00b9 * a = 1\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nhM : Monoid M\nh : \u2200 (a : M), IsUnit a\na : M\n\u22a2 \u2191(IsUnit.unit (_ : IsUnit a))\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrw [Units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one]\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nhM : CommMonoid M\nh : \u2200 (a : M), IsUnit a\na : M\n\u22a2 a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nchange \u2191(h a).unit\u207b\u00b9 * a = 1\n[GOAL]\n\u03b1 : Type u\nM : Type u_1\nhM : CommMonoid M\nh : \u2200 (a : M), IsUnit a\na : M\n\u22a2 \u2191(IsUnit.unit (_ : IsUnit a))\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrw [Units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.Units", "llama_tokens": 6715, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430436757312, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.5644946458351024}}
{"text": "[GOAL]\nm\u271d n\u271d m n : \u2115\nh : m \u2264 n\n\u22a2 m ! \u2223 n !\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nm\u271d n\u271d m n : \u2115\nh\u271d : m \u2264 n\nh : m \u2264 zero\n\u22a2 m ! \u2223 zero !\n[PROOFSTEP]\nsimp [Nat.eq_zero_of_le_zero h]\n[GOAL]\ncase succ\nm\u271d n\u271d\u00b9 m n\u271d : \u2115\nh\u271d : m \u2264 n\u271d\nn : \u2115\nIH : m \u2264 n \u2192 m ! \u2223 n !\nh : m \u2264 succ n\n\u22a2 m ! \u2223 (succ n)!\n[PROOFSTEP]\nobtain rfl | hl := h.eq_or_lt\n[GOAL]\ncase succ.inl\nm n\u271d\u00b9 n\u271d n : \u2115\nh\u271d : succ n \u2264 n\u271d\nIH : succ n \u2264 n \u2192 (succ n)! \u2223 n !\nh : succ n \u2264 succ n\n\u22a2 (succ n)! \u2223 (succ n)!\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.inr\nm\u271d n\u271d\u00b9 m n\u271d : \u2115\nh\u271d : m \u2264 n\u271d\nn : \u2115\nIH : m \u2264 n \u2192 m ! \u2223 n !\nh : m \u2264 succ n\nhl : m < succ n\n\u22a2 m ! \u2223 (succ n)!\n[PROOFSTEP]\nexact (IH (le_of_lt_succ hl)).mul_left _\n[GOAL]\nm\u271d n m : \u2115\n\u22a2 m ! * succ m ^ 0 \u2264 (m + 0)!\n[PROOFSTEP]\nsimp\n[GOAL]\nm\u271d n\u271d m n : \u2115\n\u22a2 m ! * succ m ^ (n + 1) \u2264 (m + (n + 1))!\n[PROOFSTEP]\nrw [\u2190 add_assoc, \u2190 Nat.succ_eq_add_one (m + n), Nat.factorial_succ, pow_succ', mul_comm (_ + 1), mul_comm (succ m), \u2190\n  mul_assoc]\n[GOAL]\nm\u271d n\u271d m n : \u2115\n\u22a2 m ! * succ m ^ n * succ m \u2264 (m + n)! * (m + n + 1)\n[PROOFSTEP]\nexact\n  mul_le_mul factorial_mul_pow_le_factorial (Nat.succ_le_succ (Nat.le_add_right _ _)) (Nat.zero_le _) (Nat.zero_le _)\n[GOAL]\nm n : \u2115\nhn : 0 < n\n\u22a2 n ! < m ! \u2194 n < m\n[PROOFSTEP]\nrefine' \u27e8fun h => not_le.mp fun hmn => not_le_of_lt h (factorial_le hmn), fun h => _\u27e9\n[GOAL]\nm n : \u2115\nhn : 0 < n\nh : n < m\n\u22a2 n ! < m !\n[PROOFSTEP]\nhave : \u2200 {n}, 0 < n \u2192 n ! < n.succ ! := by\n  intro k hk\n  rw [factorial_succ, succ_mul, lt_add_iff_pos_left]\n  exact mul_pos hk k.factorial_pos\n[GOAL]\nm n : \u2115\nhn : 0 < n\nh : n < m\n\u22a2 \u2200 {n : \u2115}, 0 < n \u2192 n ! < (succ n)!\n[PROOFSTEP]\nintro k hk\n[GOAL]\nm n : \u2115\nhn : 0 < n\nh : n < m\nk : \u2115\nhk : 0 < k\n\u22a2 k ! < (succ k)!\n[PROOFSTEP]\nrw [factorial_succ, succ_mul, lt_add_iff_pos_left]\n[GOAL]\nm n : \u2115\nhn : 0 < n\nh : n < m\nk : \u2115\nhk : 0 < k\n\u22a2 0 < k * k !\n[PROOFSTEP]\nexact mul_pos hk k.factorial_pos\n[GOAL]\nm n : \u2115\nhn : 0 < n\nh : n < m\nthis : \u2200 {n : \u2115}, 0 < n \u2192 n ! < (succ n)!\n\u22a2 n ! < m !\n[PROOFSTEP]\ninduction' h with k hnk ih generalizing hn\n[GOAL]\ncase refl\nm n : \u2115\nhn\u271d : 0 < n\nthis : \u2200 {n : \u2115}, 0 < n \u2192 n ! < (succ n)!\nhn : 0 < n\n\u22a2 n ! < (succ n)!\n[PROOFSTEP]\nexact this hn\n[GOAL]\ncase step\nm n : \u2115\nhn\u271d : 0 < n\nthis : \u2200 {n : \u2115}, 0 < n \u2192 n ! < (succ n)!\nk : \u2115\nhnk : Nat.le (succ n) k\nih : 0 < n \u2192 n ! < k !\nhn : 0 < n\n\u22a2 n ! < (succ k)!\n[PROOFSTEP]\nexact (ih hn).trans (this <| hn.trans <| lt_of_succ_le hnk)\n[GOAL]\nm n : \u2115\n\u22a2 n ! = 1 \u2194 n \u2264 1\n[PROOFSTEP]\napply Iff.intro\n[GOAL]\ncase mp\nm n : \u2115\n\u22a2 n ! = 1 \u2192 n \u2264 1\n[PROOFSTEP]\nintro\n[GOAL]\ncase mpr\nm n : \u2115\n\u22a2 n \u2264 1 \u2192 n ! = 1\n[PROOFSTEP]\nintro\n[GOAL]\ncase mp\nm n : \u2115\na\u271d : n ! = 1\n\u22a2 n \u2264 1\n[PROOFSTEP]\nrw [\u2190 not_lt, \u2190 one_lt_factorial, \u2039n ! = 1\u203a]\n[GOAL]\ncase mp\nm n : \u2115\na\u271d : n ! = 1\n\u22a2 \u00ac1 < 1\n[PROOFSTEP]\napply lt_irrefl\n[GOAL]\ncase mpr\nm n : \u2115\na\u271d : n \u2264 1\n\u22a2 n ! = 1\n[PROOFSTEP]\ncases \u2039n \u2264 1\u203a\n[GOAL]\ncase mpr.refl\nm : \u2115\n\u22a2 1! = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr.step\nm n : \u2115\na\u271d : Nat.le n 0\n\u22a2 n ! = 1\n[PROOFSTEP]\ncases \u2039n \u2264 0\u203a\n[GOAL]\ncase mpr.step.refl\nm : \u2115\n\u22a2 0! = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nm n : \u2115\nhn : 1 < n !\n\u22a2 n ! = m ! \u2194 n = m\n[PROOFSTEP]\nrefine' \u27e8fun h => _, congr_arg _\u27e9\n[GOAL]\nm n : \u2115\nhn : 1 < n !\nh : n ! = m !\n\u22a2 n = m\n[PROOFSTEP]\nobtain hnm | rfl | hnm := lt_trichotomy n m\n[GOAL]\ncase inl\nm n : \u2115\nhn : 1 < n !\nh : n ! = m !\nhnm : n < m\n\u22a2 n = m\n[PROOFSTEP]\nrw [\u2190 factorial_lt <| pos_of_gt <| one_lt_factorial.mp hn, h] at hnm \n[GOAL]\ncase inl\nm n : \u2115\nhn : 1 < n !\nh : n ! = m !\nhnm : m ! < m !\n\u22a2 n = m\n[PROOFSTEP]\ncases lt_irrefl _ hnm\n[GOAL]\ncase inr.inl\nn : \u2115\nhn : 1 < n !\nh : n ! = n !\n\u22a2 n = n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.inr\nm n : \u2115\nhn : 1 < n !\nh : n ! = m !\nhnm : m < n\n\u22a2 n = m\n[PROOFSTEP]\nrw [h, one_lt_factorial] at hn \n[GOAL]\ncase inr.inr\nm n : \u2115\nhn : 1 < m\nh : n ! = m !\nhnm : m < n\n\u22a2 n = m\n[PROOFSTEP]\nrw [\u2190 factorial_lt (lt_trans one_pos hn), h] at hnm \n[GOAL]\ncase inr.inr\nm n : \u2115\nhn : 1 < m\nh : n ! = m !\nhnm : m ! < m !\n\u22a2 n = m\n[PROOFSTEP]\ncases lt_irrefl _ hnm\n[GOAL]\nm n\u271d n : \u2115\nhi : 3 \u2264 n\n\u22a2 n < n !\n[PROOFSTEP]\nhave : 0 < n := (by decide : 0 < 2).trans (succ_le_iff.mp hi)\n[GOAL]\nm n\u271d n : \u2115\nhi : 3 \u2264 n\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nm n\u271d n : \u2115\nhi : 3 \u2264 n\nthis : 0 < n\n\u22a2 n < n !\n[PROOFSTEP]\nhave : 1 < pred n := le_pred_of_lt (succ_le_iff.mp hi)\n[GOAL]\nm n\u271d n : \u2115\nhi : 3 \u2264 n\nthis\u271d : 0 < n\nthis : 1 < pred n\n\u22a2 n < n !\n[PROOFSTEP]\nrw [\u2190 succ_pred_eq_of_pos \u20390 < n\u203a, factorial_succ]\n[GOAL]\nm n\u271d n : \u2115\nhi : 3 \u2264 n\nthis\u271d : 0 < n\nthis : 1 < pred n\n\u22a2 succ (pred n) < (pred n + 1) * (pred n)!\n[PROOFSTEP]\nexact lt_mul_of_one_lt_right (pred n).succ_pos ((\u20391 < pred n\u203a).trans_le (self_le_factorial _))\n[GOAL]\nm n\u271d i n : \u2115\nhi : 2 \u2264 i\n\u22a2 i + (n + 1)! < (i + n + 1)!\n[PROOFSTEP]\nrw [\u2190 Nat.succ_eq_add_one (i + _), factorial_succ (i + _), add_mul, one_mul]\n[GOAL]\nm n\u271d i n : \u2115\nhi : 2 \u2264 i\n\u22a2 i + (n + 1)! < (i + n) * (i + n)! + (i + n)!\n[PROOFSTEP]\nhave : i \u2264 i + n := le.intro rfl\n[GOAL]\nm n\u271d i n : \u2115\nhi : 2 \u2264 i\nthis : i \u2264 i + n\n\u22a2 i + (n + 1)! < (i + n) * (i + n)! + (i + n)!\n[PROOFSTEP]\nexact\n  add_lt_add_of_lt_of_le\n    (this.trans_lt\n      ((lt_mul_iff_one_lt_right ((by decide : 0 < 2).trans_le (hi.trans this))).mpr\n        (lt_iff_le_and_ne.mpr\n          \u27e8(i + n).factorial_pos, fun g =>\n            Nat.not_succ_le_self 1 ((hi.trans this).trans (factorial_eq_one.mp g.symm))\u27e9)))\n    (factorial_le ((le_of_eq (add_comm n 1)).trans ((add_le_add_iff_right n).mpr ((by decide : 1 \u2264 2).trans hi))))\n[GOAL]\nm n\u271d i n : \u2115\nhi : 2 \u2264 i\nthis : i \u2264 i + n\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nm n\u271d i n : \u2115\nhi : 2 \u2264 i\nthis : i \u2264 i + n\n\u22a2 1 \u2264 2\n[PROOFSTEP]\ndecide\n[GOAL]\nm n\u271d i n : \u2115\nhi : 2 \u2264 i\nhn : 1 \u2264 n\n\u22a2 i + n ! < (i + n)!\n[PROOFSTEP]\ncases hn\n[GOAL]\ncase refl\nm n i : \u2115\nhi : 2 \u2264 i\n\u22a2 i + 1! < (i + 1)!\n[PROOFSTEP]\nrw [factorial_one]\n[GOAL]\ncase refl\nm n i : \u2115\nhi : 2 \u2264 i\n\u22a2 i + 1 < (i + 1)!\n[PROOFSTEP]\nexact lt_factorial_self (succ_le_succ hi)\n[GOAL]\ncase step\nm n i : \u2115\nhi : 2 \u2264 i\nm\u271d : \u2115\na\u271d : Nat.le 1 m\u271d\n\u22a2 i + (succ m\u271d)! < (i + succ m\u271d)!\n[PROOFSTEP]\nexact add_factorial_succ_lt_factorial_add_succ _ hi\n[GOAL]\nm n\u271d i n : \u2115\n\u22a2 i + (n + 1)! \u2264 (i + (n + 1))!\n[PROOFSTEP]\ncases (le_or_lt (2 : \u2115) i)\n[GOAL]\ncase inl\nm n\u271d i n : \u2115\nh\u271d : 2 \u2264 i\n\u22a2 i + (n + 1)! \u2264 (i + (n + 1))!\n[PROOFSTEP]\nrw [\u2190 add_assoc]\n[GOAL]\ncase inl\nm n\u271d i n : \u2115\nh\u271d : 2 \u2264 i\n\u22a2 i + (n + 1)! \u2264 (i + n + 1)!\n[PROOFSTEP]\napply Nat.le_of_lt\n[GOAL]\ncase inl.h\nm n\u271d i n : \u2115\nh\u271d : 2 \u2264 i\n\u22a2 i + (n + 1)! < (i + n + 1)!\n[PROOFSTEP]\napply add_factorial_succ_lt_factorial_add_succ\n[GOAL]\ncase inl.h.hi\nm n\u271d i n : \u2115\nh\u271d : 2 \u2264 i\n\u22a2 2 \u2264 i\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr\nm n\u271d i n : \u2115\nh\u271d : i < 2\n\u22a2 i + (n + 1)! \u2264 (i + (n + 1))!\n[PROOFSTEP]\nmatch i with\n| 0 => simp\n|\n1 =>\n  rw [\u2190 add_assoc, \u2190 Nat.succ_eq_add_one (1 + n), factorial_succ (1 + n), add_mul, one_mul, add_comm 1 n,\n    add_le_add_iff_right]\n  apply one_le_mul\n  \u00b7 apply Nat.le_add_left\n  \u00b7 apply factorial_pos\n| succ (succ n) => contradiction\n[GOAL]\nm n\u271d i n : \u2115\nh\u271d : 0 < 2\n\u22a2 0 + (n + 1)! \u2264 (0 + (n + 1))!\n[PROOFSTEP]\nsimp\n[GOAL]\nm n\u271d i n : \u2115\nh\u271d : 1 < 2\n\u22a2 1 + (n + 1)! \u2264 (1 + (n + 1))!\n[PROOFSTEP]\nrw [\u2190 add_assoc, \u2190 Nat.succ_eq_add_one (1 + n), factorial_succ (1 + n), add_mul, one_mul, add_comm 1 n,\n  add_le_add_iff_right]\n[GOAL]\nm n\u271d i n : \u2115\nh\u271d : 1 < 2\n\u22a2 1 \u2264 (n + 1) * (n + 1)!\n[PROOFSTEP]\napply one_le_mul\n[GOAL]\ncase ha\nm n\u271d i n : \u2115\nh\u271d : 1 < 2\n\u22a2 1 \u2264 n + 1\n[PROOFSTEP]\napply Nat.le_add_left\n[GOAL]\ncase hb\nm n\u271d i n : \u2115\nh\u271d : 1 < 2\n\u22a2 1 \u2264 (n + 1)!\n[PROOFSTEP]\napply factorial_pos\n[GOAL]\nm n\u271d\u00b9 i n\u271d n : \u2115\nh\u271d : succ (succ n) < 2\n\u22a2 succ (succ n) + (n\u271d + 1)! \u2264 (succ (succ n) + (n\u271d + 1))!\n[PROOFSTEP]\ncontradiction\n[GOAL]\nm n\u271d i n : \u2115\nn1 : 1 \u2264 n\n\u22a2 i + n ! \u2264 (i + n)!\n[PROOFSTEP]\ncases' n1 with h\n[GOAL]\ncase refl\nm n i : \u2115\n\u22a2 i + 1! \u2264 (i + 1)!\n[PROOFSTEP]\nexact self_le_factorial _\n[GOAL]\ncase step\nm n i h : \u2115\na\u271d : Nat.le 1 h\n\u22a2 i + (succ h)! \u2264 (i + succ h)!\n[PROOFSTEP]\nexact add_factorial_succ_le_factorial_add_succ i h\n[GOAL]\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\n\u22a2 n ! * n ^ (m - n) \u2264 m !\n[PROOFSTEP]\nsuffices n ! * (n + 1) ^ (m - n) \u2264 m ! from by\n  apply LE.le.trans _ this\n  apply mul_le_mul_left\n  apply pow_le_pow_of_le_left (le_succ n)\n[GOAL]\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ! * n ^ (m - n) \u2264 m !\n[PROOFSTEP]\napply LE.le.trans _ this\n[GOAL]\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ! * n ^ (m - n) \u2264 n ! * (n + 1) ^ (m - n)\n[PROOFSTEP]\napply mul_le_mul_left\n[GOAL]\ncase h\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ^ (m - n) \u2264 (n + 1) ^ (m - n)\n[PROOFSTEP]\napply pow_le_pow_of_le_left (le_succ n)\n[GOAL]\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !\n[PROOFSTEP]\nhave := @Nat.factorial_mul_pow_le_factorial n (m - n)\n[GOAL]\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * succ n ^ (m - n) \u2264 (n + (m - n))!\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !\n[PROOFSTEP]\nsimp [hnm] at this \n[GOAL]\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * succ n ^ (m - n) \u2264 m !\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !\n[PROOFSTEP]\nexact this\n[GOAL]\nk : \u2115\n\u22a2 ascFactorial 0 k = k !\n[PROOFSTEP]\ninduction' k with t ht\n[GOAL]\ncase zero\n\u22a2 ascFactorial 0 zero = zero !\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nt : \u2115\nht : ascFactorial 0 t = t !\n\u22a2 ascFactorial 0 (succ t) = (succ t)!\n[PROOFSTEP]\nrw [ascFactorial, ht, zero_add, Nat.factorial_succ]\n[GOAL]\nn : \u2115\n\u22a2 (n + 1) * ascFactorial (succ n) 0 = (n + 0 + 1) * ascFactorial n 0\n[PROOFSTEP]\nrw [add_zero, ascFactorial_zero, ascFactorial_zero]\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1) * ascFactorial (succ n) (k + 1) = (n + (k + 1) + 1) * ascFactorial n (k + 1)\n[PROOFSTEP]\nrw [ascFactorial, mul_left_comm, succ_ascFactorial n k, ascFactorial, succ_add, \u2190 add_assoc, succ_eq_add_one]\n[GOAL]\nn : \u2115\n\u22a2 n ! * ascFactorial n 0 = (n + 0)!\n[PROOFSTEP]\nrw [ascFactorial, add_zero, mul_one]\n[GOAL]\nn k : \u2115\n\u22a2 n ! * ascFactorial n (k + 1) = (n + (k + 1))!\n[PROOFSTEP]\nrw [ascFactorial_succ, mul_left_comm, factorial_mul_ascFactorial n k, \u2190 add_assoc, \u2190 Nat.succ_eq_add_one (n + k),\n  factorial]\n[GOAL]\nn k : \u2115\n\u22a2 ascFactorial n k = (n + k)! / n !\n[PROOFSTEP]\napply mul_left_cancel\u2080 n.factorial_ne_zero\n[GOAL]\nn k : \u2115\n\u22a2 n ! * ascFactorial n k = n ! * ((n + k)! / n !)\n[PROOFSTEP]\nrw [factorial_mul_ascFactorial]\n[GOAL]\nn k : \u2115\n\u22a2 (n + k)! = n ! * ((n + k)! / n !)\n[PROOFSTEP]\nexact (Nat.mul_div_cancel' <| factorial_dvd_factorial <| le.intro rfl).symm\n[GOAL]\nn k : \u2115\nh : k < n\n\u22a2 (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)\n[PROOFSTEP]\nlet t := n - k.succ\n[GOAL]\nn k : \u2115\nh : k < n\nt : \u2115 := n - succ k\n\u22a2 (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)\n[PROOFSTEP]\nlet ht : t = n - k.succ := rfl\n[GOAL]\nn k : \u2115\nh : k < n\nt : \u2115 := n - succ k\nht : t = n - succ k := rfl\n\u22a2 (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)\n[PROOFSTEP]\nsuffices h' : n - k = t.succ\n[GOAL]\nn k : \u2115\nh : k < n\nt : \u2115 := n - succ k\nht : t = n - succ k := rfl\nh' : n - k = succ t\n\u22a2 (n - k) * ascFactorial (n - k) k = ascFactorial (n - (k + 1)) (k + 1)\n[PROOFSTEP]\nrw [\u2190 ht, h', succ_ascFactorial, ascFactorial_succ]\n[GOAL]\ncase h'\nn k : \u2115\nh : k < n\nt : \u2115 := n - succ k\nht : t = n - succ k := rfl\n\u22a2 n - k = succ t\n[PROOFSTEP]\nrw [ht, succ_eq_add_one, \u2190 tsub_tsub_assoc (succ_le_of_lt h) (succ_pos _), succ_sub_one]\n[GOAL]\nn : \u2115\n\u22a2 (n + 1) ^ 0 \u2264 ascFactorial n 0\n[PROOFSTEP]\nrw [ascFactorial_zero, pow_zero]\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1) ^ (k + 1) \u2264 ascFactorial n (k + 1)\n[PROOFSTEP]\nrw [pow_succ, mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1) * (n + 1) ^ k \u2264 ascFactorial n (k + 1)\n[PROOFSTEP]\nexact Nat.mul_le_mul (Nat.add_le_add_right le_self_add _) (pow_succ_le_ascFactorial _ k)\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1) ^ (k + 2) < ascFactorial n (k + 2)\n[PROOFSTEP]\nrw [pow_succ, ascFactorial, mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1) * (n + 1) ^ (k + 1) < (n + (k + 1) + 1) * ascFactorial n (k + 1)\n[PROOFSTEP]\nexact\n  Nat.mul_lt_mul (Nat.add_lt_add_right (Nat.lt_add_of_pos_right succ_pos') 1) (pow_succ_le_ascFactorial n _)\n    (pow_pos succ_pos' _)\n[GOAL]\nn : \u2115\n\u22a2 2 \u2264 0 \u2192 (n + 1) ^ 0 < ascFactorial n 0\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\nn : \u2115\n\u22a2 2 \u2264 1 \u2192 (n + 1) ^ 1 < ascFactorial n 1\n[PROOFSTEP]\nintro\n[GOAL]\nn : \u2115\na\u271d : 2 \u2264 1\n\u22a2 (n + 1) ^ 1 < ascFactorial n 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\n\u22a2 ascFactorial n 0 \u2264 (n + 0) ^ 0\n[PROOFSTEP]\nrw [ascFactorial_zero, pow_zero]\n[GOAL]\nn k : \u2115\n\u22a2 ascFactorial n (k + 1) \u2264 (n + (k + 1)) ^ (k + 1)\n[PROOFSTEP]\nrw [ascFactorial_succ, pow_succ, \u2190 add_assoc, \u2190 Nat.succ_eq_add_one (n + k), mul_comm _ (succ (n + k))]\n[GOAL]\nn k : \u2115\n\u22a2 succ (n + k) * ascFactorial n k \u2264 succ (n + k) * succ (n + k) ^ k\n[PROOFSTEP]\nexact Nat.mul_le_mul_of_nonneg_left ((ascFactorial_le_pow_add _ k).trans (Nat.pow_le_pow_of_le_left (le_succ _) _))\n[GOAL]\nn : \u2115\n\u22a2 2 \u2264 0 \u2192 ascFactorial n 0 < (n + 0) ^ 0\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\nn : \u2115\n\u22a2 2 \u2264 1 \u2192 ascFactorial n 1 < (n + 1) ^ 1\n[PROOFSTEP]\nintro\n[GOAL]\nn : \u2115\na\u271d : 2 \u2264 1\n\u22a2 ascFactorial n 1 < (n + 1) ^ 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn k : \u2115\nx\u271d : 2 \u2264 k + 2\n\u22a2 ascFactorial n (k + 2) < (n + (k + 2)) ^ (k + 2)\n[PROOFSTEP]\nrw [ascFactorial_succ, pow_succ]\n[GOAL]\nn k : \u2115\nx\u271d : 2 \u2264 k + 2\n\u22a2 (n + (k + 1) + 1) * ascFactorial n (k + 1) < (n + (k + 2)) ^ (k + 1) * (n + (k + 2))\n[PROOFSTEP]\nrw [add_assoc n (k + 1) 1, mul_comm <| (n + (k + 2)) ^ (k + 1)]\n[GOAL]\nn k : \u2115\nx\u271d : 2 \u2264 k + 2\n\u22a2 (n + (k + 1 + 1)) * ascFactorial n (k + 1) < (n + (k + 2)) * (n + (k + 2)) ^ (k + 1)\n[PROOFSTEP]\nrefine'\n  Nat.mul_lt_mul' le_rfl ((ascFactorial_le_pow_add n _).trans_lt (pow_lt_pow_of_lt_left (lt_add_one _) (succ_pos _)))\n    (succ_pos _)\n[GOAL]\nk : \u2115\n\u22a2 descFactorial 0 (succ k) = 0\n[PROOFSTEP]\nrw [descFactorial_succ, zero_tsub, zero_mul]\n[GOAL]\nn : \u2115\n\u22a2 descFactorial n 1 = n\n[PROOFSTEP]\nrw [descFactorial_succ, descFactorial_zero, mul_one, tsub_zero]\n[GOAL]\nn : \u2115\n\u22a2 descFactorial (n + 1) (0 + 1) = (n + 1) * descFactorial n 0\n[PROOFSTEP]\nrw [descFactorial_zero, descFactorial_one, mul_one]\n[GOAL]\nn k : \u2115\n\u22a2 descFactorial (n + 1) (succ k + 1) = (n + 1) * descFactorial n (succ k)\n[PROOFSTEP]\nrw [descFactorial_succ, succ_descFactorial_succ _ k, descFactorial_succ, succ_sub_succ, mul_left_comm]\n[GOAL]\nn : \u2115\n\u22a2 (n + 1 - 0) * descFactorial (n + 1) 0 = (n + 1) * descFactorial n 0\n[PROOFSTEP]\nrw [tsub_zero, descFactorial_zero, descFactorial_zero]\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1 - (k + 1)) * descFactorial (n + 1) (k + 1) = (n + 1) * descFactorial n (k + 1)\n[PROOFSTEP]\nrw [descFactorial, succ_descFactorial _ k, descFactorial_succ, succ_sub_succ, mul_left_comm]\n[GOAL]\n\u22a2 descFactorial 0 0 = 0!\n[PROOFSTEP]\nrw [descFactorial_zero, factorial_zero]\n[GOAL]\nn : \u2115\n\u22a2 descFactorial (succ n) (succ n) = (succ n)!\n[PROOFSTEP]\nrw [succ_descFactorial_succ, descFactorial_self n, factorial_succ]\n[GOAL]\nn : \u2115\n\u22a2 descFactorial n 0 = 0 \u2194 n < 0\n[PROOFSTEP]\nsimp only [descFactorial_zero, Nat.one_ne_zero, Nat.not_lt_zero]\n[GOAL]\nn k : \u2115\n\u22a2 descFactorial n (succ k) = 0 \u2194 n < succ k\n[PROOFSTEP]\nrw [descFactorial_succ, mul_eq_zero, descFactorial_eq_zero_iff_lt, lt_succ_iff, tsub_eq_zero_iff_le, lt_iff_le_and_ne,\n  or_iff_left_iff_imp, and_imp]\n[GOAL]\nn k : \u2115\n\u22a2 n \u2264 k \u2192 n \u2260 k \u2192 n \u2264 k\n[PROOFSTEP]\nexact fun h _ => h\n[GOAL]\nn : \u2115\n\u22a2 descFactorial (n + 0) 0 = ascFactorial n 0\n[PROOFSTEP]\nrw [ascFactorial_zero, descFactorial_zero]\n[GOAL]\nn k : \u2115\n\u22a2 descFactorial (n + succ k) (succ k) = ascFactorial n (succ k)\n[PROOFSTEP]\nrw [Nat.add_succ, Nat.succ_eq_add_one, Nat.succ_eq_add_one, succ_descFactorial_succ, ascFactorial_succ,\n  add_descFactorial_eq_ascFactorial _ k]\n[GOAL]\nn : \u2115\nx\u271d : 0 \u2264 n\n\u22a2 (n - 0)! * descFactorial n 0 = n !\n[PROOFSTEP]\nrw [descFactorial_zero, mul_one, tsub_zero]\n[GOAL]\nk : \u2115\nh : succ k \u2264 0\n\u22a2 (0 - succ k)! * descFactorial 0 (succ k) = 0!\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\nk : \u2115\nh : succ k \u2264 0\n\u22a2 False\n[PROOFSTEP]\nexact not_succ_le_zero k h\n[GOAL]\nn k : \u2115\nh : succ k \u2264 succ n\n\u22a2 (succ n - succ k)! * descFactorial (succ n) (succ k) = (succ n)!\n[PROOFSTEP]\nrw [succ_descFactorial_succ, succ_sub_succ, \u2190 mul_assoc, mul_comm (n - k)!, mul_assoc,\n  factorial_mul_descFactorial (Nat.succ_le_succ_iff.1 h), factorial_succ]\n[GOAL]\nn k : \u2115\nh : k \u2264 n\n\u22a2 descFactorial n k = n ! / (n - k)!\n[PROOFSTEP]\napply mul_left_cancel\u2080 (factorial_ne_zero (n - k))\n[GOAL]\nn k : \u2115\nh : k \u2264 n\n\u22a2 (n - k)! * descFactorial n k = (n - k)! * (n ! / (n - k)!)\n[PROOFSTEP]\nrw [factorial_mul_descFactorial h]\n[GOAL]\nn k : \u2115\nh : k \u2264 n\n\u22a2 n ! = (n - k)! * (n ! / (n - k)!)\n[PROOFSTEP]\nexact (Nat.mul_div_cancel' <| factorial_dvd_factorial <| Nat.sub_le n k).symm\n[GOAL]\nn : \u2115\n\u22a2 (n + 1 - 0) ^ 0 \u2264 descFactorial n 0\n[PROOFSTEP]\nrw [descFactorial_zero, pow_zero]\n[GOAL]\nn k : \u2115\n\u22a2 (n + 1 - (k + 1)) ^ (k + 1) \u2264 descFactorial n (k + 1)\n[PROOFSTEP]\nrw [descFactorial_succ, pow_succ, succ_sub_succ, mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 (n - k) * (n - k) ^ k \u2264 (n - k) * descFactorial n k\n[PROOFSTEP]\napply Nat.mul_le_mul_of_nonneg_left\n[GOAL]\ncase h\u2081\nn k : \u2115\n\u22a2 (n - k) ^ k \u2264 descFactorial n k\n[PROOFSTEP]\nexact (le_trans (Nat.pow_le_pow_of_le_left (tsub_le_tsub_right (le_succ _) _) k) (pow_sub_le_descFactorial n k))\n[GOAL]\nn : \u2115\nh : 0 + 2 \u2264 n\n\u22a2 (n - (0 + 1)) ^ (0 + 2) < descFactorial n (0 + 2)\n[PROOFSTEP]\nrw [descFactorial_succ, pow_succ, pow_one, descFactorial_one]\n[GOAL]\nn : \u2115\nh : 0 + 2 \u2264 n\n\u22a2 (n - (0 + 1)) * (n - (0 + 1)) < (n - 1) * n\n[PROOFSTEP]\nexact Nat.mul_lt_mul_of_pos_left (tsub_lt_self (lt_of_lt_of_le zero_lt_two h) zero_lt_one) (tsub_pos_of_lt h)\n[GOAL]\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 (n - (k + 1 + 1)) ^ (k + 1 + 2) < descFactorial n (k + 1 + 2)\n[PROOFSTEP]\nrw [descFactorial_succ, pow_succ, mul_comm]\n[GOAL]\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 (n - (k + 1 + 1)) * (n - (k + 1 + 1)) ^ (k + 2) < (n - (k + 2)) * descFactorial n (k + 2)\n[PROOFSTEP]\napply Nat.mul_lt_mul_of_pos_left\n[GOAL]\ncase h\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 (n - (k + 1 + 1)) ^ (k + 2) < descFactorial n (k + 2)\n[PROOFSTEP]\nrefine' ((Nat.pow_le_pow_of_le_left (tsub_le_tsub_right (le_succ n) _) _).trans_lt _)\n[GOAL]\ncase h\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 (succ n - (k + 1 + 1)) ^ (k + 2) < descFactorial n (k + 2)\n[PROOFSTEP]\nrw [succ_sub_succ]\n[GOAL]\ncase h\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 (n - (k + 1)) ^ (k + 2) < descFactorial n (k + 2)\n[PROOFSTEP]\nexact pow_sub_lt_descFactorial' ((le_succ _).trans h)\n[GOAL]\ncase hk\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 n - (k + 1 + 1) > 0\n[PROOFSTEP]\napply tsub_pos_of_lt\n[GOAL]\ncase hk.h\nn k : \u2115\nh : k + 1 + 2 \u2264 n\n\u22a2 k + 1 + 1 < n\n[PROOFSTEP]\napply h\n[GOAL]\nn : \u2115\n\u22a2 2 \u2264 0 \u2192 0 \u2264 n \u2192 (n + 1 - 0) ^ 0 < descFactorial n 0\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\nn : \u2115\n\u22a2 2 \u2264 1 \u2192 1 \u2264 n \u2192 (n + 1 - 1) ^ 1 < descFactorial n 1\n[PROOFSTEP]\nintro\n[GOAL]\nn : \u2115\na\u271d : 2 \u2264 1\n\u22a2 1 \u2264 n \u2192 (n + 1 - 1) ^ 1 < descFactorial n 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn k : \u2115\nx\u271d : 2 \u2264 k + 2\nh : k + 2 \u2264 n\n\u22a2 (n + 1 - (k + 2)) ^ (k + 2) < descFactorial n (k + 2)\n[PROOFSTEP]\nrw [succ_sub_succ]\n[GOAL]\nn k : \u2115\nx\u271d : 2 \u2264 k + 2\nh : k + 2 \u2264 n\n\u22a2 (n - (k + 1)) ^ (k + 2) < descFactorial n (k + 2)\n[PROOFSTEP]\nexact pow_sub_lt_descFactorial' h\n[GOAL]\nn : \u2115\n\u22a2 descFactorial n 0 \u2264 n ^ 0\n[PROOFSTEP]\nrw [descFactorial_zero, pow_zero]\n[GOAL]\nn k : \u2115\n\u22a2 descFactorial n (k + 1) \u2264 n ^ (k + 1)\n[PROOFSTEP]\nrw [descFactorial_succ, pow_succ, mul_comm _ n]\n[GOAL]\nn k : \u2115\n\u22a2 (n - k) * descFactorial n k \u2264 n * n ^ k\n[PROOFSTEP]\nexact Nat.mul_le_mul (Nat.sub_le _ _) (descFactorial_le_pow _ k)\n[GOAL]\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 2 \u2264 0 \u2192 descFactorial n 0 < n ^ 0\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 2 \u2264 1 \u2192 descFactorial n 1 < n ^ 1\n[PROOFSTEP]\nintro\n[GOAL]\nn : \u2115\nhn : 1 \u2264 n\na\u271d : 2 \u2264 1\n\u22a2 descFactorial n 1 < n ^ 1\n[PROOFSTEP]\ncontradiction\n[GOAL]\nn : \u2115\nhn : 1 \u2264 n\nk : \u2115\nx\u271d : 2 \u2264 k + 2\n\u22a2 descFactorial n (k + 2) < n ^ (k + 2)\n[PROOFSTEP]\nrw [descFactorial_succ, pow_succ', mul_comm, mul_comm n]\n[GOAL]\nn : \u2115\nhn : 1 \u2264 n\nk : \u2115\nx\u271d : 2 \u2264 k + 2\n\u22a2 descFactorial n (k + 1) * (n - (k + 1)) < n ^ (k + 1) * n\n[PROOFSTEP]\nexact Nat.mul_lt_mul' (descFactorial_le_pow _ _) (tsub_lt_self hn k.zero_lt_succ) (pow_pos (Nat.lt_of_succ_le hn) _)\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Factorial.Basic", "llama_tokens": 11088, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.831143031127974, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.5644946427101906}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\nn1 n2 k e1 e2 t1 t2 : \u03b1\nh1 : n1 * e1 = t1\nh2 : n2 * e2 = t2\nh3 : n1 * n2 = k\n\u22a2 k * (e1 * e2) = t1 * t2\n[PROOFSTEP]\nrw [\u2190 h3, mul_comm n1, mul_assoc n2, \u2190 mul_assoc n1, h1, \u2190 mul_assoc n2, mul_comm n2, mul_assoc, h2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\nn1 n2 k e1 e2 t1 : \u03b1\nh1 : n1 * e1 = t1\nh2 : n2 / e2 = 1\nh3 : n1 * n2 = k\n\u22a2 k * (e1 / e2) = t1\n[PROOFSTEP]\nrw [\u2190 h3, mul_assoc, mul_div_left_comm, h2, \u2190 mul_assoc, h1, mul_comm, one_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\nn e e' : \u03b1\nh : n * e = e'\nh2 : n \u2260 0\n\u22a2 e * n = e'\n[PROOFSTEP]\nrwa [mul_comm] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nn e1 e2 t1 t2 : \u03b1\nh1 : n * e1 = t1\nh2 : n * e2 = t2\n\u22a2 n * (e1 + e2) = t1 + t2\n[PROOFSTEP]\nsimp [left_distrib, *]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nn e1 e2 t1 t2 : \u03b1\nh1 : n * e1 = t1\nh2 : n * e2 = t2\n\u22a2 n * (e1 - e2) = t1 - t2\n[PROOFSTEP]\nsimp [left_distrib, *, sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nn e t : \u03b1\nh1 : n * e = t\n\u22a2 n * -e = -t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 (a < b) = (1 / gcd * (bd * a') < 1 / gcd * (ad * b'))\n[PROOFSTEP]\nrw [mul_lt_mul_left, \u2190 ha, \u2190 hb, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm bd, mul_lt_mul_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < ad * bd\n[PROOFSTEP]\nexact mul_pos had hbd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < 1 / gcd\n[PROOFSTEP]\nexact one_div_pos.2 hgcd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 (a \u2264 b) = (1 / gcd * (bd * a') \u2264 1 / gcd * (ad * b'))\n[PROOFSTEP]\nrw [mul_le_mul_left, \u2190 ha, \u2190 hb, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm bd, mul_le_mul_left]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < ad * bd\n[PROOFSTEP]\nexact mul_pos had hbd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < 1 / gcd\n[PROOFSTEP]\nexact one_div_pos.2 hgcd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a = b) = (1 / gcd * (bd * a') = 1 / gcd * (ad * b'))\n[PROOFSTEP]\nrw [\u2190 ha, \u2190 hb, \u2190 mul_assoc bd, \u2190 mul_assoc ad, mul_comm bd]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a = b) = (1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b))\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 a = b \u2194 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 a = b \u2192 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase a.mp\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nhb : bd * a = b'\n\u22a2 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * a)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b) \u2192 a = b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)\n\u22a2 a = b\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc] at h \n[GOAL]\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 a = b\n[PROOFSTEP]\nrefine' mul_left_cancel\u2080 (mul_ne_zero _ _) h\n[GOAL]\ncase a.mpr.refine'_1\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd * ad \u2260 0\ncase a.mpr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0\n[PROOFSTEP]\napply mul_ne_zero\n[GOAL]\ncase a.mpr.refine'_1.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd \u2260 0\ncase a.mpr.refine'_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\ncase a.mpr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0\n[PROOFSTEP]\napply div_ne_zero\n[GOAL]\ncase a.mpr.refine'_1.ha.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 \u2260 0\ncase a.mpr.refine'_1.ha.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 gcd \u2260 0\ncase a.mpr.refine'_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\ncase a.mpr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\ncase a.mpr.refine'_1.ha.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 gcd \u2260 0\ncase a.mpr.refine'_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\ncase a.mpr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0\n[PROOFSTEP]\nall_goals assumption\n[GOAL]\ncase a.mpr.refine'_1.ha.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 gcd \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase a.mpr.refine'_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase a.mpr.refine'_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a \u2260 b) = (1 / gcd * (bd * a') \u2260 1 / gcd * (ad * b'))\n[PROOFSTEP]\nclassical rw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a \u2260 b) = (1 / gcd * (bd * a') \u2260 1 / gcd * (ad * b'))\n[PROOFSTEP]\nrw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]\n", "meta": {"mathlib_filename": "Mathlib.Tactic.CancelDenoms", "llama_tokens": 4813, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956580903722561, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.5641368149169795}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nx\u271d : Set C(\u03b1, \u03b2)\n\u22a2 x\u271d \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u} \u2194 x\u271d \u2208 image2 CompactOpen.gen {s | IsCompact s} {t | IsOpen t}\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 CompactOpen.gen s u \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\ndsimp [mem_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 \u2203 s_1 x u_1 x, CompactOpen.gen s u = CompactOpen.gen s_1 u_1\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\ns : Set \u03b1\nu : Set \u03b3\n\u22a2 comp g \u207b\u00b9' CompactOpen.gen s u = CompactOpen.gen s (\u2191g \u207b\u00b9' u)\n[PROOFSTEP]\next \u27e8f, _\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\ns : Set \u03b1\nu : Set \u03b3\nf : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d : Continuous f\n\u22a2 mk f \u2208 comp g \u207b\u00b9' CompactOpen.gen s u \u2194 mk f \u2208 CompactOpen.gen s (\u2191g \u207b\u00b9' u)\n[PROOFSTEP]\nchange g \u2218 f '' s \u2286 u \u2194 f '' s \u2286 g \u207b\u00b9' u\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\ns : Set \u03b1\nu : Set \u03b3\nf : \u03b1 \u2192 \u03b2\ncontinuous_toFun\u271d : Continuous f\n\u22a2 \u2191g \u2218 f '' s \u2286 u \u2194 f '' s \u2286 \u2191g \u207b\u00b9' u\n[PROOFSTEP]\nrw [image_comp, image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nm : Set C(\u03b1, \u03b3)\nx\u271d : m \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u}\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b3\nhu : IsOpen u\nhm : m = CompactOpen.gen s u\n\u22a2 IsOpen (comp g \u207b\u00b9' m)\n[PROOFSTEP]\nrw [hm, preimage_gen g]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nm : Set C(\u03b1, \u03b3)\nx\u271d : m \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u}\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b3\nhu : IsOpen u\nhm : m = CompactOpen.gen s u\n\u22a2 IsOpen (CompactOpen.gen s (\u2191g \u207b\u00b9' u))\n[PROOFSTEP]\nexact ContinuousMap.isOpen_gen hs (hu.preimage g.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nhg : Inducing \u2191g\n\u22a2 compactOpen = TopologicalSpace.induced (comp g) compactOpen\n[PROOFSTEP]\nsimp only [compactOpen_eq, induced_generateFrom_eq, image_image2, preimage_gen, hg.setOf_isOpen, image2_image_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ns : Set \u03b1\nx\u271d\u00b9 : IsCompact s\nu : Set \u03b3\nx\u271d : IsOpen u\n\u22a2 (fun g => comp g f) \u207b\u00b9' CompactOpen.gen s u = CompactOpen.gen (\u2191f '' s) u\n[PROOFSTEP]\next \u27e8g, _\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng\u271d : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ns : Set \u03b1\nx\u271d\u00b9 : IsCompact s\nu : Set \u03b3\nx\u271d : IsOpen u\ng : \u03b2 \u2192 \u03b3\ncontinuous_toFun\u271d : Continuous g\n\u22a2 mk g \u2208 (fun g => comp g f) \u207b\u00b9' CompactOpen.gen s u \u2194 mk g \u2208 CompactOpen.gen (\u2191f '' s) u\n[PROOFSTEP]\nchange g \u2218 f '' s \u2286 u \u2194 g '' (f '' s) \u2286 u\n[GOAL]\ncase h.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng\u271d : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ns : Set \u03b1\nx\u271d\u00b9 : IsCompact s\nu : Set \u03b3\nx\u271d : IsOpen u\ng : \u03b2 \u2192 \u03b3\ncontinuous_toFun\u271d : Continuous g\n\u22a2 g \u2218 \u2191f '' s \u2286 u \u2194 g '' (\u2191f '' s) \u2286 u\n[PROOFSTEP]\nrw [Set.image_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\nm : Set C(\u03b1, \u03b3)\nx\u271d : m \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u}\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b3\nhu : IsOpen u\nhm : m = CompactOpen.gen s u\n\u22a2 IsOpen ((fun g => comp g f) \u207b\u00b9' m)\n[PROOFSTEP]\nrw [hm, image_gen f hs hu]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\nm : Set C(\u03b1, \u03b3)\nx\u271d : m \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u}\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b3\nhu : IsOpen u\nhm : m = CompactOpen.gen s u\n\u22a2 IsOpen (CompactOpen.gen (\u2191f '' s) u)\n[PROOFSTEP]\nexact ContinuousMap.isOpen_gen (hs.image f.2) hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\n\u22a2 \u2200 (s : Set C(\u03b1, \u03b3)), s \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u} \u2192 IsOpen ((fun x => comp x.snd x.fst) \u207b\u00b9' s)\n[PROOFSTEP]\nrintro M \u27e8K, hK, U, hU, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u22a2 IsOpen ((fun x => comp x.snd x.fst) \u207b\u00b9' CompactOpen.gen K U)\n[PROOFSTEP]\nconv =>\n  congr\n  rw [CompactOpen.gen, preimage_setOf_eq]\n    --congr\n  ext; dsimp [setOf]\n  rw [image_comp, image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n| IsOpen ((fun x => comp x.snd x.fst) \u207b\u00b9' CompactOpen.gen K U)\n[PROOFSTEP]\n  congr\n  rw [CompactOpen.gen, preimage_setOf_eq]\n    --congr\n  ext; dsimp [setOf]\n  rw [image_comp, image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n| IsOpen ((fun x => comp x.snd x.fst) \u207b\u00b9' CompactOpen.gen K U)\n[PROOFSTEP]\n  congr\n  rw [CompactOpen.gen, preimage_setOf_eq]\n    --congr\n  ext; dsimp [setOf]\n  rw [image_comp, image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n| IsOpen ((fun x => comp x.snd x.fst) \u207b\u00b9' CompactOpen.gen K U)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n| (fun x => comp x.snd x.fst) \u207b\u00b9' CompactOpen.gen K U\n[PROOFSTEP]\nrw [CompactOpen.gen, preimage_setOf_eq]\n  --congr\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n| {a | \u2191(comp a.snd a.fst) '' K \u2286 U}\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\nx\u271d : C(\u03b1, \u03b2) \u00d7 C(\u03b2, \u03b3)\n| setOf (fun a => \u2191(comp a.snd a.fst) '' K \u2286 U) x\u271d\n[PROOFSTEP]\ndsimp [setOf]\n[GOAL]\ncase a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\nx\u271d : C(\u03b1, \u03b2) \u00d7 C(\u03b2, \u03b3)\n| \u2191x\u271d.snd \u2218 \u2191x\u271d.fst '' K \u2286 U\n[PROOFSTEP]\nrw [image_comp, image_subset_iff]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u22a2 IsOpen fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U\n[PROOFSTEP]\nrw [isOpen_iff_forall_mem_open]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u22a2 \u2200 (x : C(\u03b1, \u03b2) \u00d7 C(\u03b2, \u03b3)),\n    (x \u2208 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U) \u2192 \u2203 t, (t \u2286 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U) \u2227 IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nrintro \u27e8\u03c6\u2080, \u03c8\u2080\u27e9 H\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\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u03c6\u2080 : C(\u03b1, \u03b2)\n\u03c8\u2080 : C(\u03b2, \u03b3)\nH : (\u03c6\u2080, \u03c8\u2080) \u2208 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U\n\u22a2 \u2203 t, (t \u2286 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U) \u2227 IsOpen t \u2227 (\u03c6\u2080, \u03c8\u2080) \u2208 t\n[PROOFSTEP]\nobtain \u27e8L, hL, hKL, hLU\u27e9 := exists_compact_between (hK.image \u03c6\u2080.2) (hU.preimage \u03c8\u2080.2) H\n[GOAL]\ncase intro.intro.intro.intro.mk.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u03c6\u2080 : C(\u03b1, \u03b2)\n\u03c8\u2080 : C(\u03b2, \u03b3)\nH : (\u03c6\u2080, \u03c8\u2080) \u2208 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U\nL : Set \u03b2\nhL : IsCompact L\nhKL : \u03c6\u2080.toFun '' K \u2286 interior L\nhLU : L \u2286 \u03c8\u2080.toFun \u207b\u00b9' U\n\u22a2 \u2203 t, (t \u2286 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U) \u2227 IsOpen t \u2227 (\u03c6\u2080, \u03c8\u2080) \u2208 t\n[PROOFSTEP]\nuse{\u03c6 : C(\u03b1, \u03b2) | \u03c6 '' K \u2286 interior L} \u00d7\u02e2\n    {\u03c8 : C(\u03b2, \u03b3) | \u03c8 '' L \u2286 U}\n      -- porting note: typing hint `: \u03c6 '' K \u2286 interior L` wasn't previously required\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u03c6\u2080 : C(\u03b1, \u03b2)\n\u03c8\u2080 : C(\u03b2, \u03b3)\nH : (\u03c6\u2080, \u03c8\u2080) \u2208 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U\nL : Set \u03b2\nhL : IsCompact L\nhKL : \u03c6\u2080.toFun '' K \u2286 interior L\nhLU : L \u2286 \u03c8\u2080.toFun \u207b\u00b9' U\n\u22a2 ({\u03c6 | \u2191\u03c6 '' K \u2286 interior L} \u00d7\u02e2 {\u03c8 | \u2191\u03c8 '' L \u2286 U} \u2286 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U) \u2227\n    IsOpen ({\u03c6 | \u2191\u03c6 '' K \u2286 interior L} \u00d7\u02e2 {\u03c8 | \u2191\u03c8 '' L \u2286 U}) \u2227\n      (\u03c6\u2080, \u03c8\u2080) \u2208 {\u03c6 | \u2191\u03c6 '' K \u2286 interior L} \u00d7\u02e2 {\u03c8 | \u2191\u03c8 '' L \u2286 U}\n[PROOFSTEP]\nuse fun \u27e8\u03c6, \u03c8\u27e9 \u27e8(h\u03c6 : \u03c6 '' K \u2286 interior L), h\u03c8\u27e9 => subset_trans h\u03c6 (interior_subset.trans <| image_subset_iff.mp h\u03c8)\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u03c6\u2080 : C(\u03b1, \u03b2)\n\u03c8\u2080 : C(\u03b2, \u03b3)\nH : (\u03c6\u2080, \u03c8\u2080) \u2208 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U\nL : Set \u03b2\nhL : IsCompact L\nhKL : \u03c6\u2080.toFun '' K \u2286 interior L\nhLU : L \u2286 \u03c8\u2080.toFun \u207b\u00b9' U\n\u22a2 IsOpen ({\u03c6 | \u2191\u03c6 '' K \u2286 interior L} \u00d7\u02e2 {\u03c8 | \u2191\u03c8 '' L \u2286 U}) \u2227 (\u03c6\u2080, \u03c8\u2080) \u2208 {\u03c6 | \u2191\u03c6 '' K \u2286 interior L} \u00d7\u02e2 {\u03c8 | \u2191\u03c8 '' L \u2286 U}\n[PROOFSTEP]\nuse(ContinuousMap.isOpen_gen hK isOpen_interior).prod (ContinuousMap.isOpen_gen hL hU)\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ng : C(\u03b2, \u03b3)\nf : C(\u03b1, \u03b2)\ninst\u271d : LocallyCompactSpace \u03b2\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b3\nhU : IsOpen U\n\u03c6\u2080 : C(\u03b1, \u03b2)\n\u03c8\u2080 : C(\u03b2, \u03b3)\nH : (\u03c6\u2080, \u03c8\u2080) \u2208 fun x => \u2191x.fst '' K \u2286 \u2191x.snd \u207b\u00b9' U\nL : Set \u03b2\nhL : IsCompact L\nhKL : \u03c6\u2080.toFun '' K \u2286 interior L\nhLU : L \u2286 \u03c8\u2080.toFun \u207b\u00b9' U\n\u22a2 (\u03c6\u2080, \u03c8\u2080) \u2208 {\u03c6 | \u2191\u03c6 '' K \u2286 interior L} \u00d7\u02e2 {\u03c8 | \u2191\u03c8 '' L \u2286 U}\n[PROOFSTEP]\nexact mem_prod.mpr \u27e8hKL, image_subset_iff.mpr hLU\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace \u03b1\nx\u271d : C(\u03b1, \u03b2) \u00d7 \u03b1\nn : Set \u03b2\nf : C(\u03b1, \u03b2)\nx : \u03b1\nhn : n \u2208 \ud835\udcdd (\u2191(f, x).fst (f, x).snd)\nv : Set \u03b2\nvn : v \u2286 n\nvo : IsOpen v\nfxv : \u2191(f, x).fst (f, x).snd \u2208 v\nthis\u271d\u00b2 : v \u2208 \ud835\udcdd (\u2191f x)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nsv : s \u2286 \u2191f \u207b\u00b9' v\nsc : IsCompact s\nu : Set \u03b1\nus : u \u2286 s\nuo : IsOpen u\nxu : x \u2208 u\nw : Set (C(\u03b1, \u03b2) \u00d7 \u03b1) := CompactOpen.gen s v \u00d7\u02e2 u\nthis\u271d\u00b9 : w \u2286 (fun p => \u2191p.fst p.snd) \u207b\u00b9' n\nthis\u271d : IsOpen w\nthis : (f, x) \u2208 w\n\u22a2 w \u2286 (fun p => \u2191p.fst p.snd) \u207b\u00b9' n\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace \u03b1\nx\u271d : C(\u03b1, \u03b2) \u00d7 \u03b1\nn : Set \u03b2\nf : C(\u03b1, \u03b2)\nx : \u03b1\nhn : n \u2208 \ud835\udcdd (\u2191(f, x).fst (f, x).snd)\nv : Set \u03b2\nvn : v \u2286 n\nvo : IsOpen v\nfxv : \u2191(f, x).fst (f, x).snd \u2208 v\nthis\u271d\u00b2 : v \u2208 \ud835\udcdd (\u2191f x)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nsv : s \u2286 \u2191f \u207b\u00b9' v\nsc : IsCompact s\nu : Set \u03b1\nus : u \u2286 s\nuo : IsOpen u\nxu : x \u2208 u\nw : Set (C(\u03b1, \u03b2) \u00d7 \u03b1) := CompactOpen.gen s v \u00d7\u02e2 u\nthis\u271d\u00b9 : w \u2286 (fun p => \u2191p.fst p.snd) \u207b\u00b9' n\nthis\u271d : IsOpen w\nthis : (f, x) \u2208 w\n\u22a2 IsOpen w\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace \u03b1\nx\u271d : C(\u03b1, \u03b2) \u00d7 \u03b1\nn : Set \u03b2\nf : C(\u03b1, \u03b2)\nx : \u03b1\nhn : n \u2208 \ud835\udcdd (\u2191(f, x).fst (f, x).snd)\nv : Set \u03b2\nvn : v \u2286 n\nvo : IsOpen v\nfxv : \u2191(f, x).fst (f, x).snd \u2208 v\nthis\u271d\u00b2 : v \u2208 \ud835\udcdd (\u2191f x)\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nsv : s \u2286 \u2191f \u207b\u00b9' v\nsc : IsCompact s\nu : Set \u03b1\nus : u \u2286 s\nuo : IsOpen u\nxu : x \u2208 u\nw : Set (C(\u03b1, \u03b2) \u00d7 \u03b1) := CompactOpen.gen s v \u00d7\u02e2 u\nthis\u271d\u00b9 : w \u2286 (fun p => \u2191p.fst p.snd) \u207b\u00b9' n\nthis\u271d : IsOpen w\nthis : (f, x) \u2208 w\n\u22a2 (f, x) \u2208 w\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\na : \u03b1\n\u22a2 Continuous fun f => \u2191f a\n[PROOFSTEP]\nrefine continuous_def.2 fun U hU \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\na : \u03b1\nU : Set ((fun x => \u03b2) a)\nhU : IsOpen U\n\u22a2 IsOpen ((fun f => \u2191f a) \u207b\u00b9' U)\n[PROOFSTEP]\nconvert ContinuousMap.isOpen_gen (isCompact_singleton (a := a)) hU using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\na : \u03b1\nU : Set ((fun x => \u03b2) a)\nhU : IsOpen U\n\u22a2 (fun f => \u2191f a) \u207b\u00b9' U = CompactOpen.gen {a} U\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\na : \u03b1\nU : Set ((fun x => \u03b2) a)\nhU : IsOpen U\nx\u271d : C(\u03b1, \u03b2)\n\u22a2 x\u271d \u2208 (fun f => \u2191f a) \u207b\u00b9' U \u2194 x\u271d \u2208 CompactOpen.gen {a} U\n[PROOFSTEP]\nsimp [CompactOpen.gen]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\n\u22a2 compactOpen \u2264 TopologicalSpace.induced (restrict s) compactOpen\n[PROOFSTEP]\nsimp only [induced_generateFrom_eq, ContinuousMap.compactOpen]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\n\u22a2 TopologicalSpace.generateFrom {m | \u2203 s x u x, m = CompactOpen.gen s u} \u2264\n    TopologicalSpace.generateFrom (preimage (restrict s) '' {m | \u2203 s_1 x u x, m = CompactOpen.gen s_1 u})\n[PROOFSTEP]\napply TopologicalSpace.generateFrom_anti\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\n\u22a2 preimage (restrict s) '' {m | \u2203 s_1 x u x, m = CompactOpen.gen s_1 u} \u2286 {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\nrintro b \u27e8a, \u27e8c, hc, u, hu, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nc : Set \u2191s\nhc : IsCompact c\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 restrict s \u207b\u00b9' CompactOpen.gen c u \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\nrefine' \u27e8(\u2191) '' c, hc.image continuous_subtype_val, u, hu, _\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nc : Set \u2191s\nhc : IsCompact c\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 restrict s \u207b\u00b9' CompactOpen.gen c u = CompactOpen.gen (Subtype.val '' c) u\n[PROOFSTEP]\next f\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nc : Set \u2191s\nhc : IsCompact c\nu : Set \u03b2\nhu : IsOpen u\nf : C(\u03b1, \u03b2)\n\u22a2 f \u2208 restrict s \u207b\u00b9' CompactOpen.gen c u \u2194 f \u2208 CompactOpen.gen (Subtype.val '' c) u\n[PROOFSTEP]\nsimp only [CompactOpen.gen, mem_setOf_eq, mem_preimage, ContinuousMap.coe_restrict]\n[GOAL]\ncase h.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nc : Set \u2191s\nhc : IsCompact c\nu : Set \u03b2\nhu : IsOpen u\nf : C(\u03b1, \u03b2)\n\u22a2 (fun a => (\u2191f \u2218 Subtype.val) a) '' c \u2286 u \u2194 (fun a => \u2191f a) '' (Subtype.val '' c) \u2286 u\n[PROOFSTEP]\nrw [image_comp f ((\u2191) : s \u2192 \u03b1)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 compactOpen = \u2a05 (s : Set \u03b1) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 compactOpen \u2264 \u2a05 (s : Set \u03b1) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen\n[PROOFSTEP]\nrefine' le_iInf\u2082 _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 \u2200 (i : Set \u03b1), IsCompact i \u2192 compactOpen \u2264 TopologicalSpace.induced (restrict i) compactOpen\n[PROOFSTEP]\nexact fun s _ => compactOpen_le_induced s\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 \u2a05 (s : Set \u03b1) (_ : IsCompact s), TopologicalSpace.induced (restrict s) compactOpen \u2264 compactOpen\n[PROOFSTEP]\nsimp only [\u2190 generateFrom_iUnion, induced_generateFrom_eq, ContinuousMap.compactOpen]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 TopologicalSpace.generateFrom\n      (\u22c3 (i : Set \u03b1) (_ : IsCompact i), preimage (restrict i) '' {m | \u2203 s x u x, m = CompactOpen.gen s u}) \u2264\n    TopologicalSpace.generateFrom {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\napply TopologicalSpace.generateFrom_anti\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 {m | \u2203 s x u x, m = CompactOpen.gen s u} \u2286\n    \u22c3 (i : Set \u03b1) (_ : IsCompact i), preimage (restrict i) '' {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\nrintro _ \u27e8s, hs, u, hu, rfl\u27e9\n[GOAL]\ncase refine'_2.h.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 CompactOpen.gen s u \u2208\n    \u22c3 (i : Set \u03b1) (_ : IsCompact i), preimage (restrict i) '' {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\nrw [mem_iUnion\u2082]\n[GOAL]\ncase refine'_2.h.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 \u2203 i j, CompactOpen.gen s u \u2208 preimage (restrict i) '' {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\nrefine' \u27e8s, hs, _, \u27e8univ, isCompact_iff_isCompact_univ.mp hs, u, hu, rfl\u27e9, _\u27e9\n[GOAL]\ncase refine'_2.h.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\n\u22a2 restrict s \u207b\u00b9' CompactOpen.gen univ u = CompactOpen.gen s u\n[PROOFSTEP]\next f\n[GOAL]\ncase refine'_2.h.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\nf : C(\u03b1, \u03b2)\n\u22a2 f \u2208 restrict s \u207b\u00b9' CompactOpen.gen univ u \u2194 f \u2208 CompactOpen.gen s u\n[PROOFSTEP]\nsimp only [CompactOpen.gen, mem_setOf_eq, mem_preimage, ContinuousMap.coe_restrict]\n[GOAL]\ncase refine'_2.h.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\nf : C(\u03b1, \u03b2)\n\u22a2 (fun a => (\u2191f \u2218 Subtype.val) a) '' univ \u2286 u \u2194 (fun a => \u2191f a) '' s \u2286 u\n[PROOFSTEP]\nrw [image_comp f ((\u2191) : s \u2192 \u03b1)]\n[GOAL]\ncase refine'_2.h.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nhs : IsCompact s\nu : Set \u03b2\nhu : IsOpen u\nf : C(\u03b1, \u03b2)\n\u22a2 \u2191f '' (Subtype.val '' univ) \u2286 u \u2194 (fun a => \u2191f a) '' s \u2286 u\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\n\u22a2 Continuous fun F => restrict s F\n[PROOFSTEP]\nrw [continuous_iff_le_induced]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\n\u22a2 compactOpen \u2264 TopologicalSpace.induced (fun F => restrict s F) compactOpen\n[PROOFSTEP]\nexact compactOpen_le_induced s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : C(\u03b1, \u03b2)\n\u22a2 \ud835\udcdd f = \u2a05 (s : Set \u03b1) (_ : IsCompact s), Filter.comap (restrict s) (\ud835\udcdd (restrict s f))\n[PROOFSTEP]\nrw [compactOpen_eq_sInf_induced]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : C(\u03b1, \u03b2)\n\u22a2 \ud835\udcdd f = \u2a05 (s : Set \u03b1) (_ : IsCompact s), Filter.comap (restrict s) (\ud835\udcdd (restrict s f))\n[PROOFSTEP]\nsimp [nhds_iInf, nhds_induced]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_4\nl : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 Filter.Tendsto F l (\ud835\udcdd f) \u2194\n    \u2200 (s : Set \u03b1), IsCompact s \u2192 Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (restrict s f))\n[PROOFSTEP]\nrw [compactOpen_eq_sInf_induced]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u03b9 : Type u_4\nl : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 Filter.Tendsto F l (\ud835\udcdd f) \u2194\n    \u2200 (s : Set \u03b1), IsCompact s \u2192 Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (restrict s f))\n[PROOFSTEP]\nsimp [nhds_iInf, nhds_induced, Filter.tendsto_comap_iff, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\n\u22a2 (\u2203 f, Filter.Tendsto F l (\ud835\udcdd f)) \u2194 \u2200 (s : Set \u03b1), IsCompact s \u2192 \u2203 f, Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd f)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\n\u22a2 (\u2203 f, Filter.Tendsto F l (\ud835\udcdd f)) \u2192 \u2200 (s : Set \u03b1), IsCompact s \u2192 \u2203 f, Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd f)\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9 s _\n[GOAL]\ncase mp.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : C(\u03b1, \u03b2)\nhf : Filter.Tendsto F l (\ud835\udcdd f)\ns : Set \u03b1\nhs\u271d : IsCompact s\n\u22a2 \u2203 f, Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd f)\n[PROOFSTEP]\nexact \u27e8f.restrict s, tendsto_compactOpen_restrict hf s\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\n\u22a2 (\u2200 (s : Set \u03b1), IsCompact s \u2192 \u2203 f, Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd f)) \u2192 \u2203 f, Filter.Tendsto F l (\ud835\udcdd f)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh : \u2200 (s : Set \u03b1), IsCompact s \u2192 \u2203 f, Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd f)\n\u22a2 \u2203 f, Filter.Tendsto F l (\ud835\udcdd f)\n[PROOFSTEP]\nchoose f hf using h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\n\u22a2 \u2203 f, Filter.Tendsto F l (\ud835\udcdd f)\n[PROOFSTEP]\nhave h :\n  \u2200 (s\u2081) (hs\u2081 : IsCompact s\u2081) (s\u2082) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    f s\u2081 hs\u2081 \u27e8x, hxs\u2081\u27e9 = f s\u2082 hs\u2082 \u27e8x, hxs\u2082\u27e9 :=\n  by\n  rintro s\u2081 hs\u2081 s\u2082 hs\u2082 x hxs\u2081 hxs\u2082\n  haveI := isCompact_iff_compactSpace.mp hs\u2081\n  haveI := isCompact_iff_compactSpace.mp hs\u2082\n  have h\u2081 := (continuous_eval_const (\u27e8x, hxs\u2081\u27e9 : s\u2081)).continuousAt.tendsto.comp (hf s\u2081 hs\u2081)\n  have h\u2082 := (continuous_eval_const (\u27e8x, hxs\u2082\u27e9 : s\u2082)).continuousAt.tendsto.comp (hf s\u2082 hs\u2082)\n  exact tendsto_nhds_unique h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\n\u22a2 \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n[PROOFSTEP]\nrintro s\u2081 hs\u2081 s\u2082 hs\u2082 x hxs\u2081 hxs\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\ns\u2081 : Set \u03b1\nhs\u2081 : IsCompact s\u2081\ns\u2082 : Set \u03b1\nhs\u2082 : IsCompact s\u2082\nx : \u03b1\nhxs\u2081 : x \u2208 s\u2081\nhxs\u2082 : x \u2208 s\u2082\n\u22a2 \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n[PROOFSTEP]\nhaveI := isCompact_iff_compactSpace.mp hs\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\ns\u2081 : Set \u03b1\nhs\u2081 : IsCompact s\u2081\ns\u2082 : Set \u03b1\nhs\u2082 : IsCompact s\u2082\nx : \u03b1\nhxs\u2081 : x \u2208 s\u2081\nhxs\u2082 : x \u2208 s\u2082\nthis : CompactSpace \u2191s\u2081\n\u22a2 \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n[PROOFSTEP]\nhaveI := isCompact_iff_compactSpace.mp hs\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\ns\u2081 : Set \u03b1\nhs\u2081 : IsCompact s\u2081\ns\u2082 : Set \u03b1\nhs\u2082 : IsCompact s\u2082\nx : \u03b1\nhxs\u2081 : x \u2208 s\u2081\nhxs\u2082 : x \u2208 s\u2082\nthis\u271d : CompactSpace \u2191s\u2081\nthis : CompactSpace \u2191s\u2082\n\u22a2 \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n[PROOFSTEP]\nhave h\u2081 := (continuous_eval_const (\u27e8x, hxs\u2081\u27e9 : s\u2081)).continuousAt.tendsto.comp (hf s\u2081 hs\u2081)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\ns\u2081 : Set \u03b1\nhs\u2081 : IsCompact s\u2081\ns\u2082 : Set \u03b1\nhs\u2082 : IsCompact s\u2082\nx : \u03b1\nhxs\u2081 : x \u2208 s\u2081\nhxs\u2082 : x \u2208 s\u2082\nthis\u271d : CompactSpace \u2191s\u2081\nthis : CompactSpace \u2191s\u2082\nh\u2081 :\n  Filter.Tendsto ((fun f => \u2191f { val := x, property := hxs\u2081 }) \u2218 fun i => restrict s\u2081 (F i)) l\n    (\ud835\udcdd (\u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 }))\n\u22a2 \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n[PROOFSTEP]\nhave h\u2082 := (continuous_eval_const (\u27e8x, hxs\u2082\u27e9 : s\u2082)).continuousAt.tendsto.comp (hf s\u2082 hs\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\ns\u2081 : Set \u03b1\nhs\u2081 : IsCompact s\u2081\ns\u2082 : Set \u03b1\nhs\u2082 : IsCompact s\u2082\nx : \u03b1\nhxs\u2081 : x \u2208 s\u2081\nhxs\u2082 : x \u2208 s\u2082\nthis\u271d : CompactSpace \u2191s\u2081\nthis : CompactSpace \u2191s\u2082\nh\u2081 :\n  Filter.Tendsto ((fun f => \u2191f { val := x, property := hxs\u2081 }) \u2218 fun i => restrict s\u2081 (F i)) l\n    (\ud835\udcdd (\u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 }))\nh\u2082 :\n  Filter.Tendsto ((fun f => \u2191f { val := x, property := hxs\u2082 }) \u2218 fun i => restrict s\u2082 (F i)) l\n    (\ud835\udcdd (\u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }))\n\u22a2 \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n[PROOFSTEP]\nexact tendsto_nhds_unique h\u2081 h\u2082\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n\u22a2 \u2203 f, Filter.Tendsto F l (\ud835\udcdd f)\n[PROOFSTEP]\nhave hs : \u2200 x : \u03b1, \u2203 (s : _), IsCompact s \u2227 s \u2208 \ud835\udcdd x := by\n  intro x\n  obtain \u27e8s, hs, hs'\u27e9 := exists_compact_mem_nhds x\n  exact \u27e8s, hs, hs'\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\n\u22a2 \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \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\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nx : \u03b1\n\u22a2 \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\n[PROOFSTEP]\nobtain \u27e8s, hs, hs'\u27e9 := exists_compact_mem_nhds x\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nx : \u03b1\ns : Set \u03b1\nhs : IsCompact s\nhs' : s \u2208 \ud835\udcdd x\n\u22a2 \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact \u27e8s, hs, hs'\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nhs : \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\n\u22a2 \u2203 f, Filter.Tendsto F l (\ud835\udcdd f)\n[PROOFSTEP]\nrefine \u27e8liftCover' _ _ h hs, ?_\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nhs : \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\n\u22a2 Filter.Tendsto F l\n    (\ud835\udcdd\n      (liftCover'\n        (fun s => \u2200 \u2983f : Filter \u03b1\u2984 [inst : Filter.NeBot f], f \u2264 Filter.principal s \u2192 \u2203 a, a \u2208 s \u2227 ClusterPt a f)\n        (fun s hs => f s hs) h hs))\n[PROOFSTEP]\nrw [tendsto_compactOpen_iff_forall]\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nhs : \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\n\u22a2 \u2200 (s : Set \u03b1),\n    IsCompact s \u2192\n      Filter.Tendsto (fun i => restrict s (F i)) l\n        (\ud835\udcdd\n          (restrict s\n            (liftCover'\n              (fun s => \u2200 \u2983f : Filter \u03b1\u2984 [inst : Filter.NeBot f], f \u2264 Filter.principal s \u2192 \u2203 a, a \u2208 s \u2227 ClusterPt a f)\n              (fun s hs => f s hs) h hs)))\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nhs\u271d : \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 Filter.Tendsto (fun i => restrict s (F i)) l\n    (\ud835\udcdd\n      (restrict s\n        (liftCover'\n          (fun s => \u2200 \u2983f : Filter \u03b1\u2984 [inst : Filter.NeBot f], f \u2264 Filter.principal s \u2192 \u2203 a, a \u2208 s \u2227 ClusterPt a f)\n          (fun s hs => f s hs) h hs\u271d)))\n[PROOFSTEP]\nrw [liftCover_restrict']\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nhs\u271d : \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s ?m.52935))\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b2\n\u03b9 : Type u_4\nl : Filter \u03b9\ninst\u271d : Filter.NeBot l\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nf : (s : Set \u03b1) \u2192 IsCompact s \u2192 C(\u2191s, \u03b2)\nhf : \u2200 (s : Set \u03b1) (hs : IsCompact s), Filter.Tendsto (fun i => restrict s (F i)) l (\ud835\udcdd (f s hs))\nh :\n  \u2200 (s\u2081 : Set \u03b1) (hs\u2081 : IsCompact s\u2081) (s\u2082 : Set \u03b1) (hs\u2082 : IsCompact s\u2082) (x : \u03b1) (hxs\u2081 : x \u2208 s\u2081) (hxs\u2082 : x \u2208 s\u2082),\n    \u2191(f s\u2081 hs\u2081) { val := x, property := hxs\u2081 } = \u2191(f s\u2082 hs\u2082) { val := x, property := hxs\u2082 }\nhs\u271d : \u2200 (x : \u03b1), \u2203 s, IsCompact s \u2227 s \u2208 \ud835\udcdd x\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 s \u2208 fun s => \u2200 \u2983f : Filter \u03b1\u2984 [inst : Filter.NeBot f], f \u2264 Filter.principal s \u2192 \u2203 a, a \u2208 s \u2227 ClusterPt a f\n[PROOFSTEP]\nexact hf s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ny : \u03b2\ns : Set \u03b1\n\u22a2 \u2191(coev \u03b1 \u03b2 y) '' s = {y} \u00d7\u02e2 s\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\n\u22a2 \u2200 (s : Set C(\u03b1, \u03b2 \u00d7 \u03b1)), s \u2208 {m | \u2203 s x u x, m = CompactOpen.gen s u} \u2192 IsOpen (coev \u03b1 \u03b2 \u207b\u00b9' s)\n[PROOFSTEP]\nrintro _ \u27e8s, sc, u, uo, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\n\u22a2 IsOpen (coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u)\n[PROOFSTEP]\nrw [isOpen_iff_forall_mem_open]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\n\u22a2 \u2200 (x : \u03b2), x \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u \u2192 \u2203 t, t \u2286 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u \u2227 IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\n\u22a2 \u2203 t, t \u2286 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u \u2227 IsOpen t \u2227 y \u2208 t\n[PROOFSTEP]\nhave hy' : (\u2191(coev \u03b1 \u03b2 y) '' s \u2286 u) := hy\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy' : \u2191(coev \u03b1 \u03b2 y) '' s \u2286 u\n\u22a2 \u2203 t, t \u2286 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u \u2227 IsOpen t \u2227 y \u2208 t\n[PROOFSTEP]\nrw [image_coev s] at hy' \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy' : {y} \u00d7\u02e2 s \u2286 u\n\u22a2 \u2203 t, t \u2286 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u \u2227 IsOpen t \u2227 y \u2208 t\n[PROOFSTEP]\nrcases generalized_tube_lemma isCompact_singleton sc uo hy' with \u27e8v, w, vo, _, yv, sw, vwu\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\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy' : {y} \u00d7\u02e2 s \u2286 u\nv : Set \u03b2\nw : Set \u03b1\nvo : IsOpen v\nleft\u271d : IsOpen w\nyv : {y} \u2286 v\nsw : s \u2286 w\nvwu : v \u00d7\u02e2 w \u2286 u\n\u22a2 \u2203 t, t \u2286 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u \u2227 IsOpen t \u2227 y \u2208 t\n[PROOFSTEP]\nrefine' \u27e8v, _, vo, singleton_subset_iff.mp yv\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\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy' : {y} \u00d7\u02e2 s \u2286 u\nv : Set \u03b2\nw : Set \u03b1\nvo : IsOpen v\nleft\u271d : IsOpen w\nyv : {y} \u2286 v\nsw : s \u2286 w\nvwu : v \u00d7\u02e2 w \u2286 u\n\u22a2 v \u2286 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\n[PROOFSTEP]\nintro y' hy'\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\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy'\u271d : {y} \u00d7\u02e2 s \u2286 u\nv : Set \u03b2\nw : Set \u03b1\nvo : IsOpen v\nleft\u271d : IsOpen w\nyv : {y} \u2286 v\nsw : s \u2286 w\nvwu : v \u00d7\u02e2 w \u2286 u\ny' : \u03b2\nhy' : y' \u2208 v\n\u22a2 y' \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\n[PROOFSTEP]\nchange coev \u03b1 \u03b2 y' '' s \u2286 u\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\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy'\u271d : {y} \u00d7\u02e2 s \u2286 u\nv : Set \u03b2\nw : Set \u03b1\nvo : IsOpen v\nleft\u271d : IsOpen w\nyv : {y} \u2286 v\nsw : s \u2286 w\nvwu : v \u00d7\u02e2 w \u2286 u\ny' : \u03b2\nhy' : y' \u2208 v\n\u22a2 \u2191(coev \u03b1 \u03b2 y') '' s \u2286 u\n[PROOFSTEP]\nrw [image_coev s]\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\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ns : Set \u03b1\nsc : IsCompact s\nu : Set (\u03b2 \u00d7 \u03b1)\nuo : IsOpen u\ny : \u03b2\nhy : y \u2208 coev \u03b1 \u03b2 \u207b\u00b9' CompactOpen.gen s u\nhy'\u271d : {y} \u00d7\u02e2 s \u2286 u\nv : Set \u03b2\nw : Set \u03b1\nvo : IsOpen v\nleft\u271d : IsOpen w\nyv : {y} \u2286 v\nsw : s \u2286 w\nvwu : v \u00d7\u02e2 w \u2286 u\ny' : \u03b2\nhy' : y' \u2208 v\n\u22a2 {y'} \u00d7\u02e2 s \u2286 u\n[PROOFSTEP]\nexact (prod_mono (singleton_subset_iff.mpr hy') sw).trans vwu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : C(\u03b1 \u00d7 \u03b2, \u03b3)\n\u22a2 curry' f = comp f \u2218 coev \u03b2 \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : C(\u03b1 \u00d7 \u03b2, \u03b3)\nx\u271d : \u03b1\na\u271d : \u03b2\n\u22a2 \u2191(curry' f x\u271d) a\u271d = \u2191((comp f \u2218 coev \u03b2 \u03b1) x\u271d) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace (\u03b1 \u00d7 \u03b2)\n\u22a2 Continuous curry\n[PROOFSTEP]\napply continuous_of_continuous_uncurry\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace (\u03b1 \u00d7 \u03b2)\n\u22a2 Continuous (Function.uncurry fun x y => \u2191(curry x) y)\n[PROOFSTEP]\napply continuous_of_continuous_uncurry\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace (\u03b1 \u00d7 \u03b2)\n\u22a2 Continuous (Function.uncurry fun x y => \u2191(Function.uncurry (fun x y => \u2191(curry x) y) x) y)\n[PROOFSTEP]\nrw [\u2190 (Homeomorph.prodAssoc _ _ _).symm.comp_continuous_iff']\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : LocallyCompactSpace (\u03b1 \u00d7 \u03b2)\n\u22a2 Continuous\n    ((Function.uncurry fun x y => \u2191(Function.uncurry (fun x y => \u2191(curry x) y) x) y) \u2218\n      \u2191(Homeomorph.symm (Homeomorph.prodAssoc C(\u03b1 \u00d7 \u03b2, \u03b3) \u03b1 \u03b2)))\n[PROOFSTEP]\nexact continuous_eval'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\n\u22a2 Continuous uncurry\n[PROOFSTEP]\napply continuous_of_continuous_uncurry\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\n\u22a2 Continuous (Function.uncurry fun x y => \u2191(uncurry x) y)\n[PROOFSTEP]\nrw [\u2190 (Homeomorph.prodAssoc _ _ _).comp_continuous_iff']\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\n\u22a2 Continuous ((Function.uncurry fun x y => \u2191(uncurry x) y) \u2218 \u2191(Homeomorph.prodAssoc C(\u03b1, C(\u03b2, \u03b3)) \u03b1 \u03b2))\n[PROOFSTEP]\napply continuous_eval'.comp (continuous_eval'.prod_map continuous_id)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\n\u22a2 Function.LeftInverse uncurry ContinuousMap.curry\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nx\u271d : C(\u03b1 \u00d7 \u03b2, \u03b3)\n\u22a2 uncurry (ContinuousMap.curry x\u271d) = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nx\u271d : C(\u03b1 \u00d7 \u03b2, \u03b3)\na\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 \u2191(uncurry (ContinuousMap.curry x\u271d)) a\u271d = \u2191x\u271d a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\n\u22a2 Function.RightInverse uncurry ContinuousMap.curry\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nx\u271d : C(\u03b1, C(\u03b2, \u03b3))\n\u22a2 ContinuousMap.curry (uncurry x\u271d) = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : LocallyCompactSpace \u03b1\ninst\u271d : LocallyCompactSpace \u03b2\nx\u271d : C(\u03b1, C(\u03b2, \u03b3))\na\u271d\u00b9 : \u03b1\na\u271d : \u03b2\n\u22a2 \u2191(\u2191(ContinuousMap.curry (uncurry x\u271d)) a\u271d\u00b9) a\u271d = \u2191(\u2191x\u271d a\u271d\u00b9) a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : Unique \u03b1\nf : C(\u03b1, \u03b2)\n\u22a2 const \u03b1 ((fun f => \u2191f default) f) = f\n[PROOFSTEP]\next a\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : Unique \u03b1\nf : C(\u03b1, \u03b2)\na : \u03b1\n\u22a2 \u2191(const \u03b1 ((fun f => \u2191f default) f)) a = \u2191f a\n[PROOFSTEP]\nrw [Unique.eq_default a]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : Unique \u03b1\nf : C(\u03b1, \u03b2)\na : \u03b1\n\u22a2 \u2191(const \u03b1 ((fun f => \u2191f default) f)) default = \u2191f default\n[PROOFSTEP]\nrfl\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\n\u22a2 Continuous g\n[PROOFSTEP]\nlet Gf : C(X\u2080, C(Y, Z)) := ContinuousMap.curry \u27e8_, hg\u27e9\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\n\u22a2 Continuous g\n[PROOFSTEP]\nhave h : \u2200 x : X, Continuous fun y => g (x, y) := by\n  intro x\n  obtain \u27e8x\u2080, rfl\u27e9 := hf.surjective x\n  exact (Gf x\u2080).continuous\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\n\u22a2 \u2200 (x : X), Continuous fun y => g (x, y)\n[PROOFSTEP]\nintro x\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nx : X\n\u22a2 Continuous fun y => g (x, y)\n[PROOFSTEP]\nobtain \u27e8x\u2080, rfl\u27e9 := hf.surjective x\n[GOAL]\ncase intro\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nx\u2080 : X\u2080\n\u22a2 Continuous fun y => g (f x\u2080, y)\n[PROOFSTEP]\nexact (Gf x\u2080).continuous\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : \u2200 (x : X), Continuous fun y => g (x, y)\n\u22a2 Continuous g\n[PROOFSTEP]\nlet G : X \u2192 C(Y, Z) := fun x => \u27e8_, h x\u27e9\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : \u2200 (x : X), Continuous fun y => g (x, y)\nG : X \u2192 C(Y, Z) := fun x => mk fun y => g (x, y)\n\u22a2 Continuous g\n[PROOFSTEP]\nhave : Continuous G := by\n  rw [hf.continuous_iff]\n  exact Gf.continuous\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : \u2200 (x : X), Continuous fun y => g (x, y)\nG : X \u2192 C(Y, Z) := fun x => mk fun y => g (x, y)\n\u22a2 Continuous G\n[PROOFSTEP]\nrw [hf.continuous_iff]\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : \u2200 (x : X), Continuous fun y => g (x, y)\nG : X \u2192 C(Y, Z) := fun x => mk fun y => g (x, y)\n\u22a2 Continuous (G \u2218 f)\n[PROOFSTEP]\nexact Gf.continuous\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : X \u00d7 Y \u2192 Z\nhg : Continuous fun p => g (f p.fst, p.snd)\nGf : C(X\u2080, C(Y, Z)) := curry (mk fun p => g (f p.fst, p.snd))\nh : \u2200 (x : X), Continuous fun y => g (x, y)\nG : X \u2192 C(Y, Z) := fun x => mk fun y => g (x, y)\nthis : Continuous G\n\u22a2 Continuous g\n[PROOFSTEP]\nexact ContinuousMap.continuous_uncurry_of_continuous \u27e8G, this\u27e9\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : Y \u00d7 X \u2192 Z\nhg : Continuous fun p => g (p.fst, f p.snd)\n\u22a2 Continuous g\n[PROOFSTEP]\nhave : Continuous fun p : X\u2080 \u00d7 Y => g ((Prod.swap p).1, f (Prod.swap p).2) := hg.comp continuous_swap\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : Y \u00d7 X \u2192 Z\nhg : Continuous fun p => g (p.fst, f p.snd)\nthis : Continuous fun p => g ((Prod.swap p).fst, f (Prod.swap p).snd)\n\u22a2 Continuous g\n[PROOFSTEP]\nhave : Continuous fun p : X\u2080 \u00d7 Y => (g \u2218 Prod.swap) (f p.1, p.2) := this\n[GOAL]\nX\u2080 : Type u_1\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\u2080\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : LocallyCompactSpace Y\nf : X\u2080 \u2192 X\nhf : QuotientMap f\ng : Y \u00d7 X \u2192 Z\nhg : Continuous fun p => g (p.fst, f p.snd)\nthis\u271d : Continuous fun p => g ((Prod.swap p).fst, f (Prod.swap p).snd)\nthis : Continuous fun p => (g \u2218 Prod.swap) (f p.fst, p.snd)\n\u22a2 Continuous g\n[PROOFSTEP]\nexact (hf.continuous_lift_prod_left this).comp continuous_swap\n", "meta": {"mathlib_filename": "Mathlib.Topology.CompactOpen", "llama_tokens": 27184, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519527982093666, "lm_q2_score": 0.6619228691808012, "lm_q1q2_score": 0.5639270405973561}}
{"text": "[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 \u2045x\u271d\u00b2 + x\u271d\u00b9, x\u271d\u2046 = \u2045x\u271d\u00b2, x\u271d\u2046 + \u2045x\u271d\u00b9, x\u271d\u2046\n[PROOFSTEP]\nsimp only [Ring.lie_def, right_distrib, left_distrib]\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d - (x\u271d * x\u271d\u00b2 + x\u271d * x\u271d\u00b9) = x\u271d\u00b2 * x\u271d - x\u271d * x\u271d\u00b2 + (x\u271d\u00b9 * x\u271d - x\u271d * x\u271d\u00b9)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 x\u271d\u00b2 * x\u271d + x\u271d\u00b9 * x\u271d - (x\u271d * x\u271d\u00b2 + x\u271d * x\u271d\u00b9) = x\u271d\u00b2 * x\u271d - x\u271d * x\u271d\u00b2 + (x\u271d\u00b9 * x\u271d - x\u271d * x\u271d\u00b9)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 \u2045x\u271d\u00b2, x\u271d\u00b9 + x\u271d\u2046 = \u2045x\u271d\u00b2, x\u271d\u00b9\u2046 + \u2045x\u271d\u00b2, x\u271d\u2046\n[PROOFSTEP]\nsimp only [Ring.lie_def, right_distrib, left_distrib]\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d - (x\u271d\u00b9 * x\u271d\u00b2 + x\u271d * x\u271d\u00b2) = x\u271d\u00b2 * x\u271d\u00b9 - x\u271d\u00b9 * x\u271d\u00b2 + (x\u271d\u00b2 * x\u271d - x\u271d * x\u271d\u00b2)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 x\u271d\u00b2 * x\u271d\u00b9 + x\u271d\u00b2 * x\u271d - (x\u271d\u00b9 * x\u271d\u00b2 + x\u271d * x\u271d\u00b2) = x\u271d\u00b2 * x\u271d\u00b9 - x\u271d\u00b9 * x\u271d\u00b2 + (x\u271d\u00b2 * x\u271d - x\u271d * x\u271d\u00b2)\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type v\ninst\u271d : Ring A\n\u22a2 \u2200 (x : A), \u2045x, x\u2046 = 0\n[PROOFSTEP]\nsimp only [Ring.lie_def, forall_const, sub_self]\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 \u2045x\u271d\u00b2, \u2045x\u271d\u00b9, x\u271d\u2046\u2046 = \u2045\u2045x\u271d\u00b2, x\u271d\u00b9\u2046, x\u271d\u2046 + \u2045x\u271d\u00b9, \u2045x\u271d\u00b2, x\u271d\u2046\u2046\n[PROOFSTEP]\nsimp only [Ring.lie_def, mul_sub_left_distrib, mul_sub_right_distrib, mul_assoc]\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d) - x\u271d\u00b2 * (x\u271d * x\u271d\u00b9) - (x\u271d\u00b9 * (x\u271d * x\u271d\u00b2) - x\u271d * (x\u271d\u00b9 * x\u271d\u00b2)) =\n    x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d) - x\u271d\u00b9 * (x\u271d\u00b2 * x\u271d) - (x\u271d * (x\u271d\u00b2 * x\u271d\u00b9) - x\u271d * (x\u271d\u00b9 * x\u271d\u00b2)) +\n      (x\u271d\u00b9 * (x\u271d\u00b2 * x\u271d) - x\u271d\u00b9 * (x\u271d * x\u271d\u00b2) - (x\u271d\u00b2 * (x\u271d * x\u271d\u00b9) - x\u271d * (x\u271d\u00b2 * x\u271d\u00b9)))\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type v\ninst\u271d : Ring A\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : A\n\u22a2 x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d) - x\u271d\u00b2 * (x\u271d * x\u271d\u00b9) - (x\u271d\u00b9 * (x\u271d * x\u271d\u00b2) - x\u271d * (x\u271d\u00b9 * x\u271d\u00b2)) =\n    x\u271d\u00b2 * (x\u271d\u00b9 * x\u271d) - x\u271d\u00b9 * (x\u271d\u00b2 * x\u271d) - (x\u271d * (x\u271d\u00b2 * x\u271d\u00b9) - x\u271d * (x\u271d\u00b9 * x\u271d\u00b2)) +\n      (x\u271d\u00b9 * (x\u271d\u00b2 * x\u271d) - x\u271d\u00b9 * (x\u271d * x\u271d\u00b2) - (x\u271d\u00b2 * (x\u271d * x\u271d\u00b9) - x\u271d * (x\u271d\u00b2 * x\u271d\u00b9)))\n[PROOFSTEP]\nabel\n[GOAL]\nA : Type v\ninst\u271d\u00b2 : Ring A\nM : Type w\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module A M\n\u22a2 \u2200 (x y : A) (m : M), \u2045x, \u2045y, m\u2046\u2046 = \u2045\u2045x, y\u2046, m\u2046 + \u2045y, \u2045x, m\u2046\u2046\n[PROOFSTEP]\nsimp [LieRing.of_associative_ring_bracket, sub_smul, mul_smul, sub_add_cancel]\n[GOAL]\nA : Type v\ninst\u271d\u00b2 : Ring A\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra R A\nt : R\nx y : A\n\u22a2 \u2045x, t \u2022 y\u2046 = t \u2022 \u2045x, y\u2046\n[PROOFSTEP]\nrw [LieRing.of_associative_ring_bracket, LieRing.of_associative_ring_bracket, Algebra.mul_smul_comm,\n  Algebra.smul_mul_assoc, smul_sub]\n[GOAL]\nA : Type v\ninst\u271d\u2076 : Ring A\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Algebra R A\nB : Type w\nC : Type w\u2081\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring C\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : Algebra R C\nf : A \u2192\u2090[R] B\ng : B \u2192\u2090[R] C\nsrc\u271d : A \u2192\u2097[R] B := toLinearMap f\nx\u271d\u00b9 x\u271d : A\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\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) }.toAddHom\n      \u2045x\u271d\u00b9, x\u271d\u2046 =\n    \u2045AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\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) }.toAddHom\n        x\u271d\u00b9,\n      AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\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) }.toAddHom\n        x\u271d\u2046\n[PROOFSTEP]\nsimp [LieRing.of_associative_ring_bracket]\n[GOAL]\nA : Type v\ninst\u271d\u2076 : Ring A\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Algebra R A\nB : Type w\nC : Type w\u2081\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring C\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : Algebra R C\nf\u271d : A \u2192\u2090[R] B\ng\u271d : B \u2192\u2090[R] C\nf g : A \u2192\u2090[R] B\nh : toLieHom f = toLieHom g\n\u22a2 f = g\n[PROOFSTEP]\next a\n[GOAL]\ncase H\nA : Type v\ninst\u271d\u2076 : Ring A\nR : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : Algebra R A\nB : Type w\nC : Type w\u2081\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Ring C\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : Algebra R C\nf\u271d : A \u2192\u2090[R] B\ng\u271d : B \u2192\u2090[R] C\nf g : A \u2192\u2090[R] B\nh : toLieHom f = toLieHom g\na : A\n\u22a2 \u2191f a = \u2191g a\n[PROOFSTEP]\nexact LieHom.congr_fun h a\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\n\u22a2 (fun x =>\n        { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n      (x + y) =\n    (fun x =>\n          { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n        x +\n      (fun x =>\n          { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n        y\n[PROOFSTEP]\next m\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nm : M\n\u22a2 \u2191((fun x =>\n            { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n          (x + y))\n      m =\n    \u2191((fun x =>\n              { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n            x +\n          (fun x =>\n              { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n            y)\n      m\n[PROOFSTEP]\napply add_lie\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nt : R\nx : L\n\u22a2 AddHom.toFun\n      {\n        toFun := fun x =>\n          { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n        map_add' :=\n          (_ :\n            \u2200 (x y : L),\n              (fun x =>\n                    {\n                      toAddHom :=\n                        { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                  (x + y) =\n                (fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                    x +\n                  (fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                    y) }\n      (t \u2022 x) =\n    \u2191(RingHom.id R) t \u2022\n      AddHom.toFun\n        {\n          toFun := fun x =>\n            { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : L),\n                (fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                    (x + y) =\n                  (fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                      x +\n                    (fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                      y) }\n        x\n[PROOFSTEP]\next m\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nt : R\nx : L\nm : M\n\u22a2 \u2191(AddHom.toFun\n          {\n            toFun := fun x =>\n              { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n            map_add' :=\n              (_ :\n                \u2200 (x y : L),\n                  (fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                      (x + y) =\n                    (fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                        x +\n                      (fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                        y) }\n          (t \u2022 x))\n      m =\n    \u2191(\u2191(RingHom.id R) t \u2022\n          AddHom.toFun\n            {\n              toFun := fun x =>\n                { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n              map_add' :=\n                (_ :\n                  \u2200 (x y : L),\n                    (fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                        (x + y) =\n                      (fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                          x +\n                        (fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                          y) }\n            x)\n      m\n[PROOFSTEP]\napply smul_lie\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            {\n              toFun := fun x =>\n                { toAddHom := { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n              map_add' :=\n                (_ :\n                  \u2200 (x y : L),\n                    (fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                        (x + y) =\n                      (fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                          x +\n                        (fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                          y) },\n          map_smul' :=\n            (_ :\n              \u2200 (t : R) (x : L),\n                AddHom.toFun\n                    {\n                      toFun := fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : L),\n                            (fun x =>\n                                  {\n                                    toAddHom :=\n                                      { toFun := fun m => \u2045x, m\u2046,\n                                        map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                (x + y) =\n                              (fun x =>\n                                    {\n                                      toAddHom :=\n                                        { toFun := fun m => \u2045x, m\u2046,\n                                          map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                  x +\n                                (fun x =>\n                                    {\n                                      toAddHom :=\n                                        { toFun := fun m => \u2045x, m\u2046,\n                                          map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                  y) }\n                    (t \u2022 x) =\n                  \u2191(RingHom.id R) t \u2022\n                    AddHom.toFun\n                      {\n                        toFun := fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : L),\n                              (fun x =>\n                                    {\n                                      toAddHom :=\n                                        { toFun := fun m => \u2045x, m\u2046,\n                                          map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                  (x + y) =\n                                (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    x +\n                                  (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    y) }\n                      x) }.toAddHom\n      \u2045x, y\u2046 =\n    \u2045AddHom.toFun\n        {\n            toAddHom :=\n              {\n                toFun := fun x =>\n                  {\n                    toAddHom :=\n                      { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : L),\n                      (fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                          (x + y) =\n                        (fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                            x +\n                          (fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                            y) },\n            map_smul' :=\n              (_ :\n                \u2200 (t : R) (x : L),\n                  AddHom.toFun\n                      {\n                        toFun := fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : L),\n                              (fun x =>\n                                    {\n                                      toAddHom :=\n                                        { toFun := fun m => \u2045x, m\u2046,\n                                          map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                  (x + y) =\n                                (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    x +\n                                  (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    y) }\n                      (t \u2022 x) =\n                    \u2191(RingHom.id R) t \u2022\n                      AddHom.toFun\n                        {\n                          toFun := fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : L),\n                                (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    (x + y) =\n                                  (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      x +\n                                    (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      y) }\n                        x) }.toAddHom\n        x,\n      AddHom.toFun\n        {\n            toAddHom :=\n              {\n                toFun := fun x =>\n                  {\n                    toAddHom :=\n                      { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : L),\n                      (fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                          (x + y) =\n                        (fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                            x +\n                          (fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                            y) },\n            map_smul' :=\n              (_ :\n                \u2200 (t : R) (x : L),\n                  AddHom.toFun\n                      {\n                        toFun := fun x =>\n                          {\n                            toAddHom :=\n                              { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : L),\n                              (fun x =>\n                                    {\n                                      toAddHom :=\n                                        { toFun := fun m => \u2045x, m\u2046,\n                                          map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                  (x + y) =\n                                (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    x +\n                                  (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    y) }\n                      (t \u2022 x) =\n                    \u2191(RingHom.id R) t \u2022\n                      AddHom.toFun\n                        {\n                          toFun := fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : L),\n                                (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    (x + y) =\n                                  (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      x +\n                                    (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      y) }\n                        x) }.toAddHom\n        y\u2046\n[PROOFSTEP]\next m\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nx y : L\nm : M\n\u22a2 \u2191(AddHom.toFun\n          {\n              toAddHom :=\n                {\n                  toFun := fun x =>\n                    {\n                      toAddHom :=\n                        { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                      map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : L),\n                        (fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                            (x + y) =\n                          (fun x =>\n                                {\n                                  toAddHom :=\n                                    { toFun := fun m => \u2045x, m\u2046,\n                                      map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                              x +\n                            (fun x =>\n                                {\n                                  toAddHom :=\n                                    { toFun := fun m => \u2045x, m\u2046,\n                                      map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                              y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (t : R) (x : L),\n                    AddHom.toFun\n                        {\n                          toFun := fun x =>\n                            {\n                              toAddHom :=\n                                { toFun := fun m => \u2045x, m\u2046,\n                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : L),\n                                (fun x =>\n                                      {\n                                        toAddHom :=\n                                          { toFun := fun m => \u2045x, m\u2046,\n                                            map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                    (x + y) =\n                                  (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      x +\n                                    (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      y) }\n                        (t \u2022 x) =\n                      \u2191(RingHom.id R) t \u2022\n                        AddHom.toFun\n                          {\n                            toFun := fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : L),\n                                  (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      (x + y) =\n                                    (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        x +\n                                      (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        y) }\n                          x) }.toAddHom\n          \u2045x, y\u2046)\n      m =\n    \u2191\u2045AddHom.toFun\n            {\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : L),\n                          (fun x =>\n                                {\n                                  toAddHom :=\n                                    { toFun := fun m => \u2045x, m\u2046,\n                                      map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                              (x + y) =\n                            (fun x =>\n                                  {\n                                    toAddHom :=\n                                      { toFun := fun m => \u2045x, m\u2046,\n                                        map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                x +\n                              (fun x =>\n                                  {\n                                    toAddHom :=\n                                      { toFun := fun m => \u2045x, m\u2046,\n                                        map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (t : R) (x : L),\n                      AddHom.toFun\n                          {\n                            toFun := fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : L),\n                                  (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      (x + y) =\n                                    (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        x +\n                                      (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        y) }\n                          (t \u2022 x) =\n                        \u2191(RingHom.id R) t \u2022\n                          AddHom.toFun\n                            {\n                              toFun := fun x =>\n                                {\n                                  toAddHom :=\n                                    { toFun := fun m => \u2045x, m\u2046,\n                                      map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : L),\n                                    (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        (x + y) =\n                                      (fun x =>\n                                            {\n                                              toAddHom :=\n                                                { toFun := fun m => \u2045x, m\u2046,\n                                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                          x +\n                                        (fun x =>\n                                            {\n                                              toAddHom :=\n                                                { toFun := fun m => \u2045x, m\u2046,\n                                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                          y) }\n                            x) }.toAddHom\n            x,\n          AddHom.toFun\n            {\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun m => \u2045x, m\u2046, map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                        map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : L),\n                          (fun x =>\n                                {\n                                  toAddHom :=\n                                    { toFun := fun m => \u2045x, m\u2046,\n                                      map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                              (x + y) =\n                            (fun x =>\n                                  {\n                                    toAddHom :=\n                                      { toFun := fun m => \u2045x, m\u2046,\n                                        map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                x +\n                              (fun x =>\n                                  {\n                                    toAddHom :=\n                                      { toFun := fun m => \u2045x, m\u2046,\n                                        map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                    map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (t : R) (x : L),\n                      AddHom.toFun\n                          {\n                            toFun := fun x =>\n                              {\n                                toAddHom :=\n                                  { toFun := fun m => \u2045x, m\u2046,\n                                    map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : L),\n                                  (fun x =>\n                                        {\n                                          toAddHom :=\n                                            { toFun := fun m => \u2045x, m\u2046,\n                                              map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                          map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                      (x + y) =\n                                    (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        x +\n                                      (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        y) }\n                          (t \u2022 x) =\n                        \u2191(RingHom.id R) t \u2022\n                          AddHom.toFun\n                            {\n                              toFun := fun x =>\n                                {\n                                  toAddHom :=\n                                    { toFun := fun m => \u2045x, m\u2046,\n                                      map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                  map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) },\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : L),\n                                    (fun x =>\n                                          {\n                                            toAddHom :=\n                                              { toFun := fun m => \u2045x, m\u2046,\n                                                map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                            map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                        (x + y) =\n                                      (fun x =>\n                                            {\n                                              toAddHom :=\n                                                { toFun := fun m => \u2045x, m\u2046,\n                                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                          x +\n                                        (fun x =>\n                                            {\n                                              toAddHom :=\n                                                { toFun := fun m => \u2045x, m\u2046,\n                                                  map_add' := (_ : \u2200 (m n : M), \u2045x, m + n\u2046 = \u2045x, m\u2046 + \u2045x, n\u2046) },\n                                              map_smul' := (_ : \u2200 (t : R) (m : M), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046) })\n                                          y) }\n                            x) }.toAddHom\n            y\u2046\n      m\n[PROOFSTEP]\napply lie_lie\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u22a2 toEndomorphism R (Module.End R M) M = LieHom.id\n[PROOFSTEP]\next g m\n[GOAL]\ncase h.h\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\ng : Module.End R M\nm : M\n\u22a2 \u2191(\u2191(toEndomorphism R (Module.End R M) M) g) m = \u2191(\u2191LieHom.id g) m\n[PROOFSTEP]\nsimp [lie_eq_smul]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN : LieSubmodule R L M\nx : L\n\u22a2 Submodule.map (\u2191(toEndomorphism R L M) x) \u2191N \u2264 \u2191N\n[PROOFSTEP]\nrintro n \u27e8m, hm, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN : LieSubmodule R L M\nx : L\nm : M\nhm : m \u2208 \u2191\u2191N\n\u22a2 \u2191(\u2191(toEndomorphism R L M) x) m \u2208 \u2191N\n[PROOFSTEP]\nexact N.lie_mem hm\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN : LieSubmodule R L M\nx : L\nm : M\nhm : m \u2208 \u2191N\n\u22a2 \u2191(LinearMap.comp (\u2191(toEndomorphism R L M) x) (Submodule.subtype \u2191N)) { val := m, property := hm } \u2208 \u2191N\n[PROOFSTEP]\nsimpa using N.lie_mem hm\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN : LieSubmodule R L M\nx : L\nh :\n  optParam\n    (\u2200 (m : M) (hm : m \u2208 \u2191N),\n      \u2191(LinearMap.comp (\u2191(toEndomorphism R L M) x) (Submodule.subtype \u2191N)) { val := m, property := hm } \u2208 \u2191N)\n    (_ :\n      \u2200 (m : M) (hm : m \u2208 \u2191N),\n        \u2191(LinearMap.comp (\u2191(toEndomorphism R L M) x) (Submodule.subtype \u2191N)) { val := m, property := hm } \u2208 \u2191N)\n\u22a2 LinearMap.restrict (\u2191(toEndomorphism R L M) x) h = \u2191(toEndomorphism R L { x // x \u2208 \u2191N }) x\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN : LieSubmodule R L M\nx : L\nh :\n  optParam\n    (\u2200 (m : M) (hm : m \u2208 \u2191N),\n      \u2191(LinearMap.comp (\u2191(toEndomorphism R L M) x) (Submodule.subtype \u2191N)) { val := m, property := hm } \u2208 \u2191N)\n    (_ :\n      \u2200 (m : M) (hm : m \u2208 \u2191N),\n        \u2191(LinearMap.comp (\u2191(toEndomorphism R L M) x) (Submodule.subtype \u2191N)) { val := m, property := hm } \u2208 \u2191N)\nx\u271d : { x // x \u2208 \u2191N }\n\u22a2 \u2191(\u2191(LinearMap.restrict (\u2191(toEndomorphism R L M) x) h) x\u271d) = \u2191(\u2191(\u2191(toEndomorphism R L { x // x \u2208 \u2191N }) x) x\u271d)\n[PROOFSTEP]\nsimp [LinearMap.restrict_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\n\u22a2 \u2191(ad R A) = LinearMap.mulLeft R - LinearMap.mulRight R\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.h\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\ninst\u271d\u2076 : LieAlgebra R L\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : LieRingModule L M\ninst\u271d\u00b2 : LieModule R L M\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\n\u22a2 \u2191(\u2191(ad R A) a) b = \u2191((LinearMap.mulLeft R - LinearMap.mulRight R) a) b\n[PROOFSTEP]\nsimp [LieRing.of_associative_ring_bracket]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nx : { x // x \u2208 K }\n\u22a2 LinearMap.comp (\u2191(ad R L) \u2191x) \u2191(incl K) = LinearMap.comp (\u2191(incl K)) (\u2191(ad R { x // x \u2208 K }) x)\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nK : LieSubalgebra R L\nx y : { x // x \u2208 K }\n\u22a2 \u2191(LinearMap.comp (\u2191(ad R L) \u2191x) \u2191(incl K)) y = \u2191(LinearMap.comp (\u2191(incl K)) (\u2191(ad R { x // x \u2208 K }) x)) y\n[PROOFSTEP]\nsimp only [ad_apply, LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, LinearMap.coe_comp, LieSubalgebra.coe_bracket,\n  Function.comp_apply]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhx :\n  x \u2208\n    { toAddSubmonoid := src\u271d.toAddSubmonoid,\n            smul_mem' :=\n              (_ :\n                \u2200 (c : R) {x : A}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n  y \u2208\n    { toAddSubmonoid := src\u271d.toAddSubmonoid,\n            smul_mem' :=\n              (_ :\n                \u2200 (c : R) {x : A}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, y\u2046 \u2208\n    { toAddSubmonoid := src\u271d.toAddSubmonoid,\n            smul_mem' :=\n              (_ :\n                \u2200 (c : R) {x : A}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n[PROOFSTEP]\nchange \u2045x, y\u2046 \u2208 A'\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhx :\n  x \u2208\n    { toAddSubmonoid := src\u271d.toAddSubmonoid,\n            smul_mem' :=\n              (_ :\n                \u2200 (c : R) {x : A}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhy :\n  y \u2208\n    { toAddSubmonoid := src\u271d.toAddSubmonoid,\n            smul_mem' :=\n              (_ :\n                \u2200 (c : R) {x : A}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\n\u22a2 \u2045x, y\u2046 \u2208 A'\n[PROOFSTEP]\nchange x \u2208 A' at hx \n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhy :\n  y \u2208\n    { toAddSubmonoid := src\u271d.toAddSubmonoid,\n            smul_mem' :=\n              (_ :\n                \u2200 (c : R) {x : A}, x \u2208 src\u271d.carrier \u2192 c \u2022 x \u2208 src\u271d.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier\nhx : x \u2208 A'\n\u22a2 \u2045x, y\u2046 \u2208 A'\n[PROOFSTEP]\nchange y \u2208 A' at hy \n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhx : x \u2208 A'\nhy : y \u2208 A'\n\u22a2 \u2045x, y\u2046 \u2208 A'\n[PROOFSTEP]\nrw [LieRing.of_associative_ring_bracket]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhx : x \u2208 A'\nhy : y \u2208 A'\n\u22a2 x * y - y * x \u2208 A'\n[PROOFSTEP]\nhave hxy := A'.mul_mem hx hy\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhx : x \u2208 A'\nhy : y \u2208 A'\nhxy : x * y \u2208 A'\n\u22a2 x * y - y * x \u2208 A'\n[PROOFSTEP]\nhave hyx := A'.mul_mem hy hx\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : CommRing R\nA : Type v\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nA' : Subalgebra R A\nsrc\u271d : (fun x => Submodule R A) A' := \u2191Subalgebra.toSubmodule A'\nx y : A\nhx : x \u2208 A'\nhy : y \u2208 A'\nhxy : x * y \u2208 A'\nhyx : y * x \u2208 A'\n\u22a2 x * y - y * x \u2208 A'\n[PROOFSTEP]\nexact Submodule.sub_mem (Subalgebra.toSubmodule A') hxy hyx\n[GOAL]\nR : Type u\nM\u2081 : Type v\nM\u2082 : Type w\ninst\u271d\u2074 : CommRing R\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 \u2191(conj e) \u2045f, g\u2046 = \u2045\u2191(conj e) f, \u2191(conj e) g\u2046\n[PROOFSTEP]\nsimp only [LieRing.of_associative_ring_bracket, LinearMap.mul_eq_comp, e.conj_comp, LinearEquiv.map_sub]\n[GOAL]\nR : Type u\nA\u2081 : Type v\nA\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\u2081\ninst\u271d\u00b2 : Ring A\u2082\ninst\u271d\u00b9 : Algebra R A\u2081\ninst\u271d : Algebra R A\u2082\ne : A\u2081 \u2243\u2090[R] A\u2082\nsrc\u271d : A\u2081 \u2243\u2097[R] A\u2082 := toLinearEquiv e\nx y : A\u2081\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := e.toFun,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A\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 : A\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 := e.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A\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 : A\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 := e.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A\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 : A\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]\nhave : e.toEquiv.toFun = e := rfl\n[GOAL]\nR : Type u\nA\u2081 : Type v\nA\u2082 : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\u2081\ninst\u271d\u00b2 : Ring A\u2082\ninst\u271d\u00b9 : Algebra R A\u2081\ninst\u271d : Algebra R A\u2082\ne : A\u2081 \u2243\u2090[R] A\u2082\nsrc\u271d : A\u2081 \u2243\u2097[R] A\u2082 := toLinearEquiv e\nx y : A\u2081\nthis : e.toFun = \u2191e\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := e.toFun,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A\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 : A\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 := e.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A\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 : A\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 := e.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A\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 : A\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_rw [LieRing.of_associative_ring_bracket, this, map_sub, map_mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.OfAssociative", "llama_tokens": 24312, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.865224072151174, "lm_q2_score": 0.6513548511303336, "lm_q1q2_score": 0.563567896710409}}
{"text": "[GOAL]\n\u22a2 {x | Liouville x} = \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nx : \u211d\n\u22a2 x \u2208 {x | Liouville x} \u2194 x \u2208 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b}\n[PROOFSTEP]\nsimp only [mem_iInter, mem_iUnion, Liouville, mem_setOf_eq, exists_prop, mem_diff, mem_singleton_iff, mem_ball,\n  Real.dist_eq, and_comm]\n[GOAL]\n\u22a2 IsG\u03b4 {x | Liouville x}\n[PROOFSTEP]\nrw [setOf_liouville_eq_iInter_iUnion]\n[GOAL]\n\u22a2 IsG\u03b4 (\u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b})\n[PROOFSTEP]\nrefine isG\u03b4_iInter fun n => IsOpen.isG\u03b4 ?_\n[GOAL]\nn : \u2115\n\u22a2 IsOpen (\u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b})\n[PROOFSTEP]\nrefine isOpen_iUnion fun a => isOpen_iUnion fun b => isOpen_iUnion fun _hb => ?_\n[GOAL]\nn : \u2115\na b : \u2124\n_hb : 1 < b\n\u22a2 IsOpen (ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b})\n[PROOFSTEP]\nexact isOpen_ball.inter isClosed_singleton.isOpen_compl\n[GOAL]\n\u22a2 {x | Liouville x} = {x | Irrational x} \u2229 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nrefine Subset.antisymm ?_ ?_\n[GOAL]\ncase refine_1\n\u22a2 {x | Liouville x} \u2286 {x | Irrational x} \u2229 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nrefine subset_inter (fun x hx => hx.irrational) ?_\n[GOAL]\ncase refine_1\n\u22a2 {x | Liouville x} \u2286 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nrw [setOf_liouville_eq_iInter_iUnion]\n[GOAL]\ncase refine_1\n\u22a2 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b} \u2286\n    \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nexact iInter_mono fun n => iUnion\u2082_mono fun a b => iUnion_mono fun _hb => diff_subset _ _\n[GOAL]\ncase refine_2\n\u22a2 {x | Irrational x} \u2229 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \u2286 {x | Liouville x}\n[PROOFSTEP]\nsimp only [inter_iInter, inter_iUnion, setOf_liouville_eq_iInter_iUnion]\n[GOAL]\ncase refine_2\n\u22a2 \u22c2 (i : \u2115), \u22c3 (i_1 : \u2124) (i_2 : \u2124) (_ : 1 < i_2), {x | Irrational x} \u2229 ball (\u2191i_1 / \u2191i_2) (1 / \u2191i_2 ^ i) \u2286\n    \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b}\n[PROOFSTEP]\nrefine iInter_mono fun n => iUnion\u2082_mono fun a b => iUnion_mono fun hb => ?_\n[GOAL]\ncase refine_2\nn : \u2115\na b : \u2124\nhb : 1 < b\n\u22a2 {x | Irrational x} \u2229 ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \u2286 ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b}\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\ncase refine_2\nn : \u2115\na b : \u2124\nhb : 1 < b\n\u22a2 ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \u2229 {x | Irrational x} \u2286 ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \\ {\u2191a / \u2191b}\n[PROOFSTEP]\nexact diff_subset_diff Subset.rfl (singleton_subset_iff.2 \u27e8a / b, by norm_cast\u27e9)\n[GOAL]\nn : \u2115\na b : \u2124\nhb : 1 < b\n\u22a2 \u2191(\u2191a / \u2191b) = \u2191a / \u2191b\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u22a2 \u2200\u1da0 (x : \u211d) in residual \u211d, Liouville x\n[PROOFSTEP]\nrw [Filter.Eventually, setOf_liouville_eq_irrational_inter_iInter_iUnion]\n[GOAL]\n\u22a2 {x | Irrational x} \u2229 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n) \u2208 residual \u211d\n[PROOFSTEP]\nrefine eventually_residual_irrational.and ?_\n[GOAL]\n\u22a2 \u2200\u1da0 (x : \u211d) in residual \u211d, x \u2208 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nrefine eventually_residual.2 \u27e8_, ?_, Rat.denseEmbedding_coe_real.dense.mono ?_, Subset.rfl\u27e9\n[GOAL]\ncase refine_1\n\u22a2 IsG\u03b4 (\u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n))\n[PROOFSTEP]\nexact\n  isG\u03b4_iInter fun n => IsOpen.isG\u03b4 <| isOpen_iUnion fun a => isOpen_iUnion fun b => isOpen_iUnion fun _hb => isOpen_ball\n[GOAL]\ncase refine_2\n\u22a2 range Rat.cast \u2286 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nrintro _ \u27e8r, rfl\u27e9\n[GOAL]\ncase refine_2.intro\nr : \u211a\n\u22a2 \u2191r \u2208 \u22c2 (n : \u2115), \u22c3 (a : \u2124) (b : \u2124) (_ : 1 < b), ball (\u2191a / \u2191b) (1 / \u2191b ^ n)\n[PROOFSTEP]\nsimp only [mem_iInter, mem_iUnion]\n[GOAL]\ncase refine_2.intro\nr : \u211a\n\u22a2 \u2200 (i : \u2115), \u2203 i_1 i_2 i_3, \u2191r \u2208 ball (\u2191i_1 / \u2191i_2) (1 / \u2191i_2 ^ i)\n[PROOFSTEP]\nrefine fun n => \u27e8r.num * 2, r.den * 2, ?_, ?_\u27e9\n[GOAL]\ncase refine_2.intro.refine_1\nr : \u211a\nn : \u2115\n\u22a2 1 < \u2191r.den * 2\n[PROOFSTEP]\nhave := Int.ofNat_le.2 r.pos\n[GOAL]\ncase refine_2.intro.refine_1\nr : \u211a\nn : \u2115\nthis : \u2191(Nat.succ 0) \u2264 \u2191r.den\n\u22a2 1 < \u2191r.den * 2\n[PROOFSTEP]\nrw [Int.ofNat_one] at this \n[GOAL]\ncase refine_2.intro.refine_1\nr : \u211a\nn : \u2115\nthis : 1 \u2264 \u2191r.den\n\u22a2 1 < \u2191r.den * 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine_2.intro.refine_2\nr : \u211a\nn : \u2115\n\u22a2 \u2191r \u2208 ball (\u2191(r.num * 2) / \u2191(\u2191r.den * 2)) (1 / \u2191(\u2191r.den * 2) ^ n)\n[PROOFSTEP]\nconvert @mem_ball_self \u211d _ (r : \u211d) _ _\n[GOAL]\ncase h.e'_5.h.e'_3\nr : \u211a\nn : \u2115\n\u22a2 \u2191(r.num * 2) / \u2191(\u2191r.den * 2) = \u2191r\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_5.h.e'_3\nr : \u211a\nn : \u2115\n\u22a2 \u2191r.num * 2 / (\u2191r.den * 2) = \u2191r\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_5.h.e'_3\nr : \u211a\nn : \u2115\n\u22a2 Rat.divInt (r.num * 2) \u2191(r.den * 2) = r\n[PROOFSTEP]\nsimp [Rat.divInt_mul_right (two_ne_zero), Rat.mkRat_self]\n[GOAL]\ncase refine_2.intro.refine_2.convert_2\nr : \u211a\nn : \u2115\n\u22a2 0 < 1 / \u2191(\u2191r.den * 2) ^ n\n[PROOFSTEP]\nrefine' one_div_pos.2 (pow_pos (Int.cast_pos.2 _) _)\n[GOAL]\ncase refine_2.intro.refine_2.convert_2\nr : \u211a\nn : \u2115\n\u22a2 0 < \u2191r.den * 2\n[PROOFSTEP]\nexact mul_pos (Int.coe_nat_pos.2 r.pos) zero_lt_two\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Liouville.Residual", "llama_tokens": 3034, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.817574471748733, "lm_q2_score": 0.6893056231680122, "lm_q1q2_score": 0.5635586807350188}}
{"text": "[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u22a2 restrictScalars F (normalClosure F K L) = \u2a06 (x : K), adjoin F (rootSet (minpoly F x) L)\n[PROOFSTEP]\nclassical\nhave hi : \u2200 x : K, IsIntegral F x := fun x \u21a6 (isIntegral_algebraMap_iff (algebraMap K L).injective).mp (h.isIntegral _)\nrefine' le_antisymm (iSup_le _) (iSup_le fun x => adjoin_le_iff.mpr fun y hy => _)\n\u00b7 rintro f _ \u27e8x, rfl\u27e9\n  refine' le_iSup (fun x => adjoin F ((minpoly F x).rootSet L)) x (subset_adjoin F ((minpoly F x).rootSet L) _)\n  rw [mem_rootSet_of_ne (minpoly.ne_zero (hi x)), AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,\n    Polynomial.aeval_algHom_apply, minpoly.aeval, map_zero]\n\u00b7 rw [Polynomial.rootSet, Finset.mem_coe, Multiset.mem_toFinset] at hy \n  let g := (algHomAdjoinIntegralEquiv F (hi x)).symm \u27e8y, hy\u27e9\n  refine'\n    le_iSup (fun f : K \u2192\u2090[F] L => f.fieldRange) ((g.liftNormal L).comp (toAlgHom F K L))\n      \u27e8x, (g.liftNormal_commutes L (AdjoinSimple.gen F x)).trans _\u27e9\n  rw [Algebra.id.map_eq_id, RingHom.id_apply]\n    -- Porting note: in mathlib3 this next `apply` closed the goal.\n        -- Now it can't find a proof by unification, so we have to do it ourselves.\n  apply PowerBasis.lift_gen\n  change aeval y (minpoly F (AdjoinSimple.gen F x)) = 0\n  exact minpoly_gen (hi x) \u25b8 aeval_eq_zero_of_mem_rootSet (Multiset.mem_toFinset.mpr hy)\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u22a2 restrictScalars F (normalClosure F K L) = \u2a06 (x : K), adjoin F (rootSet (minpoly F x) L)\n[PROOFSTEP]\nhave hi : \u2200 x : K, IsIntegral F x := fun x \u21a6 (isIntegral_algebraMap_iff (algebraMap K L).injective).mp (h.isIntegral _)\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\n\u22a2 restrictScalars F (normalClosure F K L) = \u2a06 (x : K), adjoin F (rootSet (minpoly F x) L)\n[PROOFSTEP]\nrefine' le_antisymm (iSup_le _) (iSup_le fun x => adjoin_le_iff.mpr fun y hy => _)\n[GOAL]\ncase refine'_1\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\n\u22a2 \u2200 (i : K \u2192\u2090[F] L), AlgHom.fieldRange i \u2264 \u2a06 (x : K), adjoin F (rootSet (minpoly F x) L)\n[PROOFSTEP]\nrintro f _ \u27e8x, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nf : K \u2192\u2090[F] L\nx : K\n\u22a2 \u2191\u2191f x \u2208 \u2a06 (x : K), adjoin F (rootSet (minpoly F x) L)\n[PROOFSTEP]\nrefine' le_iSup (fun x => adjoin F ((minpoly F x).rootSet L)) x (subset_adjoin F ((minpoly F x).rootSet L) _)\n[GOAL]\ncase refine'_1.intro\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nf : K \u2192\u2090[F] L\nx : K\n\u22a2 \u2191\u2191f x \u2208 rootSet (minpoly F x) L\n[PROOFSTEP]\nrw [mem_rootSet_of_ne (minpoly.ne_zero (hi x)), AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,\n  Polynomial.aeval_algHom_apply, minpoly.aeval, map_zero]\n[GOAL]\ncase refine'_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 rootSet (minpoly F x) L\n\u22a2 y \u2208 \u2191(restrictScalars F (normalClosure F K L))\n[PROOFSTEP]\nrw [Polynomial.rootSet, Finset.mem_coe, Multiset.mem_toFinset] at hy \n[GOAL]\ncase refine'_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x))\n\u22a2 y \u2208 \u2191(restrictScalars F (normalClosure F K L))\n[PROOFSTEP]\nlet g := (algHomAdjoinIntegralEquiv F (hi x)).symm \u27e8y, hy\u27e9\n[GOAL]\ncase refine'_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x))\ng : (fun x_1 => { x_2 // x_2 \u2208 F\u27eex\u27ef } \u2192\u2090[F] L) { val := y, property := hy } :=\n  \u2191(algHomAdjoinIntegralEquiv F (_ : IsIntegral F x)).symm { val := y, property := hy }\n\u22a2 y \u2208 \u2191(restrictScalars F (normalClosure F K L))\n[PROOFSTEP]\nrefine'\n  le_iSup (fun f : K \u2192\u2090[F] L => f.fieldRange) ((g.liftNormal L).comp (toAlgHom F K L))\n    \u27e8x, (g.liftNormal_commutes L (AdjoinSimple.gen F x)).trans _\u27e9\n[GOAL]\ncase refine'_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x))\ng : (fun x_1 => { x_2 // x_2 \u2208 F\u27eex\u27ef } \u2192\u2090[F] L) { val := y, property := hy } :=\n  \u2191(algHomAdjoinIntegralEquiv F (_ : IsIntegral F x)).symm { val := y, property := hy }\n\u22a2 \u2191(algebraMap L L) (\u2191g (AdjoinSimple.gen F x)) = y\n[PROOFSTEP]\nrw [Algebra.id.map_eq_id, RingHom.id_apply]\n  -- Porting note: in mathlib3 this next `apply` closed the goal.\n      -- Now it can't find a proof by unification, so we have to do it ourselves.\n[GOAL]\ncase refine'_2\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x))\ng : (fun x_1 => { x_2 // x_2 \u2208 F\u27eex\u27ef } \u2192\u2090[F] L) { val := y, property := hy } :=\n  \u2191(algHomAdjoinIntegralEquiv F (_ : IsIntegral F x)).symm { val := y, property := hy }\n\u22a2 \u2191g (AdjoinSimple.gen F x) = y\n[PROOFSTEP]\napply PowerBasis.lift_gen\n[GOAL]\ncase refine'_2.hy\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x))\ng : (fun x_1 => { x_2 // x_2 \u2208 F\u27eex\u27ef } \u2192\u2090[F] L) { val := y, property := hy } :=\n  \u2191(algHomAdjoinIntegralEquiv F (_ : IsIntegral F x)).symm { val := y, property := hy }\n\u22a2 \u2191(aeval\n          \u2191(\u2191(Equiv.subtypeEquiv (Equiv.refl L)\n                    (_ :\n                      \u2200 (x_1 : L),\n                        \u2191(aeval x_1) (minpoly F (adjoin.powerBasis (_ : IsIntegral F x)).gen) = 0 \u2194\n                          \u2191(Equiv.refl L) x_1 \u2208\n                            roots\n                              (Polynomial.map (algebraMap F L)\n                                (minpoly F (adjoin.powerBasis (_ : IsIntegral F x)).gen)))).symm\n              (\u2191(Equiv.subtypeEquiv (Equiv.refl L)\n                      (_ :\n                        \u2200 (x_1 : L),\n                          x_1 \u2208\n                              roots\n                                (Polynomial.map (algebraMap F L)\n                                  (minpoly F (adjoin.powerBasis (_ : IsIntegral F x)).gen)) \u2194\n                            \u2191(Equiv.refl L) x_1 \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x)))).symm\n                { val := y, property := hy })))\n      (minpoly F (adjoin.powerBasis (_ : IsIntegral F x)).gen) =\n    0\n[PROOFSTEP]\nchange aeval y (minpoly F (AdjoinSimple.gen F x)) = 0\n[GOAL]\ncase refine'_2.hy\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\nhi : \u2200 (x : K), IsIntegral F x\nx : K\ny : L\nhy : y \u2208 roots (Polynomial.map (algebraMap F L) (minpoly F x))\ng : (fun x_1 => { x_2 // x_2 \u2208 F\u27eex\u27ef } \u2192\u2090[F] L) { val := y, property := hy } :=\n  \u2191(algHomAdjoinIntegralEquiv F (_ : IsIntegral F x)).symm { val := y, property := hy }\n\u22a2 \u2191(aeval y) (minpoly F (AdjoinSimple.gen F x)) = 0\n[PROOFSTEP]\nexact minpoly_gen (hi x) \u25b8 aeval_eq_zero_of_mem_rootSet (Multiset.mem_toFinset.mpr hy)\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u22a2 Normal F { x // x \u2208 normalClosure F K L }\n[PROOFSTEP]\nlet \u03d5 := algebraMap K L\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u03d5 : K \u2192+* L := algebraMap K L\n\u22a2 Normal F { x // x \u2208 normalClosure F K L }\n[PROOFSTEP]\nrw [\u2190 IntermediateField.restrictScalars_normal, restrictScalars_eq_iSup_adjoin]\n  -- Porting note: use the `(_)` trick to obtain an instance by unification.\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u03d5 : K \u2192+* L := algebraMap K L\n\u22a2 Normal F { x // x \u2208 \u2a06 (x : K), adjoin F (rootSet (minpoly F x) L) }\n[PROOFSTEP]\napply IntermediateField.normal_iSup (h := _)\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u03d5 : K \u2192+* L := algebraMap K L\n\u22a2 \u2200 (i : K), Normal F { x // x \u2208 adjoin F (rootSet (minpoly F i) L) }\n[PROOFSTEP]\nintro x\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u03d5 : K \u2192+* L := algebraMap K L\nx : K\n\u22a2 Normal F { x_1 // x_1 \u2208 adjoin F (rootSet (minpoly F x) L) }\n[PROOFSTEP]\napply Normal.of_isSplittingField (p := minpoly F x) (hFEp := _)\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra F L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsScalarTower F K L\nh : Normal F L\n\u03d5 : K \u2192+* L := algebraMap K L\nx : K\n\u22a2 IsSplittingField F { x_1 // x_1 \u2208 adjoin F (rootSet (minpoly F x) L) } (minpoly F x)\n[PROOFSTEP]\nexact\n  adjoin_rootSet_isSplittingField\n    ((minpoly.eq_of_algebraMap_eq \u03d5.injective ((isIntegral_algebraMap_iff \u03d5.injective).mp (h.isIntegral (\u03d5 x)))\n          rfl).symm \u25b8\n      h.splits _)\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra F L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsScalarTower F K L\ninst\u271d : FiniteDimensional F K\n\u22a2 FiniteDimensional F { x // x \u2208 normalClosure F K L }\n[PROOFSTEP]\nhaveI : \u2200 f : K \u2192\u2090[F] L, FiniteDimensional F f.fieldRange := fun f => f.toLinearMap.finiteDimensional_range\n[GOAL]\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra F L\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsScalarTower F K L\ninst\u271d : FiniteDimensional F K\nthis : \u2200 (f : K \u2192\u2090[F] L), FiniteDimensional F { x // x \u2208 AlgHom.fieldRange f }\n\u22a2 FiniteDimensional F { x // x \u2208 normalClosure F K L }\n[PROOFSTEP]\napply IntermediateField.finiteDimensional_iSup_of_finite\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.NormalClosure", "llama_tokens": 5448, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7981867873410141, "lm_q2_score": 0.7057850340255386, "lm_q1q2_score": 0.563348288862213}}
{"text": "[GOAL]\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx\u271d y : X\nh : IsMetricSeparated s t\nr : ENNReal\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\nx : X\nhx1 : x \u2208 s\nhx2 : x \u2208 t\n\u22a2 r = 0\n[PROOFSTEP]\nsimpa using hr x hx1 x hx2\n[GOAL]\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx y : X\ns' : Set X\nh : IsMetricSeparated s t\nh' : IsMetricSeparated s' t\n\u22a2 IsMetricSeparated (s \u222a s') t\n[PROOFSTEP]\nrcases h, h' with \u27e8\u27e8r, r0, hr\u27e9, \u27e8r', r0', hr'\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx y : X\ns' : Set X\nr : ENNReal\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\nr' : ENNReal\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 IsMetricSeparated (s \u222a s') t\n[PROOFSTEP]\nrefine' \u27e8min r r', _, fun x hx y hy => hx.elim _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx y : X\ns' : Set X\nr : ENNReal\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\nr' : ENNReal\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 min r r' \u2260 0\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero] at r0 r0' \u22a2\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx y : X\ns' : Set X\nr : ENNReal\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\nr' : ENNReal\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 0 < min r r'\n[PROOFSTEP]\nexact lt_min r0 r0'\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx\u271d y\u271d : X\ns' : Set X\nr : ENNReal\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\nr' : ENNReal\nr0' : r' \u2260 0\nhr' : \u2200 (x : X), x \u2208 s' \u2192 \u2200 (y : X), y \u2208 t \u2192 r' \u2264 edist x y\nx : X\nhx : x \u2208 s \u222a s'\ny : X\nhy : y \u2208 t\n\u22a2 x \u2208 s \u2192 min r r' \u2264 edist x y\n[PROOFSTEP]\nexact fun hx => (min_le_left _ _).trans (hr _ hx _ hy)\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\nX : Type u_1\ninst\u271d : EMetricSpace X\ns t : Set X\nx\u271d y\u271d : X\ns' : Set X\nr : ENNReal\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\nr' : ENNReal\nr0' : r' \u2260 0\nhr' : \u2200 (x : X), x \u2208 s' \u2192 \u2200 (y : X), y \u2208 t \u2192 r' \u2264 edist x y\nx : X\nhx : x \u2208 s \u222a s'\ny : X\nhy : y \u2208 t\n\u22a2 x \u2208 s' \u2192 min r r' \u2264 edist x y\n[PROOFSTEP]\nexact fun hx => (min_le_right _ _).trans (hr' _ hx _ hy)\n[GOAL]\nX : Type u_1\ninst\u271d : EMetricSpace X\ns\u271d t\u271d : Set X\nx y : X\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Finite I\ns : \u03b9 \u2192 Set X\nt : Set X\n\u22a2 IsMetricSeparated (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) t \u2194 \u2200 (i : \u03b9), i \u2208 I \u2192 IsMetricSeparated (s i) t\n[PROOFSTEP]\nrefine' Finite.induction_on hI (by simp) @fun i I _ _ hI => _\n[GOAL]\nX : Type u_1\ninst\u271d : EMetricSpace X\ns\u271d t\u271d : Set X\nx y : X\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Finite I\ns : \u03b9 \u2192 Set X\nt : Set X\n\u22a2 IsMetricSeparated (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), s i) t \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsMetricSeparated (s i) t\n[PROOFSTEP]\nsimp\n[GOAL]\nX : Type u_1\ninst\u271d : EMetricSpace X\ns\u271d t\u271d : Set X\nx y : X\n\u03b9 : Type u_2\nI\u271d : Set \u03b9\nhI\u271d : Set.Finite I\u271d\ns : \u03b9 \u2192 Set X\nt : Set X\ni : \u03b9\nI : Set \u03b9\nx\u271d\u00b9 : \u00aci \u2208 I\nx\u271d : Set.Finite I\nhI : IsMetricSeparated (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) t \u2194 \u2200 (i : \u03b9), i \u2208 I \u2192 IsMetricSeparated (s i) t\n\u22a2 IsMetricSeparated (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i I), s i_1) t \u2194\n    \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 IsMetricSeparated (s i_1) t\n[PROOFSTEP]\nrw [biUnion_insert, ball_insert_iff, union_left_iff, hI]\n[GOAL]\nX : Type u_1\ninst\u271d : EMetricSpace X\ns\u271d t\u271d : Set X\nx y : X\n\u03b9 : Type u_2\nI : Set \u03b9\nhI : Set.Finite I\ns : Set X\nt : \u03b9 \u2192 Set X\n\u22a2 IsMetricSeparated s (\u22c3 (i : \u03b9) (_ : i \u2208 I), t i) \u2194 \u2200 (i : \u03b9), i \u2208 I \u2192 IsMetricSeparated s (t i)\n[PROOFSTEP]\nsimpa only [@comm _ _ s] using finite_iUnion_left_iff hI\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.MetricSeparated", "llama_tokens": 2065, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696748, "lm_q2_score": 0.7122321903471563, "lm_q1q2_score": 0.563041140401938}}
{"text": "[GOAL]\nA : Type u\u2081\nB : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} A\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} B\ninst\u271d\u2075 : HasEqualizers A\ninst\u271d\u2074 : HasImages A\ninst\u271d\u00b3 : StrongEpiCategory B\ninst\u271d\u00b2 : HasImages B\nL : A \u2964 B\ninst\u271d\u00b9 : {X Y Z : A} \u2192 (f : X \u27f6 Z) \u2192 (g : Y \u27f6 Z) \u2192 PreservesLimit (cospan f g) L\ninst\u271d : {X Y Z : A} \u2192 (f : X \u27f6 Y) \u2192 (g : X \u27f6 Z) \u2192 PreservesColimit (span f g) L\nX Y : A\nf : X \u27f6 Y\n\u22a2 L.map (factorThruImage f) \u226b L.map (image.\u03b9 f) = L.map f\n[PROOFSTEP]\nrw [\u2190 L.map_comp, Limits.image.fac]\n[GOAL]\nA : Type u\u2081\nB : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} A\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} B\ninst\u271d\u2075 : HasEqualizers A\ninst\u271d\u2074 : HasImages A\ninst\u271d\u00b3 : StrongEpiCategory B\ninst\u271d\u00b2 : HasImages B\nL : A \u2964 B\ninst\u271d\u00b9 : {X Y Z : A} \u2192 (f : X \u27f6 Z) \u2192 (g : Y \u27f6 Z) \u2192 PreservesLimit (cospan f g) L\ninst\u271d : {X Y Z : A} \u2192 (f : X \u27f6 Y) \u2192 (g : X \u27f6 Z) \u2192 PreservesColimit (span f g) L\nX Y : A\nf : X \u27f6 Y\n\u22a2 factorThruImage (L.map f) \u226b (iso L f).hom = L.map (factorThruImage f)\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\u2081\nB : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} A\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} B\ninst\u271d\u2075 : HasEqualizers A\ninst\u271d\u2074 : HasImages A\ninst\u271d\u00b3 : StrongEpiCategory B\ninst\u271d\u00b2 : HasImages B\nL : A \u2964 B\ninst\u271d\u00b9 : {X Y Z : A} \u2192 (f : X \u27f6 Z) \u2192 (g : Y \u27f6 Z) \u2192 PreservesLimit (cospan f g) L\ninst\u271d : {X Y Z : A} \u2192 (f : X \u27f6 Y) \u2192 (g : X \u27f6 Z) \u2192 PreservesColimit (span f g) L\nX Y : A\nf : X \u27f6 Y\n\u22a2 (iso L f).hom \u226b L.map (image.\u03b9 f) = image.\u03b9 (L.map f)\n[PROOFSTEP]\nrw [iso_hom, image.lift_fac]\n[GOAL]\nA : Type u\u2081\nB : Type u\u2082\ninst\u271d\u2077 : Category.{v\u2081, u\u2081} A\ninst\u271d\u2076 : Category.{v\u2082, u\u2082} B\ninst\u271d\u2075 : HasEqualizers A\ninst\u271d\u2074 : HasImages A\ninst\u271d\u00b3 : StrongEpiCategory B\ninst\u271d\u00b2 : HasImages B\nL : A \u2964 B\ninst\u271d\u00b9 : {X Y Z : A} \u2192 (f : X \u27f6 Z) \u2192 (g : Y \u27f6 Z) \u2192 PreservesLimit (cospan f g) L\ninst\u271d : {X Y Z : A} \u2192 (f : X \u27f6 Y) \u2192 (g : X \u27f6 Z) \u2192 PreservesColimit (span f g) L\nX Y : A\nf : X \u27f6 Y\n\u22a2 (iso L f).inv \u226b image.\u03b9 (L.map f) = L.map (image.\u03b9 f)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Images", "llama_tokens": 976, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8596637541053281, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.562989261312147}}
{"text": "[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nx : E\nI : Type u_3\ninst\u271d : TopologicalSpace I\nh\u271d : IsEvenlyCovered f (f x) I\ne : Trivialization I f := Classical.choose (_ : \u2203 t, f x \u2208 t.baseSet)\nh : f x \u2208 (Classical.choose (_ : \u2203 t, f x \u2208 t.baseSet)).baseSet := Classical.choose_spec h\u271d.right\nhe : (f x, (\u2191e x).snd) = \u2191e x := Trivialization.mk_proj_snd' e h\n\u22a2 (f x, (\u2191e x).snd) \u2208 e.target\n[PROOFSTEP]\nrwa [he, e.mem_target, e.coe_fst (e.mem_source.mpr h)]\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nhf : IsCoveringMapOn f s\n\u22a2 IsLocallyHomeomorphOn f (f \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' IsLocallyHomeomorphOn.mk f (f \u207b\u00b9' s) fun x hx => _\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : E), y \u2208 e.source \u2192 f y = \u2191e y\n[PROOFSTEP]\nlet e := (hf (f x) hx).toTrivialization\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x \u2208 f \u207b\u00b9' s\ne : Trivialization (\u2191(f \u207b\u00b9' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))\n\u22a2 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : E), y \u2208 e.source \u2192 f y = \u2191e y\n[PROOFSTEP]\nhave h := (hf (f x) hx).mem_toTrivialization_baseSet\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x \u2208 f \u207b\u00b9' s\ne : Trivialization (\u2191(f \u207b\u00b9' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))\nh : f x \u2208 (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))).baseSet\n\u22a2 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : E), y \u2208 e.source \u2192 f y = \u2191e y\n[PROOFSTEP]\nlet he := e.mem_source.2 h\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x \u2208 f \u207b\u00b9' s\ne : Trivialization (\u2191(f \u207b\u00b9' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))\nh : f x \u2208 (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))).baseSet\nhe : x \u2208 e.source := Iff.mpr (Trivialization.mem_source e) h\n\u22a2 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : E), y \u2208 e.source \u2192 f y = \u2191e y\n[PROOFSTEP]\nrefine'\n  \u27e8e.toLocalHomeomorph.trans\n      { toFun := fun p => p.1\n        invFun := fun p => \u27e8p, x, rfl\u27e9\n        source := e.baseSet \u00d7\u02e2 ({\u27e8x, rfl\u27e9} : Set (f \u207b\u00b9' {f x}))\n        target := e.baseSet\n        open_source := e.open_baseSet.prod (singletons_open_iff_discrete.2 (hf (f x) hx).1 \u27e8x, rfl\u27e9)\n        open_target := e.open_baseSet\n        map_source' := fun p => And.left\n        map_target' := fun p hp => \u27e8hp, rfl\u27e9\n        left_inv' := fun p hp => Prod.ext rfl hp.2.symm\n        right_inv' := fun p _ => rfl\n        continuous_toFun := continuous_fst.continuousOn\n        continuous_invFun := (continuous_id'.prod_mk continuous_const).continuousOn },\n    \u27e8he, by rwa [e.toLocalHomeomorph.symm_symm, e.proj_toFun x he], (hf (f x) hx).toTrivialization_apply\u27e9, fun p h =>\n    (e.proj_toFun p h.1).symm\u27e9\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\nhf : IsCoveringMapOn f s\nx : E\nhx : x \u2208 f \u207b\u00b9' s\ne : Trivialization (\u2191(f \u207b\u00b9' {f x})) f := IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))\nh : f x \u2208 (IsEvenlyCovered.toTrivialization (_ : IsEvenlyCovered f (f x) \u2191(f \u207b\u00b9' {f x}))).baseSet\nhe : x \u2208 e.source := Iff.mpr (Trivialization.mem_source e) h\n\u22a2 (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e.toLocalHomeomorph)) x).fst \u2208 e.baseSet\n[PROOFSTEP]\nrwa [e.toLocalHomeomorph.symm_symm, e.proj_toFun x he]\n[GOAL]\nE : Type u_1\nX : Type u_2\ninst\u271d\u00b9 : TopologicalSpace E\ninst\u271d : TopologicalSpace X\nf : E \u2192 X\ns : Set X\n\u22a2 IsCoveringMap f \u2194 IsCoveringMapOn f Set.univ\n[PROOFSTEP]\nsimp only [IsCoveringMap, IsCoveringMapOn, Set.mem_univ, forall_true_left]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Covering", "llama_tokens": 1915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.7025300698514778, "lm_q1q2_score": 0.5625102081619905}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 LeftMoves (powHalf n) = PUnit\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 LeftMoves (powHalf Nat.zero) = PUnit\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 LeftMoves (powHalf (Nat.succ n\u271d)) = PUnit\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\ni : LeftMoves (powHalf n)\n\u22a2 moveLeft (powHalf n) i = 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\ni : LeftMoves (powHalf Nat.zero)\n\u22a2 moveLeft (powHalf Nat.zero) i = 0\n[PROOFSTEP]\ncases i\n[GOAL]\ncase succ\nn\u271d : \u2115\ni : LeftMoves (powHalf (Nat.succ n\u271d))\n\u22a2 moveLeft (powHalf (Nat.succ n\u271d)) i = 0\n[PROOFSTEP]\ncases i\n[GOAL]\ncase zero.unit\n\u22a2 moveLeft (powHalf Nat.zero) PUnit.unit = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.unit\nn\u271d : \u2115\n\u22a2 moveLeft (powHalf (Nat.succ n\u271d)) PUnit.unit = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 Unique (LeftMoves (powHalf n))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 Unique (LeftMoves (powHalf Nat.zero))\n[PROOFSTEP]\nexact PUnit.unique\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 Unique (LeftMoves (powHalf (Nat.succ n\u271d)))\n[PROOFSTEP]\nexact PUnit.unique\n[GOAL]\n\u22a2 birthday (powHalf 1) = 2\n[PROOFSTEP]\nrw [birthday_def]\n[GOAL]\n\u22a2 max (Ordinal.lsub fun i => birthday (moveLeft (powHalf 1) i))\n      (Ordinal.lsub fun i => birthday (moveRight (powHalf 1) i)) =\n    2\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u22a2 max (Ordinal.lsub fun i => birthday (moveLeft (powHalf 1) i)) (Ordinal.lsub fun i => birthday 1) = 2\n[PROOFSTEP]\nsimpa using Order.le_succ (1 : Ordinal)\n[GOAL]\nn : \u2115\n\u22a2 Numeric (powHalf n)\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u22a2 Numeric (powHalf Nat.zero)\n[PROOFSTEP]\nexact numeric_one\n[GOAL]\ncase succ\nn : \u2115\nhn : Numeric (powHalf n)\n\u22a2 Numeric (powHalf (Nat.succ n))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase succ.left\nn : \u2115\nhn : Numeric (powHalf n)\n\u22a2 \u2200 (i j : PUnit), OfNat.ofNat 0 i < (fun x => powHalf n) j\n[PROOFSTEP]\nsimpa using hn.moveLeft_lt default\n[GOAL]\ncase succ.right\nn : \u2115\nhn : Numeric (powHalf n)\n\u22a2 (\u2200 (i : PUnit), Numeric (OfNat.ofNat 0 i)) \u2227 \u2200 (j : PUnit), Numeric ((fun x => powHalf n) j)\n[PROOFSTEP]\nexact \u27e8fun _ => numeric_zero, fun _ => hn\u27e9\n[GOAL]\nn : \u2115\n\u22a2 powHalf n \u2264 1\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u22a2 powHalf Nat.zero \u2264 1\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase succ\nn : \u2115\nhn : powHalf n \u2264 1\n\u22a2 powHalf (Nat.succ n) \u2264 1\n[PROOFSTEP]\nexact (powHalf_succ_le_powHalf n).trans hn\n[GOAL]\nn : \u2115\n\u22a2 0 < powHalf n\n[PROOFSTEP]\nrw [\u2190 lf_iff_lt numeric_zero (numeric_powHalf n), zero_lf_le]\n[GOAL]\nn : \u2115\n\u22a2 \u2203 i, 0 \u2264 moveLeft (powHalf n) i\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 powHalf (n + 1) + powHalf (n + 1) \u2248 powHalf n\n[PROOFSTEP]\ninduction' n using Nat.strong_induction_on with n hn\n[GOAL]\ncase h\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf (n + 1) + powHalf (n + 1) \u2248 powHalf n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf (n + 1) + powHalf (n + 1) \u2264 powHalf n\n[PROOFSTEP]\nrw [le_iff_forall_lf]\n[GOAL]\ncase h.right\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf n \u2264 powHalf (n + 1) + powHalf (n + 1)\n[PROOFSTEP]\nrw [le_iff_forall_lf]\n[GOAL]\ncase h.left\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 (\u2200 (i : LeftMoves (powHalf (n + 1) + powHalf (n + 1))), moveLeft (powHalf (n + 1) + powHalf (n + 1)) i \u29cf powHalf n) \u2227\n    \u2200 (j : RightMoves (powHalf n)), powHalf (n + 1) + powHalf (n + 1) \u29cf moveRight (powHalf n) j\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.right\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 (\u2200 (i : LeftMoves (powHalf n)), moveLeft (powHalf n) i \u29cf powHalf (n + 1) + powHalf (n + 1)) \u2227\n    \u2200 (j : RightMoves (powHalf (n + 1) + powHalf (n + 1))), powHalf n \u29cf moveRight (powHalf (n + 1) + powHalf (n + 1)) j\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left.left\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 \u2200 (i : LeftMoves (powHalf (n + 1) + powHalf (n + 1))), moveLeft (powHalf (n + 1) + powHalf (n + 1)) i \u29cf powHalf n\n[PROOFSTEP]\nrintro (\u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase h.left.left.inl.unit\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveLeft (powHalf (n + 1) + powHalf (n + 1)) (Sum.inl PUnit.unit) \u29cf powHalf n\n[PROOFSTEP]\napply lf_of_lt\n[GOAL]\ncase h.left.left.inr.unit\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveLeft (powHalf (n + 1) + powHalf (n + 1)) (Sum.inr PUnit.unit) \u29cf powHalf n\n[PROOFSTEP]\napply lf_of_lt\n[GOAL]\ncase h.left.left.inl.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveLeft (powHalf (n + 1) + powHalf (n + 1)) (Sum.inl PUnit.unit) < powHalf n\n[PROOFSTEP]\ncalc\n  0 + powHalf n.succ \u2248 powHalf n.succ := zero_add_equiv _\n  _ < powHalf n := powHalf_succ_lt_powHalf n\n[GOAL]\ncase h.left.left.inr.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveLeft (powHalf (n + 1) + powHalf (n + 1)) (Sum.inr PUnit.unit) < powHalf n\n[PROOFSTEP]\ncalc\n  powHalf n.succ + 0 \u2248 powHalf n.succ := add_zero_equiv _\n  _ < powHalf n := powHalf_succ_lt_powHalf n\n[GOAL]\ncase h.left.right\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 \u2200 (j : RightMoves (powHalf n)), powHalf (n + 1) + powHalf (n + 1) \u29cf moveRight (powHalf n) j\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase h.left.right.zero\nhn : \u2200 (m : \u2115), m < Nat.zero \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 \u2200 (j : RightMoves (powHalf Nat.zero)),\n    powHalf (Nat.zero + 1) + powHalf (Nat.zero + 1) \u29cf moveRight (powHalf Nat.zero) j\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase h.left.right.succ\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 \u2200 (j : RightMoves (powHalf (Nat.succ n))),\n    powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1) \u29cf moveRight (powHalf (Nat.succ n)) j\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase h.left.right.succ.unit\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1) \u29cf moveRight (powHalf (Nat.succ n)) PUnit.unit\n[PROOFSTEP]\napply lf_of_moveRight_le\n[GOAL]\ncase h.left.right.succ.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveRight (powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1)) ?h.left.right.succ.unit.j \u2264\n    moveRight (powHalf (Nat.succ n)) PUnit.unit\ncase h.left.right.succ.unit.j\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 RightMoves (powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1))\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.left.right.succ.unit.j\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 RightMoves (powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1))\ncase h.left.right.succ.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveRight (powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1)) ?h.left.right.succ.unit.j \u2264\n    moveRight (powHalf (Nat.succ n)) PUnit.unit\n[PROOFSTEP]\nexact Sum.inl default\n[GOAL]\ncase h.left.right.succ.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < Nat.succ n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 moveRight (powHalf (Nat.succ n + 1) + powHalf (Nat.succ n + 1)) (Sum.inl default) \u2264\n    moveRight (powHalf (Nat.succ n)) PUnit.unit\n[PROOFSTEP]\ncalc\n  powHalf n.succ + powHalf (n.succ + 1) \u2264 powHalf n.succ + powHalf n.succ :=\n    add_le_add_left (powHalf_succ_le_powHalf _) _\n  _ \u2248 powHalf n := hn _ (Nat.lt_succ_self n)\n[GOAL]\ncase h.right.left\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 \u2200 (i : LeftMoves (powHalf n)), moveLeft (powHalf n) i \u29cf powHalf (n + 1) + powHalf (n + 1)\n[PROOFSTEP]\nsimp only [powHalf_moveLeft, forall_const]\n[GOAL]\ncase h.right.left\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 0 \u29cf powHalf (n + 1) + powHalf (n + 1)\n[PROOFSTEP]\napply lf_of_lt\n[GOAL]\ncase h.right.left.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 0 < powHalf (n + 1) + powHalf (n + 1)\n[PROOFSTEP]\ncalc\n  0 \u2248 0 + 0 := (Equiv.symm (add_zero_equiv 0))\n  _ \u2264 powHalf n.succ + 0 := (add_le_add_right (zero_le_powHalf _) _)\n  _ < powHalf n.succ + powHalf n.succ := add_lt_add_left (powHalf_pos _) _\n[GOAL]\ncase h.right.right\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 \u2200 (j : RightMoves (powHalf (n + 1) + powHalf (n + 1))), powHalf n \u29cf moveRight (powHalf (n + 1) + powHalf (n + 1)) j\n[PROOFSTEP]\nrintro (\u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n[GOAL]\ncase h.right.right.inl.unit\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf n \u29cf moveRight (powHalf (n + 1) + powHalf (n + 1)) (Sum.inl PUnit.unit)\n[PROOFSTEP]\napply lf_of_lt\n[GOAL]\ncase h.right.right.inr.unit\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf n \u29cf moveRight (powHalf (n + 1) + powHalf (n + 1)) (Sum.inr PUnit.unit)\n[PROOFSTEP]\napply lf_of_lt\n[GOAL]\ncase h.right.right.inl.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf n < moveRight (powHalf (n + 1) + powHalf (n + 1)) (Sum.inl PUnit.unit)\n[PROOFSTEP]\ncalc\n  powHalf n \u2248 powHalf n + 0 := (Equiv.symm (add_zero_equiv _))\n  _ < powHalf n + powHalf n.succ := add_lt_add_left (powHalf_pos _) _\n[GOAL]\ncase h.right.right.inr.unit.h\nn : \u2115\nhn : \u2200 (m : \u2115), m < n \u2192 powHalf (m + 1) + powHalf (m + 1) \u2248 powHalf m\n\u22a2 powHalf n < moveRight (powHalf (n + 1) + powHalf (n + 1)) (Sum.inr PUnit.unit)\n[PROOFSTEP]\ncalc\n  powHalf n \u2248 0 + powHalf n := (Equiv.symm (zero_add_equiv _))\n  _ < powHalf n.succ + powHalf n := add_lt_add_right (powHalf_pos _) _\n[GOAL]\nn : \u2115\n\u22a2 2 \u2022 powHalf (Nat.succ n) = powHalf n\n[PROOFSTEP]\nrw [two_nsmul]\n[GOAL]\nn : \u2115\n\u22a2 powHalf (Nat.succ n) + powHalf (Nat.succ n) = powHalf n\n[PROOFSTEP]\nexact Quotient.sound (PGame.add_powHalf_succ_self_eq_powHalf n)\n[GOAL]\nn : \u2115\n\u22a2 2 ^ n \u2022 powHalf n = 1\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u22a2 2 ^ Nat.zero \u2022 powHalf Nat.zero = 1\n[PROOFSTEP]\nsimp only [Nat.zero_eq, pow_zero, powHalf_zero, one_smul]\n[GOAL]\ncase succ\nn : \u2115\nhn : 2 ^ n \u2022 powHalf n = 1\n\u22a2 2 ^ Nat.succ n \u2022 powHalf (Nat.succ n) = 1\n[PROOFSTEP]\nrw [\u2190 hn, \u2190 double_powHalf_succ_eq_powHalf n, smul_smul (2 ^ n) 2 (powHalf n.succ), mul_comm, pow_succ]\n[GOAL]\nn k : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + k) = powHalf k\n[PROOFSTEP]\ninduction' k with k hk\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.zero) = powHalf Nat.zero\n[PROOFSTEP]\nsimp only [add_zero, Surreal.nsmul_pow_two_powHalf, Nat.zero_eq, eq_self_iff_true, Surreal.powHalf_zero]\n[GOAL]\ncase succ\nn k : \u2115\nhk : 2 ^ n \u2022 powHalf (n + k) = powHalf k\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k)\n[PROOFSTEP]\nrw [\u2190 double_powHalf_succ_eq_powHalf (n + k), \u2190 double_powHalf_succ_eq_powHalf k, smul_algebra_smul_comm] at hk \n[GOAL]\ncase succ\nn k : \u2115\nhk : 2 \u2022 2 ^ n \u2022 powHalf (Nat.succ (n + k)) = 2 \u2022 powHalf (Nat.succ k)\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k)\n[PROOFSTEP]\nrwa [\u2190 zsmul_eq_zsmul_iff' two_ne_zero]\n[GOAL]\nm : \u2124\nn k : \u2115\n\u22a2 (m * 2 ^ n) \u2022 powHalf (n + k) = m \u2022 powHalf k\n[PROOFSTEP]\nrw [mul_zsmul]\n[GOAL]\nm : \u2124\nn k : \u2115\n\u22a2 m \u2022 2 ^ n \u2022 powHalf (n + k) = m \u2022 powHalf k\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nm : \u2124\nn k : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + k) = powHalf k\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase e_a\nm : \u2124\nn k : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + k) = powHalf k\n[PROOFSTEP]\nexact nsmul_pow_two_powHalf' n k\n[GOAL]\nm\u2081 m\u2082 : \u2124\ny\u2081 y\u2082 : \u2115\nh\u2082 : m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082\n\u22a2 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nrevert m\u2081 m\u2082\n[GOAL]\ny\u2081 y\u2082 : \u2115\n\u22a2 \u2200 {m\u2081 m\u2082 : \u2124}, m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082 \u2192 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nwlog h : y\u2081 \u2264 y\u2082\n[GOAL]\ncase inr\ny\u2081 y\u2082 : \u2115\nthis : \u2200 {y\u2081 y\u2082 : \u2115}, y\u2081 \u2264 y\u2082 \u2192 \u2200 {m\u2081 m\u2082 : \u2124}, m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082 \u2192 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\nh : \u00acy\u2081 \u2264 y\u2082\n\u22a2 \u2200 {m\u2081 m\u2082 : \u2124}, m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082 \u2192 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nintro m\u2081 m\u2082 aux\n[GOAL]\ncase inr\ny\u2081 y\u2082 : \u2115\nthis : \u2200 {y\u2081 y\u2082 : \u2115}, y\u2081 \u2264 y\u2082 \u2192 \u2200 {m\u2081 m\u2082 : \u2124}, m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082 \u2192 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\nh : \u00acy\u2081 \u2264 y\u2082\nm\u2081 m\u2082 : \u2124\naux : m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082\n\u22a2 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nexact (this (le_of_not_le h) aux.symm).symm\n[GOAL]\ny\u2081 y\u2082 : \u2115\nh : y\u2081 \u2264 y\u2082\n\u22a2 \u2200 {m\u2081 m\u2082 : \u2124}, m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082 \u2192 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nintro m\u2081 m\u2082 h\u2082\n[GOAL]\ny\u2081 y\u2082 : \u2115\nh : y\u2081 \u2264 y\u2082\nm\u2081 m\u2082 : \u2124\nh\u2082 : m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ y\u2082\n\u22a2 m\u2081 \u2022 powHalf y\u2082 = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := le_iff_exists_add.mp h\n[GOAL]\ncase intro\ny\u2081 : \u2115\nm\u2081 m\u2082 : \u2124\nc : \u2115\nh : y\u2081 \u2264 y\u2081 + c\nh\u2082 : m\u2081 * 2 ^ y\u2081 = m\u2082 * 2 ^ (y\u2081 + c)\n\u22a2 m\u2081 \u2022 powHalf (y\u2081 + c) = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nrw [add_comm, pow_add, \u2190 mul_assoc, mul_eq_mul_right_iff] at h\u2082 \n[GOAL]\ncase intro\ny\u2081 : \u2115\nm\u2081 m\u2082 : \u2124\nc : \u2115\nh : y\u2081 \u2264 y\u2081 + c\nh\u2082 : m\u2081 = m\u2082 * 2 ^ c \u2228 2 ^ y\u2081 = 0\n\u22a2 m\u2081 \u2022 powHalf (y\u2081 + c) = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\ncases' h\u2082 with h\u2082 h\u2082\n[GOAL]\ncase intro.inl\ny\u2081 : \u2115\nm\u2081 m\u2082 : \u2124\nc : \u2115\nh : y\u2081 \u2264 y\u2081 + c\nh\u2082 : m\u2081 = m\u2082 * 2 ^ c\n\u22a2 m\u2081 \u2022 powHalf (y\u2081 + c) = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nrw [h\u2082, add_comm, zsmul_pow_two_powHalf m\u2082 c y\u2081]\n[GOAL]\ncase intro.inr\ny\u2081 : \u2115\nm\u2081 m\u2082 : \u2124\nc : \u2115\nh : y\u2081 \u2264 y\u2081 + c\nh\u2082 : 2 ^ y\u2081 = 0\n\u22a2 m\u2081 \u2022 powHalf (y\u2081 + c) = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nhave := Nat.one_le_pow y\u2081 2 Nat.succ_pos'\n[GOAL]\ncase intro.inr\ny\u2081 : \u2115\nm\u2081 m\u2082 : \u2124\nc : \u2115\nh : y\u2081 \u2264 y\u2081 + c\nh\u2082 : 2 ^ y\u2081 = 0\nthis : 1 \u2264 2 ^ y\u2081\n\u22a2 m\u2081 \u2022 powHalf (y\u2081 + c) = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nnorm_cast at h\u2082 \n[GOAL]\ncase intro.inr\ny\u2081 : \u2115\nm\u2081 m\u2082 : \u2124\nc : \u2115\nh : y\u2081 \u2264 y\u2081 + c\nthis : 1 \u2264 2 ^ y\u2081\nh\u2082 : 2 ^ y\u2081 = 0\n\u22a2 m\u2081 \u2022 powHalf (y\u2081 + c) = m\u2082 \u2022 powHalf y\u2081\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : Localization.Away 2\n\u22a2 \u2200 {a c : \u2124} {b d : { x // x \u2208 Submonoid.powers 2 }},\n    \u2191(Localization.r (Submonoid.powers 2)) (a, b) (c, d) \u2192 a \u2022 powHalf (Submonoid.log b) = c \u2022 powHalf (Submonoid.log d)\n[PROOFSTEP]\nintro m\u2081 m\u2082 n\u2081 n\u2082 h\u2081\n[GOAL]\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nobtain \u27e8\u27e8n\u2083, y\u2083, hn\u2083\u27e9, h\u2082\u27e9 := Localization.r_iff_exists.mp h\u2081\n[GOAL]\ncase intro.mk.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u2082 :\n  \u2191{ val := n\u2083, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = n\u2083) } * (\u2191(m\u2082, n\u2082).snd * (m\u2081, n\u2081).fst) =\n    \u2191{ val := n\u2083, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = n\u2083) } * (\u2191(m\u2081, n\u2081).snd * (m\u2082, n\u2082).fst)\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, mul_eq_mul_left_iff] at h\u2082 \n[GOAL]\ncase intro.mk.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u2082 : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082 \u2228 n\u2083 = 0\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\ncases h\u2082\n[GOAL]\ncase intro.mk.intro.inl\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nobtain \u27e8a\u2081, ha\u2081\u27e9 := n\u2081.prop\n[GOAL]\ncase intro.mk.intro.inl.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : (fun x x_1 => x ^ x_1) 2 a\u2081 = \u2191n\u2081\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nobtain \u27e8a\u2082, ha\u2082\u27e9 := n\u2082.prop\n[GOAL]\ncase intro.mk.intro.inl.intro.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : (fun x x_1 => x ^ x_1) 2 a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : (fun x x_1 => x ^ x_1) 2 a\u2082 = \u2191n\u2082\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nsimp only at ha\u2081 ha\u2082 \u22a2\n[GOAL]\ncase intro.mk.intro.inl.intro.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : 2 ^ a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : 2 ^ a\u2082 = \u2191n\u2082\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nhave hn\u2081 : n\u2081 = Submonoid.pow 2 a\u2081 := Subtype.ext ha\u2081.symm\n[GOAL]\ncase intro.mk.intro.inl.intro.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : 2 ^ a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : 2 ^ a\u2082 = \u2191n\u2082\nhn\u2081 : n\u2081 = Submonoid.pow 2 a\u2081\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nhave hn\u2082 : n\u2082 = Submonoid.pow 2 a\u2082 := Subtype.ext ha\u2082.symm\n[GOAL]\ncase intro.mk.intro.inl.intro.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : 2 ^ a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : 2 ^ a\u2082 = \u2191n\u2082\nhn\u2081 : n\u2081 = Submonoid.pow 2 a\u2081\nhn\u2082 : n\u2082 = Submonoid.pow 2 a\u2082\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nhave h\u2082 : 1 < (2 : \u2124).natAbs := one_lt_two\n[GOAL]\ncase intro.mk.intro.inl.intro.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : 2 ^ a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : 2 ^ a\u2082 = \u2191n\u2082\nhn\u2081 : n\u2081 = Submonoid.pow 2 a\u2081\nhn\u2082 : n\u2082 = Submonoid.pow 2 a\u2082\nh\u2082 : 1 < Int.natAbs 2\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nrw [hn\u2081, hn\u2082, Submonoid.log_pow_int_eq_self h\u2082, Submonoid.log_pow_int_eq_self h\u2082]\n[GOAL]\ncase intro.mk.intro.inl.intro.intro\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : 2 ^ a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : 2 ^ a\u2082 = \u2191n\u2082\nhn\u2081 : n\u2081 = Submonoid.pow 2 a\u2081\nhn\u2082 : n\u2082 = Submonoid.pow 2 a\u2082\nh\u2082 : 1 < Int.natAbs 2\n\u22a2 m\u2081 \u2022 powHalf a\u2081 = m\u2082 \u2022 powHalf a\u2082\n[PROOFSTEP]\napply dyadic_aux\n[GOAL]\ncase intro.mk.intro.inl.intro.intro.h\u2082\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : \u2191n\u2082 * m\u2081 = \u2191n\u2081 * m\u2082\na\u2081 : \u2115\nha\u2081 : 2 ^ a\u2081 = \u2191n\u2081\na\u2082 : \u2115\nha\u2082 : 2 ^ a\u2082 = \u2191n\u2082\nhn\u2081 : n\u2081 = Submonoid.pow 2 a\u2081\nhn\u2082 : n\u2082 = Submonoid.pow 2 a\u2082\nh\u2082 : 1 < Int.natAbs 2\n\u22a2 m\u2081 * 2 ^ a\u2082 = m\u2082 * 2 ^ a\u2081\n[PROOFSTEP]\nrwa [ha\u2081, ha\u2082, mul_comm, mul_comm m\u2082]\n[GOAL]\ncase intro.mk.intro.inr\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : n\u2083 = 0\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nhave : (1 : \u2124) \u2264 2 ^ y\u2083 := by exact_mod_cast Nat.one_le_pow y\u2083 2 Nat.succ_pos'\n[GOAL]\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : n\u2083 = 0\n\u22a2 1 \u2264 2 ^ y\u2083\n[PROOFSTEP]\nexact_mod_cast Nat.one_le_pow y\u2083 2 Nat.succ_pos'\n[GOAL]\ncase intro.mk.intro.inr\nx : Localization.Away 2\nm\u2081 m\u2082 : \u2124\nn\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }\nh\u2081 : \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082)\nn\u2083 : \u2124\ny\u2083 : \u2115\nhn\u2083 : (fun x x_1 => x ^ x_1) 2 y\u2083 = n\u2083\nh\u271d : n\u2083 = 0\nthis : 1 \u2264 2 ^ y\u2083\n\u22a2 m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : Localization.Away 2\n\u22a2 \u2200 (x y : \u2124 \u00d7 { x // x \u2208 Submonoid.powers 2 }),\n    ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk x.fst x.snd + Localization.mk y.fst y.snd) =\n      ZeroHom.toFun\n          {\n            toFun := fun x =>\n              Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n                (_ :\n                  \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                    \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                      m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n            map_zero' :=\n              (_ :\n                Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                    (_ :\n                      \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                        \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                          m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                  0 \u2022 powHalf (Submonoid.log 1)) }\n          (Localization.mk x.fst x.snd) +\n        ZeroHom.toFun\n          {\n            toFun := fun x =>\n              Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n                (_ :\n                  \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                    \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                      m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n            map_zero' :=\n              (_ :\n                Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                    (_ :\n                      \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                        \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                          m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                  0 \u2022 powHalf (Submonoid.log 1)) }\n          (Localization.mk y.fst y.snd)\n[PROOFSTEP]\nrintro \u27e8a, \u27e8b, \u27e8b', rfl\u27e9\u27e9\u27e9 \u27e8c, \u27e8d, \u27e8d', rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase mk.mk.intro.mk.mk.intro\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun x =>\n          Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n            (_ :\n              \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                  m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n        map_zero' :=\n          (_ :\n            Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                (_ :\n                  \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                    \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                      m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n              0 \u2022 powHalf (Submonoid.log 1)) }\n      (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd +\n        Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd) =\n    ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd) +\n      ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd)\n[PROOFSTEP]\nhave h\u2082 : 1 < (2 : \u2124).natAbs := one_lt_two\n[GOAL]\ncase mk.mk.intro.mk.mk.intro\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\nh\u2082 : 1 < Int.natAbs 2\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun x =>\n          Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n            (_ :\n              \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                  m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n        map_zero' :=\n          (_ :\n            Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                (_ :\n                  \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                    \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                      m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n              0 \u2022 powHalf (Submonoid.log 1)) }\n      (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd +\n        Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd) =\n    ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd) +\n      ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd)\n[PROOFSTEP]\nhave hpow\u2082 := Submonoid.log_pow_int_eq_self h\u2082\n[GOAL]\ncase mk.mk.intro.mk.mk.intro\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\nh\u2082 : 1 < Int.natAbs 2\nhpow\u2082 : \u2200 (m : \u2115), Submonoid.log (Submonoid.pow 2 m) = m\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun x =>\n          Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n            (_ :\n              \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                  m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n        map_zero' :=\n          (_ :\n            Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                (_ :\n                  \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                    \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                      m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n              0 \u2022 powHalf (Submonoid.log 1)) }\n      (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd +\n        Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd) =\n    ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd) +\n      ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd)\n[PROOFSTEP]\nsimp_rw [Submonoid.pow_apply] at hpow\u2082 \n[GOAL]\ncase mk.mk.intro.mk.mk.intro\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\nh\u2082 : 1 < Int.natAbs 2\nhpow\u2082 : \u2200 (m : \u2115), Submonoid.log { val := 2 ^ m, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = 2 ^ m) } = m\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun x =>\n          Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n            (_ :\n              \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                  m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n        map_zero' :=\n          (_ :\n            Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                (_ :\n                  \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                    \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                      m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n              0 \u2022 powHalf (Submonoid.log 1)) }\n      (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd +\n        Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd) =\n    ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).fst\n          (a,\n              { val := (fun x x_1 => x ^ x_1) 2 b',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 b') }).snd) +\n      ZeroHom.toFun\n        {\n          toFun := fun x =>\n            Localization.liftOn x (fun x y => x \u2022 powHalf (Submonoid.log y))\n              (_ :\n                \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                  \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                    m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)),\n          map_zero' :=\n            (_ :\n              Localization.liftOn 0 (fun x y => x \u2022 powHalf (Submonoid.log y))\n                  (_ :\n                    \u2200 {m\u2081 m\u2082 : \u2124} {n\u2081 n\u2082 : { x // x \u2208 Submonoid.powers 2 }},\n                      \u2191(Localization.r (Submonoid.powers 2)) (m\u2081, n\u2081) (m\u2082, n\u2082) \u2192\n                        m\u2081 \u2022 powHalf (Submonoid.log n\u2081) = m\u2082 \u2022 powHalf (Submonoid.log n\u2082)) =\n                0 \u2022 powHalf (Submonoid.log 1)) }\n        (Localization.mk\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).fst\n          (c,\n              { val := (fun x x_1 => x ^ x_1) 2 d',\n                property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = (fun x x_1 => x ^ x_1) 2 d') }).snd)\n[PROOFSTEP]\nsimp_rw [Localization.add_mk, Localization.liftOn_mk, Submonoid.log_mul (Int.pow_right_injective h\u2082), hpow\u2082]\n[GOAL]\ncase mk.mk.intro.mk.mk.intro\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\nh\u2082 : 1 < Int.natAbs 2\nhpow\u2082 : \u2200 (m : \u2115), Submonoid.log { val := 2 ^ m, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = 2 ^ m) } = m\n\u22a2 (2 ^ b' * c + 2 ^ d' * a) \u2022 powHalf (b' + d') = a \u2022 powHalf b' + c \u2022 powHalf d'\n[PROOFSTEP]\ncalc\n  (2 ^ b' * c + 2 ^ d' * a) \u2022 powHalf (b' + d') = (c * 2 ^ b') \u2022 powHalf (b' + d') + (a * 2 ^ d') \u2022 powHalf (d' + b') :=\n    by simp only [add_smul, mul_comm, add_comm]\n  _ = c \u2022 powHalf d' + a \u2022 powHalf b' := by simp only [zsmul_pow_two_powHalf]\n  _ = a \u2022 powHalf b' + c \u2022 powHalf d' := add_comm _ _\n[GOAL]\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\nh\u2082 : 1 < Int.natAbs 2\nhpow\u2082 : \u2200 (m : \u2115), Submonoid.log { val := 2 ^ m, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = 2 ^ m) } = m\n\u22a2 (2 ^ b' * c + 2 ^ d' * a) \u2022 powHalf (b' + d') = (c * 2 ^ b') \u2022 powHalf (b' + d') + (a * 2 ^ d') \u2022 powHalf (d' + b')\n[PROOFSTEP]\nsimp only [add_smul, mul_comm, add_comm]\n[GOAL]\nx y : Localization.Away 2\na : \u2124\nb' : \u2115\nc : \u2124\nd' : \u2115\nh\u2082 : 1 < Int.natAbs 2\nhpow\u2082 : \u2200 (m : \u2115), Submonoid.log { val := 2 ^ m, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) 2 y = 2 ^ m) } = m\n\u22a2 (c * 2 ^ b') \u2022 powHalf (b' + d') + (a * 2 ^ d') \u2022 powHalf (d' + b') = c \u2022 powHalf d' + a \u2022 powHalf b'\n[PROOFSTEP]\nsimp only [zsmul_pow_two_powHalf]\n[GOAL]\nm : \u2124\np : { x // x \u2208 Submonoid.powers 2 }\n\u22a2 \u2191dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m p) = m \u2022 powHalf (Submonoid.log p)\n[PROOFSTEP]\nrw [\u2190 Localization.mk_eq_mk']\n[GOAL]\nm : \u2124\np : { x // x \u2208 Submonoid.powers 2 }\n\u22a2 \u2191dyadicMap (Localization.mk m p) = m \u2022 powHalf (Submonoid.log p)\n[PROOFSTEP]\nrfl\n[GOAL]\nm : \u2124\nn : \u2115\n\u22a2 \u2191dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m (Submonoid.pow 2 n)) = m \u2022 powHalf n\n[PROOFSTEP]\nrw [dyadicMap_apply, @Submonoid.log_pow_int_eq_self 2 one_lt_two]\n[GOAL]\nm : \u2124\nn : \u2115\n\u22a2 m \u2022 powHalf (Submonoid.log (Submonoid.pow 2 n)) = m \u2022 powHalf n\n[PROOFSTEP]\nrw [@Submonoid.log_pow_int_eq_self 2 one_lt_two]\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Surreal.Dyadic", "llama_tokens": 20181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.857768108626046, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.5617478015245491}}
{"text": "[GOAL]\n\u22a2 \u2200 {X' X : Type u} {Y : AddCommGroupCat} (f : X' \u27f6 X) (g : X \u27f6 (forget AddCommGroupCat).obj Y),\n    \u2191((fun X G => FreeAbelianGroup.lift.symm) X' Y).symm (f \u226b g) =\n      free.map f \u226b \u2191((fun X G => FreeAbelianGroup.lift.symm) X Y).symm g\n[PROOFSTEP]\nintros\n[GOAL]\nX'\u271d X\u271d : Type u\nY\u271d : AddCommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget AddCommGroupCat).obj Y\u271d\n\u22a2 \u2191((fun X G => FreeAbelianGroup.lift.symm) X'\u271d Y\u271d).symm (f\u271d \u226b g\u271d) =\n    free.map f\u271d \u226b \u2191((fun X G => FreeAbelianGroup.lift.symm) X\u271d Y\u271d).symm g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w\nX'\u271d X\u271d : Type u\nY\u271d : AddCommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget AddCommGroupCat).obj Y\u271d\nx\u271d : \u2191(free.obj X'\u271d)\n\u22a2 \u2191(\u2191((fun X G => FreeAbelianGroup.lift.symm) X'\u271d Y\u271d).symm (f\u271d \u226b g\u271d)) x\u271d =\n    \u2191(free.map f\u271d \u226b \u2191((fun X G => FreeAbelianGroup.lift.symm) X\u271d Y\u271d).symm g\u271d) x\u271d\n[PROOFSTEP]\nsimp only [Equiv.symm_symm]\n[GOAL]\ncase w\nX'\u271d X\u271d : Type u\nY\u271d : AddCommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget AddCommGroupCat).obj Y\u271d\nx\u271d : \u2191(free.obj X'\u271d)\n\u22a2 \u2191(\u2191FreeAbelianGroup.lift (f\u271d \u226b g\u271d)) x\u271d = \u2191(free.map f\u271d \u226b \u2191FreeAbelianGroup.lift g\u271d) x\u271d\n[PROOFSTEP]\napply FreeAbelianGroup.lift_comp\n[GOAL]\n\u22a2 \u2200 (X : Type u),\n    { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map (\ud835\udfd9 X) =\n      \ud835\udfd9 ({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X)\n[PROOFSTEP]\nintros\n[GOAL]\nX\u271d : Type u\n\u22a2 { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map (\ud835\udfd9 X\u271d) =\n    \ud835\udfd9 ({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nX\u271d : Type u\nx\u271d : \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\n\u22a2 \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map (\ud835\udfd9 X\u271d)) x\u271d =\n    \u2191(\ud835\udfd9 ({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)) x\u271d\n[PROOFSTEP]\nrw [\u2190 FreeGroup.map.unique]\n[GOAL]\ncase w.hg\nX\u271d : Type u\nx\u271d : \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\n\u22a2 \u2200 (x : X\u271d),\n    \u2191(\ud835\udfd9 ({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)) (FreeGroup.of x) =\n      FreeGroup.of (\ud835\udfd9 X\u271d x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase w.hg\nX\u271d : Type u\nx\u271d\u00b9 : \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\nx\u271d : X\u271d\n\u22a2 \u2191(\ud835\udfd9 ({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)) (FreeGroup.of x\u271d) =\n    FreeGroup.of (\ud835\udfd9 X\u271d x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 {X Y Z : Type u} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n    { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map (f \u226b g) =\n      { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map f \u226b\n        { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map g\n[PROOFSTEP]\nintros\n[GOAL]\nX\u271d Y\u271d Z\u271d : Type u\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map (f\u271d \u226b g\u271d) =\n    { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map f\u271d \u226b\n      { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map g\u271d\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nX\u271d Y\u271d Z\u271d : Type u\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\n\u22a2 \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map (f\u271d \u226b g\u271d)) x\u271d =\n    \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map f\u271d \u226b\n          { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map g\u271d)\n      x\u271d\n[PROOFSTEP]\nrw [\u2190 FreeGroup.map.unique]\n[GOAL]\ncase w.hg\nX\u271d Y\u271d Z\u271d : Type u\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\n\u22a2 \u2200 (x : X\u271d),\n    \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map f\u271d \u226b\n            { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map g\u271d)\n        (FreeGroup.of x) =\n      FreeGroup.of ((f\u271d \u226b g\u271d) x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase w.hg\nX\u271d Y\u271d Z\u271d : Type u\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d\u00b9 : \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.obj X\u271d)\nx\u271d : X\u271d\n\u22a2 \u2191({ obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map f\u271d \u226b\n          { obj := fun \u03b1 => of (FreeGroup \u03b1), map := fun {X Y} => FreeGroup.map }.map g\u271d)\n      (FreeGroup.of x\u271d) =\n    FreeGroup.of ((f\u271d \u226b g\u271d) x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 {X' X : Type u} {Y : GroupCat} (f : X' \u27f6 X) (g : X \u27f6 (forget GroupCat).obj Y),\n    \u2191((fun X G => FreeGroup.lift.symm) X' Y).symm (f \u226b g) = free.map f \u226b \u2191((fun X G => FreeGroup.lift.symm) X Y).symm g\n[PROOFSTEP]\nintros\n[GOAL]\nX'\u271d X\u271d : Type u\nY\u271d : GroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget GroupCat).obj Y\u271d\n\u22a2 \u2191((fun X G => FreeGroup.lift.symm) X'\u271d Y\u271d).symm (f\u271d \u226b g\u271d) =\n    free.map f\u271d \u226b \u2191((fun X G => FreeGroup.lift.symm) X\u271d Y\u271d).symm g\u271d\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nX'\u271d X\u271d : Type u\nY\u271d : GroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget GroupCat).obj Y\u271d\nx\u271d : \u2191(free.obj X'\u271d)\n\u22a2 \u2191(\u2191((fun X G => FreeGroup.lift.symm) X'\u271d Y\u271d).symm (f\u271d \u226b g\u271d)) x\u271d =\n    \u2191(free.map f\u271d \u226b \u2191((fun X G => FreeGroup.lift.symm) X\u271d Y\u271d).symm g\u271d) x\u271d\n[PROOFSTEP]\nsimp only [Equiv.symm_symm]\n[GOAL]\ncase w\nX'\u271d X\u271d : Type u\nY\u271d : GroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget GroupCat).obj Y\u271d\nx\u271d : \u2191(free.obj X'\u271d)\n\u22a2 \u2191(\u2191FreeGroup.lift (f\u271d \u226b g\u271d)) x\u271d = \u2191(free.map f\u271d \u226b \u2191FreeGroup.lift g\u271d) x\u271d\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase w.h\nX'\u271d X\u271d : Type u\nY\u271d : GroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget GroupCat).obj Y\u271d\nx\u271d : \u2191(free.obj X'\u271d)\n\u22a2 \u2191(free.map f\u271d \u226b \u2191FreeGroup.lift g\u271d) x\u271d = \u2191(\u2191FreeGroup.lift (f\u271d \u226b g\u271d)) x\u271d\n[PROOFSTEP]\napply FreeGroup.lift.unique\n[GOAL]\ncase w.h.hg\nX'\u271d X\u271d : Type u\nY\u271d : GroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget GroupCat).obj Y\u271d\nx\u271d : \u2191(free.obj X'\u271d)\n\u22a2 \u2200 (x : X'\u271d), \u2191(free.map f\u271d \u226b \u2191FreeGroup.lift g\u271d) (FreeGroup.of x) = (f\u271d \u226b g\u271d) x\n[PROOFSTEP]\nintros\n[GOAL]\ncase w.h.hg\nX'\u271d X\u271d : Type u\nY\u271d : GroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget GroupCat).obj Y\u271d\nx\u271d\u00b9 : \u2191(free.obj X'\u271d)\nx\u271d : X'\u271d\n\u22a2 \u2191(free.map f\u271d \u226b \u2191FreeGroup.lift g\u271d) (FreeGroup.of x\u271d) = (f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\napply FreeGroup.lift.of\n[GOAL]\nG : GroupCat\n\u22a2 CommGroup (Abelianization \u2191G)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX\u271d Y\u271d : GroupCat\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (fun x => \u2191Abelianization.of (\u2191f x)) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nX\u271d Y\u271d : GroupCat\nf : X\u271d \u27f6 Y\u271d\n\u22a2 \u2200 (x y : \u2191X\u271d),\n    OneHom.toFun { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) }\n        (x * y) =\n      OneHom.toFun { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) } x *\n        OneHom.toFun { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) } y\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 (X : GroupCat),\n    { obj := fun G => Bundled.mk (Abelianization \u2191G),\n            map := fun {X Y} f =>\n              \u2191Abelianization.lift\n                {\n                  toOneHom :=\n                    { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                  map_mul' :=\n                    (_ :\n                      \u2200 (a a_1 : \u2191X),\n                        \u2191Abelianization.of (\u2191f (a * a_1)) =\n                          \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n        (\ud835\udfd9 X) =\n      \ud835\udfd9\n        ({ obj := fun G => Bundled.mk (Abelianization \u2191G),\n              map := fun {X Y} f =>\n                \u2191Abelianization.lift\n                  {\n                    toOneHom :=\n                      { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                    map_mul' :=\n                      (_ :\n                        \u2200 (a a_1 : \u2191X),\n                          \u2191Abelianization.of (\u2191f (a * a_1)) =\n                            \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.obj\n          X)\n[PROOFSTEP]\nintros\n[GOAL]\nX\u271d : GroupCat\n\u22a2 { obj := fun G => Bundled.mk (Abelianization \u2191G),\n          map := fun {X Y} f =>\n            \u2191Abelianization.lift\n              {\n                toOneHom :=\n                  { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (a a_1 : \u2191X),\n                      \u2191Abelianization.of (\u2191f (a * a_1)) =\n                        \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n      (\ud835\udfd9 X\u271d) =\n    \ud835\udfd9\n      ({ obj := fun G => Bundled.mk (Abelianization \u2191G),\n            map := fun {X Y} f =>\n              \u2191Abelianization.lift\n                {\n                  toOneHom :=\n                    { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                  map_mul' :=\n                    (_ :\n                      \u2200 (a a_1 : \u2191X),\n                        \u2191Abelianization.of (\u2191f (a * a_1)) =\n                          \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.obj\n        X\u271d)\n[PROOFSTEP]\nsimp only [MonoidHom.mk_coe, coe_id]\n[GOAL]\nX\u271d : GroupCat\n\u22a2 \u2191Abelianization.lift\n      {\n        toOneHom :=\n          { toFun := fun x => \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) x), map_one' := (_ : \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) 1) = 1) },\n        map_mul' :=\n          (_ :\n            \u2200 (a a_1 : \u2191X\u271d),\n              \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) (a * a_1)) =\n                \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) a) * \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) a_1)) } =\n    \ud835\udfd9 (Bundled.mk (Abelianization \u2191X\u271d))\n[PROOFSTEP]\napply (Equiv.apply_eq_iff_eq_symm_apply Abelianization.lift).mpr\n[GOAL]\nX\u271d : GroupCat\n\u22a2 {\n      toOneHom :=\n        { toFun := fun x => \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) x), map_one' := (_ : \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) 1) = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (a a_1 : \u2191X\u271d),\n            \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) (a * a_1)) =\n              \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) a) * \u2191Abelianization.of (\u2191(\ud835\udfd9 X\u271d) a_1)) } =\n    \u2191Abelianization.lift.symm (\ud835\udfd9 (Bundled.mk (Abelianization \u2191X\u271d)))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 {X Y Z : GroupCat} (f : X \u27f6 Y) (g : Y \u27f6 Z),\n    { obj := fun G => Bundled.mk (Abelianization \u2191G),\n            map := fun {X Y} f =>\n              \u2191Abelianization.lift\n                {\n                  toOneHom :=\n                    { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                  map_mul' :=\n                    (_ :\n                      \u2200 (a a_1 : \u2191X),\n                        \u2191Abelianization.of (\u2191f (a * a_1)) =\n                          \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n        (f \u226b g) =\n      { obj := fun G => Bundled.mk (Abelianization \u2191G),\n              map := fun {X Y} f =>\n                \u2191Abelianization.lift\n                  {\n                    toOneHom :=\n                      { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                    map_mul' :=\n                      (_ :\n                        \u2200 (a a_1 : \u2191X),\n                          \u2191Abelianization.of (\u2191f (a * a_1)) =\n                            \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n          f \u226b\n        { obj := fun G => Bundled.mk (Abelianization \u2191G),\n              map := fun {X Y} f =>\n                \u2191Abelianization.lift\n                  {\n                    toOneHom :=\n                      { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                    map_mul' :=\n                      (_ :\n                        \u2200 (a a_1 : \u2191X),\n                          \u2191Abelianization.of (\u2191f (a * a_1)) =\n                            \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n          g\n[PROOFSTEP]\nintros\n[GOAL]\nX\u271d Y\u271d Z\u271d : GroupCat\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun G => Bundled.mk (Abelianization \u2191G),\n          map := fun {X Y} f =>\n            \u2191Abelianization.lift\n              {\n                toOneHom :=\n                  { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (a a_1 : \u2191X),\n                      \u2191Abelianization.of (\u2191f (a * a_1)) =\n                        \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n      (f\u271d \u226b g\u271d) =\n    { obj := fun G => Bundled.mk (Abelianization \u2191G),\n            map := fun {X Y} f =>\n              \u2191Abelianization.lift\n                {\n                  toOneHom :=\n                    { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                  map_mul' :=\n                    (_ :\n                      \u2200 (a a_1 : \u2191X),\n                        \u2191Abelianization.of (\u2191f (a * a_1)) =\n                          \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n        f\u271d \u226b\n      { obj := fun G => Bundled.mk (Abelianization \u2191G),\n            map := fun {X Y} f =>\n              \u2191Abelianization.lift\n                {\n                  toOneHom :=\n                    { toFun := fun x => \u2191Abelianization.of (\u2191f x), map_one' := (_ : \u2191Abelianization.of (\u2191f 1) = 1) },\n                  map_mul' :=\n                    (_ :\n                      \u2200 (a a_1 : \u2191X),\n                        \u2191Abelianization.of (\u2191f (a * a_1)) =\n                          \u2191Abelianization.of (\u2191f a) * \u2191Abelianization.of (\u2191f a_1)) } }.map\n        g\u271d\n[PROOFSTEP]\nsimp only [coe_comp]\n[GOAL]\nX\u271d Y\u271d Z\u271d : GroupCat\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 \u2191Abelianization.lift\n      {\n        toOneHom :=\n          { toFun := fun x => \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) x),\n            map_one' := (_ : \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) 1) = 1) },\n        map_mul' :=\n          (_ :\n            \u2200 (a a_1 : \u2191X\u271d),\n              \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) (a * a_1)) =\n                \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) a) * \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) a_1)) } =\n    \u2191Abelianization.lift\n        {\n          toOneHom :=\n            { toFun := fun x => \u2191Abelianization.of (\u2191f\u271d x), map_one' := (_ : \u2191Abelianization.of (\u2191f\u271d 1) = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (a a_1 : \u2191X\u271d),\n                \u2191Abelianization.of (\u2191f\u271d (a * a_1)) = \u2191Abelianization.of (\u2191f\u271d a) * \u2191Abelianization.of (\u2191f\u271d a_1)) } \u226b\n      \u2191Abelianization.lift\n        {\n          toOneHom :=\n            { toFun := fun x => \u2191Abelianization.of (\u2191g\u271d x), map_one' := (_ : \u2191Abelianization.of (\u2191g\u271d 1) = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (a a_1 : \u2191Y\u271d),\n                \u2191Abelianization.of (\u2191g\u271d (a * a_1)) = \u2191Abelianization.of (\u2191g\u271d a) * \u2191Abelianization.of (\u2191g\u271d a_1)) }\n[PROOFSTEP]\napply (Equiv.apply_eq_iff_eq_symm_apply Abelianization.lift).mpr\n[GOAL]\nX\u271d Y\u271d Z\u271d : GroupCat\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 {\n      toOneHom :=\n        { toFun := fun x => \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) x),\n          map_one' := (_ : \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) 1) = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (a a_1 : \u2191X\u271d),\n            \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) (a * a_1)) =\n              \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) a) * \u2191Abelianization.of (\u2191(f\u271d \u226b g\u271d) a_1)) } =\n    \u2191Abelianization.lift.symm\n      (\u2191Abelianization.lift\n          {\n            toOneHom :=\n              { toFun := fun x => \u2191Abelianization.of (\u2191f\u271d x), map_one' := (_ : \u2191Abelianization.of (\u2191f\u271d 1) = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (a a_1 : \u2191X\u271d),\n                  \u2191Abelianization.of (\u2191f\u271d (a * a_1)) = \u2191Abelianization.of (\u2191f\u271d a) * \u2191Abelianization.of (\u2191f\u271d a_1)) } \u226b\n        \u2191Abelianization.lift\n          {\n            toOneHom :=\n              { toFun := fun x => \u2191Abelianization.of (\u2191g\u271d x), map_one' := (_ : \u2191Abelianization.of (\u2191g\u271d 1) = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (a a_1 : \u2191Y\u271d),\n                  \u2191Abelianization.of (\u2191g\u271d (a * a_1)) = \u2191Abelianization.of (\u2191g\u271d a) * \u2191Abelianization.of (\u2191g\u271d a_1)) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 {X' X : GroupCat} {Y : CommGroupCat} (f : X' \u27f6 X) (g : X \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y),\n    \u2191((fun G A => Abelianization.lift.symm) X' Y).symm (f \u226b g) =\n      abelianize.map f \u226b \u2191((fun G A => Abelianization.lift.symm) X Y).symm g\n[PROOFSTEP]\nintros\n[GOAL]\nX'\u271d X\u271d : GroupCat\nY\u271d : CommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y\u271d\n\u22a2 \u2191((fun G A => Abelianization.lift.symm) X'\u271d Y\u271d).symm (f\u271d \u226b g\u271d) =\n    abelianize.map f\u271d \u226b \u2191((fun G A => Abelianization.lift.symm) X\u271d Y\u271d).symm g\u271d\n[PROOFSTEP]\next1\n[GOAL]\ncase w\nX'\u271d X\u271d : GroupCat\nY\u271d : CommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y\u271d\nx\u271d : \u2191(abelianize.obj X'\u271d)\n\u22a2 \u2191(\u2191((fun G A => Abelianization.lift.symm) X'\u271d Y\u271d).symm (f\u271d \u226b g\u271d)) x\u271d =\n    \u2191(abelianize.map f\u271d \u226b \u2191((fun G A => Abelianization.lift.symm) X\u271d Y\u271d).symm g\u271d) x\u271d\n[PROOFSTEP]\nsimp only [Equiv.symm_symm]\n[GOAL]\ncase w\nX'\u271d X\u271d : GroupCat\nY\u271d : CommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y\u271d\nx\u271d : \u2191(abelianize.obj X'\u271d)\n\u22a2 \u2191(\u2191Abelianization.lift (f\u271d \u226b g\u271d)) x\u271d = \u2191(abelianize.map f\u271d \u226b \u2191Abelianization.lift g\u271d) x\u271d\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase w.h\nX'\u271d X\u271d : GroupCat\nY\u271d : CommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y\u271d\nx\u271d : \u2191(abelianize.obj X'\u271d)\n\u22a2 \u2191(abelianize.map f\u271d \u226b \u2191Abelianization.lift g\u271d) x\u271d = \u2191(\u2191Abelianization.lift (f\u271d \u226b g\u271d)) x\u271d\n[PROOFSTEP]\napply Abelianization.lift.unique\n[GOAL]\ncase w.h.h\u03c6\nX'\u271d X\u271d : GroupCat\nY\u271d : CommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y\u271d\nx\u271d : \u2191(abelianize.obj X'\u271d)\n\u22a2 \u2200 (x : \u2191X'\u271d), \u2191(abelianize.map f\u271d \u226b \u2191Abelianization.lift g\u271d) (\u2191Abelianization.of x) = \u2191(f\u271d \u226b g\u271d) x\n[PROOFSTEP]\nintros\n[GOAL]\ncase w.h.h\u03c6\nX'\u271d X\u271d : GroupCat\nY\u271d : CommGroupCat\nf\u271d : X'\u271d \u27f6 X\u271d\ng\u271d : X\u271d \u27f6 (forget\u2082 CommGroupCat GroupCat).obj Y\u271d\nx\u271d\u00b9 : \u2191(abelianize.obj X'\u271d)\nx\u271d : \u2191X'\u271d\n\u22a2 \u2191(abelianize.map f\u271d \u226b \u2191Abelianization.lift g\u271d) (\u2191Abelianization.of x\u271d) = \u2191(f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\napply Abelianization.lift.of\n[GOAL]\n\u22a2 \u2200 \u2983X Y : MonCat\u2984 (f : X \u27f6 Y),\n    (MonCat.units \u22d9 forget\u2082 GroupCat MonCat).map f \u226b (fun X => Units.coeHom \u2191X) Y =\n      (fun X => Units.coeHom \u2191X) X \u226b (\ud835\udfed MonCat).map f\n[PROOFSTEP]\nintros\n[GOAL]\nX\u271d Y\u271d : MonCat\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (MonCat.units \u22d9 forget\u2082 GroupCat MonCat).map f\u271d \u226b (fun X => Units.coeHom \u2191X) Y\u271d =\n    (fun X => Units.coeHom \u2191X) X\u271d \u226b (\ud835\udfed MonCat).map f\u271d\n[PROOFSTEP]\nexact MonoidHom.ext fun x => rfl\n[GOAL]\n\u22a2 \u2200 \u2983X Y : CommMonCat\u2984 (f : X \u27f6 Y),\n    (CommMonCat.units \u22d9 forget\u2082 CommGroupCat CommMonCat).map f \u226b (fun X => Units.coeHom \u2191X) Y =\n      (fun X => Units.coeHom \u2191X) X \u226b (\ud835\udfed CommMonCat).map f\n[PROOFSTEP]\nintros\n[GOAL]\nX\u271d Y\u271d : CommMonCat\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (CommMonCat.units \u22d9 forget\u2082 CommGroupCat CommMonCat).map f\u271d \u226b (fun X => Units.coeHom \u2191X) Y\u271d =\n    (fun X => Units.coeHom \u2191X) X\u271d \u226b (\ud835\udfed CommMonCat).map f\u271d\n[PROOFSTEP]\nexact MonoidHom.ext fun x => rfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.GroupCat.Adjunctions", "llama_tokens": 9173, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339756938818, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.5616814803045058}}
{"text": "[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\n\u22a2 Injective toAffineMap\n[PROOFSTEP]\nrintro \u27e8e, el, h\u27e9 \u27e8e', el', h'\u27e9 H\n[GOAL]\ncase mk.mk\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2243 P\u2082\nel : V\u2081 \u2243\u2097[k] V\u2082\nh : \u2200 (p : P\u2081) (v : V\u2081), \u2191e (v +\u1d65 p) = \u2191el v +\u1d65 \u2191e p\ne' : P\u2081 \u2243 P\u2082\nel' : V\u2081 \u2243\u2097[k] V\u2082\nh' : \u2200 (p : P\u2081) (v : V\u2081), \u2191e' (v +\u1d65 p) = \u2191el' v +\u1d65 \u2191e' p\nH : \u2191{ toEquiv := e, linear := el, map_vadd' := h } = \u2191{ toEquiv := e', linear := el', map_vadd' := h' }\n\u22a2 { toEquiv := e, linear := el, map_vadd' := h } = { toEquiv := e', linear := el', map_vadd' := h' }\n[PROOFSTEP]\nsimp only [(toAffineMap_mk), (AffineMap.mk.injEq), Equiv.coe_inj, LinearEquiv.toLinearMap_inj] at H \n[GOAL]\ncase mk.mk\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2243 P\u2082\nel : V\u2081 \u2243\u2097[k] V\u2082\nh : \u2200 (p : P\u2081) (v : V\u2081), \u2191e (v +\u1d65 p) = \u2191el v +\u1d65 \u2191e p\ne' : P\u2081 \u2243 P\u2082\nel' : V\u2081 \u2243\u2097[k] V\u2082\nh' : \u2200 (p : P\u2081) (v : V\u2081), \u2191e' (v +\u1d65 p) = \u2191el' v +\u1d65 \u2191e' p\nH : e = e' \u2227 el = el'\n\u22a2 { toEquiv := e, linear := el, map_vadd' := h } = { toEquiv := e', linear := el', map_vadd' := h' }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_toEquiv\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2243 P\u2082\nel : V\u2081 \u2243\u2097[k] V\u2082\nh : \u2200 (p : P\u2081) (v : V\u2081), \u2191e (v +\u1d65 p) = \u2191el v +\u1d65 \u2191e p\ne' : P\u2081 \u2243 P\u2082\nel' : V\u2081 \u2243\u2097[k] V\u2082\nh' : \u2200 (p : P\u2081) (v : V\u2081), \u2191e' (v +\u1d65 p) = \u2191el' v +\u1d65 \u2191e' p\nH : e = e' \u2227 el = el'\n\u22a2 e = e'\ncase mk.mk.e_linear\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2243 P\u2082\nel : V\u2081 \u2243\u2097[k] V\u2082\nh : \u2200 (p : P\u2081) (v : V\u2081), \u2191e (v +\u1d65 p) = \u2191el v +\u1d65 \u2191e p\ne' : P\u2081 \u2243 P\u2082\nel' : V\u2081 \u2243\u2097[k] V\u2082\nh' : \u2200 (p : P\u2081) (v : V\u2081), \u2191e' (v +\u1d65 p) = \u2191el' v +\u1d65 \u2191e' p\nH : e = e' \u2227 el = el'\n\u22a2 el = el'\n[PROOFSTEP]\nexacts [H.1, H.2]\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2192 P\u2082\ne' : V\u2081 \u2243\u2097[k] V\u2082\np : P\u2081\nh : \u2200 (p' : P\u2081), e p' = \u2191e' (p' -\u1d65 p) +\u1d65 e p\np' : P\u2081\n\u22a2 (fun q' => \u2191(LinearEquiv.symm e') (q' -\u1d65 e p) +\u1d65 p) (e p') = p'\n[PROOFSTEP]\nsimp [h p', (vadd_vsub), (vsub_vadd)]\n  -- Porting note: `simp` needs `()`\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2192 P\u2082\ne' : V\u2081 \u2243\u2097[k] V\u2082\np : P\u2081\nh : \u2200 (p' : P\u2081), e p' = \u2191e' (p' -\u1d65 p) +\u1d65 e p\nq' : P\u2082\n\u22a2 e ((fun q' => \u2191(LinearEquiv.symm e') (q' -\u1d65 e p) +\u1d65 p) q') = q'\n[PROOFSTEP]\nsimp [h (e'.symm (q' -\u1d65 e p) +\u1d65 p), (vadd_vsub), (vsub_vadd)]\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2192 P\u2082\ne' : V\u2081 \u2243\u2097[k] V\u2082\np : P\u2081\nh : \u2200 (p' : P\u2081), e p' = \u2191e' (p' -\u1d65 p) +\u1d65 e p\np' : P\u2081\nv : V\u2081\n\u22a2 \u2191{ toFun := e, invFun := fun q' => \u2191(LinearEquiv.symm e') (q' -\u1d65 e p) +\u1d65 p,\n          left_inv := (_ : \u2200 (p' : P\u2081), \u2191(LinearEquiv.symm e') (e p' -\u1d65 e p) +\u1d65 p = p'),\n          right_inv := (_ : \u2200 (q' : P\u2082), e (\u2191(LinearEquiv.symm e') (q' -\u1d65 e p) +\u1d65 p) = q') }\n      (v +\u1d65 p') =\n    \u2191e' v +\u1d65\n      \u2191{ toFun := e, invFun := fun q' => \u2191(LinearEquiv.symm e') (q' -\u1d65 e p) +\u1d65 p,\n            left_inv := (_ : \u2200 (p' : P\u2081), \u2191(LinearEquiv.symm e') (e p' -\u1d65 e p) +\u1d65 p = p'),\n            right_inv := (_ : \u2200 (q' : P\u2082), e (\u2191(LinearEquiv.symm e') (q' -\u1d65 e p) +\u1d65 p) = q') }\n        p'\n[PROOFSTEP]\nsimp [h p', h (v +\u1d65 p'), (vadd_vsub_assoc), (vadd_vadd)]\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2243\u1d43[k] P\u2082\np : P\u2082\nv : V\u2082\n\u22a2 v +\u1d65 p = \u2191e.symm.symm (\u2191(LinearEquiv.symm e.linear) v +\u1d65 \u2191e.symm p)\n[PROOFSTEP]\nrw [Equiv.symm_symm, e.map_vadd' ((Equiv.symm e.toEquiv) p) ((LinearEquiv.symm e.linear) v),\n  LinearEquiv.apply_symm_apply, Equiv.apply_symm_apply]\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\ne : P\u2081 \u2243\u1d43[k] P\u2082\ne' : P\u2082 \u2243\u1d43[k] P\u2083\np : P\u2081\nv : V\u2081\n\u22a2 \u2191(e.trans e'.toEquiv) (v +\u1d65 p) = \u2191(LinearEquiv.trans e.linear e'.linear) v +\u1d65 \u2191(e.trans e'.toEquiv) p\n[PROOFSTEP]\nsimp only [LinearEquiv.trans_apply, (coe_toEquiv), (\u00b7 \u2218 \u00b7), Equiv.coe_trans, (map_vadd)]\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V\u2081\ninst\u271d\u00b9\u2070 : Module k V\u2081\ninst\u271d\u2079 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2078 : AddCommGroup V\u2082\ninst\u271d\u2077 : Module k V\u2082\ninst\u271d\u2076 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2075 : AddCommGroup V\u2083\ninst\u271d\u2074 : Module k V\u2083\ninst\u271d\u00b3 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b2 : AddCommGroup V\u2084\ninst\u271d\u00b9 : Module k V\u2084\ninst\u271d : AffineSpace V\u2084 P\u2084\np p' : P\u2081\nv : V\u2081\n\u22a2 \u2191(Equiv.constVSub p) (v +\u1d65 p') = \u2191(LinearEquiv.neg k) v +\u1d65 \u2191(Equiv.constVSub p) p'\n[PROOFSTEP]\nsimp [(Equiv.coe_constVSub), (vsub_vadd_eq_vsub_sub), neg_add_eq_sub]\n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\ninst\u271d\u00b9\u00b2 : AddCommGroup V\u2081\ninst\u271d\u00b9\u00b9 : Module k V\u2081\ninst\u271d\u00b9\u2070 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2079 : AddCommGroup V\u2082\ninst\u271d\u2078 : Module k V\u2082\ninst\u271d\u2077 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2076 : AddCommGroup V\u2083\ninst\u271d\u2075 : Module k V\u2083\ninst\u271d\u2074 : AffineSpace V\u2083 P\u2083\ninst\u271d\u00b3 : AddCommGroup V\u2084\ninst\u271d\u00b2 : Module k V\u2084\ninst\u271d\u00b9 : AffineSpace V\u2084 P\u2084\ninst\u271d : Invertible 2\nx y : V\u2081\nh : bit0 x = bit0 y\n\u22a2 x = y\n[PROOFSTEP]\nrwa [bit0, bit0, \u2190 two_smul k x, \u2190 two_smul k y, (isUnit_of_invertible (2 : k)).smul_left_cancel] at h \n[GOAL]\nk : Type u_1\nP\u2081 : Type u_2\nP\u2082 : Type u_3\nP\u2083 : Type u_4\nP\u2084 : Type u_5\nV\u2081 : Type u_6\nV\u2082 : Type u_7\nV\u2083 : Type u_8\nV\u2084 : Type u_9\ninst\u271d\u00b9\u2074 : Ring k\ninst\u271d\u00b9\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b9\u00b2 : Module k V\u2081\ninst\u271d\u00b9\u00b9 : AffineSpace V\u2081 P\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup V\u2082\ninst\u271d\u2079 : Module k V\u2082\ninst\u271d\u2078 : AffineSpace V\u2082 P\u2082\ninst\u271d\u2077 : AddCommGroup V\u2083\ninst\u271d\u2076 : Module k V\u2083\ninst\u271d\u2075 : AffineSpace V\u2083 P\u2083\ninst\u271d\u2074 : AddCommGroup V\u2084\ninst\u271d\u00b3 : Module k V\u2084\ninst\u271d\u00b2 : AffineSpace V\u2084 P\u2084\nR' : Type u_10\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : Module R' V\u2081\nc p : P\u2081\n\u22a2 \u2191(homothety c (-1)) p = \u2191(pointReflection R' c) p\n[PROOFSTEP]\nsimp [(homothety_apply), pointReflection_apply _, (neg_vsub_eq_vsub_rev)]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "llama_tokens": 5213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722394, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.5616814638534877}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\n\u22a2 Finset.Nonempty (Finset.sigma s t) \u2194 \u2203 i, i \u2208 s \u2227 Finset.Nonempty (t i)\n[PROOFSTEP]\nsimp [Finset.Nonempty]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\n\u22a2 Finset.sigma s t = \u2205 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 t i = \u2205\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff_eq_empty, sigma_nonempty, not_exists, not_and]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\n\u22a2 Set.PairwiseDisjoint \u2191s fun i => map (Embedding.sigmaMk i) (t i)\n[PROOFSTEP]\nintro i _ j _ hij\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 (_root_.Disjoint on fun i => map (Embedding.sigmaMk i) (t i)) i j\n[PROOFSTEP]\nrw [Function.onFun, disjoint_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 \u2200 \u2983a : (x : \u03b9) \u00d7 \u03b1 x\u2984, a \u2208 map (Embedding.sigmaMk i) (t i) \u2192 \u00aca \u2208 map (Embedding.sigmaMk j) (t j)\n[PROOFSTEP]\nsimp_rw [mem_map, Function.Embedding.sigmaMk_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 \u2200 \u2983a : (x : \u03b9) \u00d7 \u03b1 x\u2984,\n    (\u2203 a_1, a_1 \u2208 t i \u2227 { fst := i, snd := a_1 } = a) \u2192 \u00ac\u2203 a_2, a_2 \u2208 t j \u2227 { fst := j, snd := a_2 } = a\n[PROOFSTEP]\nrintro _ \u27e8y, _, rfl\u27e9 \u27e8z, _, hz'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\ny : \u03b1 i\nleft\u271d\u00b9 : y \u2208 t i\nz : \u03b1 j\nleft\u271d : z \u2208 t j\nhz' : { fst := j, snd := z } = { fst := i, snd := y }\n\u22a2 False\n[PROOFSTEP]\nexact hij (congr_arg Sigma.fst hz'.symm)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d : DecidableEq ((i : \u03b9) \u00d7 \u03b1 i)\ns : Finset \u03b9\nt : (i : \u03b9) \u2192 Finset (\u03b1 i)\n\u22a2 Finset.sigma s t = Finset.biUnion s fun i => map (Embedding.sigmaMk i) (t i)\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase a.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns\u271d s\u2081 s\u2082 : Finset \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d : DecidableEq ((i : \u03b9) \u00d7 \u03b1 i)\ns : Finset \u03b9\nt : (i : \u03b9) \u2192 Finset (\u03b1 i)\nx : \u03b9\ny : \u03b1 x\n\u22a2 { fst := x, snd := y } \u2208 Finset.sigma s t \u2194\n    { fst := x, snd := y } \u2208 Finset.biUnion s fun i => map (Embedding.sigmaMk i) (t i)\n[PROOFSTEP]\nsimp [and_left_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\nf : (i : \u03b9) \u00d7 \u03b1 i \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\n\u22a2 sup (Finset.sigma s t) f = sup s fun i => sup (t i) fun b => f { fst := i, snd := b }\n[PROOFSTEP]\nsimp only [le_antisymm_iff, Finset.sup_le_iff, mem_sigma, and_imp, Sigma.forall]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\nf : (i : \u03b9) \u00d7 \u03b1 i \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b2\ninst\u271d : OrderBot \u03b2\n\u22a2 (\u2200 (a : \u03b9) (b : \u03b1 a),\n      a \u2208 s \u2192 b \u2208 t a \u2192 f { fst := a, snd := b } \u2264 sup s fun i => sup (t i) fun b => f { fst := i, snd := b }) \u2227\n    \u2200 (b : \u03b9), b \u2208 s \u2192 \u2200 (b_1 : \u03b1 b), b_1 \u2208 t b \u2192 f { fst := b, snd := b_1 } \u2264 sup (Finset.sigma s t) f\n[PROOFSTEP]\nexact\n  \u27e8fun i a hi ha => (le_sup hi).trans' <| le_sup (f := fun a => f \u27e8i, a\u27e9) ha, fun i hi a ha =>\n    le_sup <| mem_sigma.2 \u27e8hi, ha\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : Sigma \u03b1\nb : Sigma \u03b2\nx : Sigma \u03b3\n\u22a2 x \u2208 sigmaLift f a b \u2194 \u2203 ha hb, x.snd \u2208 f (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n[PROOFSTEP]\nobtain \u27e8\u27e8i, a\u27e9, j, b\u27e9 := a, b\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b2 j\n\u22a2 x \u2208 sigmaLift f { fst := i, snd := a } { fst := j, snd := b } \u2194\n    \u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := j, snd := b }.snd)\n[PROOFSTEP]\nobtain rfl | h := Decidable.eq_or_ne i j\n[GOAL]\ncase mk.mk.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\n\u22a2 x \u2208 sigmaLift f { fst := i, snd := a } { fst := i, snd := b } \u2194\n    \u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := i, snd := b }.snd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mk.inl.mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\n\u22a2 x \u2208 sigmaLift f { fst := i, snd := a } { fst := i, snd := b } \u2192\n    \u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := i, snd := b }.snd)\n[PROOFSTEP]\nsimp_rw [sigmaLift]\n[GOAL]\ncase mk.mk.inl.mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\n\u22a2 (x \u2208 if h : True then map (Embedding.sigmaMk i) (f a b) else \u2205) \u2192\n    \u2203 h h_1, x.snd \u2208 f ((_ : { fst := i, snd := a }.fst = x.fst) \u25b8 a) ((_ : { fst := i, snd := b }.fst = x.fst) \u25b8 b)\n[PROOFSTEP]\nsimp only [dite_eq_ite, ite_true, mem_map, Embedding.sigmaMk_apply, forall_exists_index, and_imp]\n[GOAL]\ncase mk.mk.inl.mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\n\u22a2 \u2200 (x_1 : \u03b3 i),\n    x_1 \u2208 f a b \u2192\n      { fst := i, snd := x_1 } = x \u2192\n        \u2203 h h_1, x.snd \u2208 f ((_ : { fst := i, snd := a }.fst = x.fst) \u25b8 a) ((_ : { fst := i, snd := b }.fst = x.fst) \u25b8 b)\n[PROOFSTEP]\nrintro x hx rfl\n[GOAL]\ncase mk.mk.inl.mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\nx : \u03b3 i\nhx : x \u2208 f a b\n\u22a2 \u2203 h h_1,\n    { fst := i, snd := x }.snd \u2208\n      f ((_ : { fst := i, snd := a }.fst = { fst := i, snd := x }.fst) \u25b8 a)\n        ((_ : { fst := i, snd := b }.fst = { fst := i, snd := x }.fst) \u25b8 b)\n[PROOFSTEP]\nexact \u27e8rfl, rfl, hx\u27e9\n[GOAL]\ncase mk.mk.inl.mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\n\u22a2 (\u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := i, snd := b }.snd)) \u2192\n    x \u2208 sigmaLift f { fst := i, snd := a } { fst := i, snd := b }\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, \u27e8\u27e9, hx\u27e9\n[GOAL]\ncase mk.mk.inl.mpr.intro.refl.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\na : \u03b1 x.1\nb : \u03b2 x.1\nhx :\n  x.snd \u2208\n    f ((_ : { fst := x.1, snd := a }.fst = { fst := x.1, snd := a }.fst) \u25b8 { fst := x.1, snd := a }.snd)\n      ((_ : { fst := x.1, snd := b }.fst = { fst := x.1, snd := b }.fst) \u25b8 { fst := x.1, snd := b }.snd)\n\u22a2 x \u2208 sigmaLift f { fst := x.1, snd := a } { fst := x.1, snd := b }\n[PROOFSTEP]\nrw [sigmaLift, dif_pos rfl, mem_map]\n[GOAL]\ncase mk.mk.inl.mpr.intro.refl.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\na : \u03b1 x.1\nb : \u03b2 x.1\nhx :\n  x.snd \u2208\n    f ((_ : { fst := x.1, snd := a }.fst = { fst := x.1, snd := a }.fst) \u25b8 { fst := x.1, snd := a }.snd)\n      ((_ : { fst := x.1, snd := b }.fst = { fst := x.1, snd := b }.fst) \u25b8 { fst := x.1, snd := b }.snd)\n\u22a2 \u2203 a_1,\n    a_1 \u2208\n        f ((_ : { fst := x.1, snd := a }.fst = { fst := x.1, snd := a }.fst) \u25b8 { fst := x.1, snd := a }.snd)\n          { fst := x.1, snd := b }.snd \u2227\n      \u2191(Embedding.sigmaMk { fst := x.1, snd := b }.fst) a_1 = x\n[PROOFSTEP]\nexact \u27e8_, hx, by simp [Sigma.ext_iff]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\na : \u03b1 x.1\nb : \u03b2 x.1\nhx :\n  x.snd \u2208\n    f ((_ : { fst := x.1, snd := a }.fst = { fst := x.1, snd := a }.fst) \u25b8 { fst := x.1, snd := a }.snd)\n      ((_ : { fst := x.1, snd := b }.fst = { fst := x.1, snd := b }.fst) \u25b8 { fst := x.1, snd := b }.snd)\n\u22a2 \u2191(Embedding.sigmaMk { fst := x.1, snd := b }.fst) x.snd = x\n[PROOFSTEP]\nsimp [Sigma.ext_iff]\n[GOAL]\ncase mk.mk.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b2 j\nh : i \u2260 j\n\u22a2 x \u2208 sigmaLift f { fst := i, snd := a } { fst := j, snd := b } \u2194\n    \u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := j, snd := b }.snd)\n[PROOFSTEP]\nrw [sigmaLift, dif_neg h]\n[GOAL]\ncase mk.mk.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b2 j\nh : i \u2260 j\n\u22a2 x \u2208 \u2205 \u2194 \u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := j, snd := b }.snd)\n[PROOFSTEP]\nrefine' iff_of_false (not_mem_empty _) _\n[GOAL]\ncase mk.mk.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b2 j\nh : i \u2260 j\n\u22a2 \u00ac\u2203 ha hb, x.snd \u2208 f (ha \u25b8 { fst := i, snd := a }.snd) (hb \u25b8 { fst := j, snd := b }.snd)\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, \u27e8\u27e9, _\u27e9\n[GOAL]\ncase mk.mk.inr.intro.refl.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\nx : Sigma \u03b3\na : \u03b1 x.1\nb : \u03b2 x.1\nh : x.1 \u2260 x.1\nh\u271d :\n  x.snd \u2208\n    f ((_ : { fst := x.1, snd := a }.fst = { fst := x.1, snd := a }.fst) \u25b8 { fst := x.1, snd := a }.snd)\n      ((_ : { fst := x.1, snd := b }.fst = { fst := x.1, snd := b }.fst) \u25b8 { fst := x.1, snd := b }.snd)\n\u22a2 False\n[PROOFSTEP]\nexact h rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\nx : \u03b3 i\n\u22a2 { fst := i, snd := x } \u2208 sigmaLift f { fst := i, snd := a } { fst := i, snd := b } \u2194 x \u2208 f a b\n[PROOFSTEP]\nrw [sigmaLift, dif_pos rfl, mem_map]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\nx : \u03b3 i\n\u22a2 (\u2203 a_1,\n      a_1 \u2208\n          f ((_ : { fst := i, snd := a }.fst = { fst := i, snd := a }.fst) \u25b8 { fst := i, snd := a }.snd)\n            { fst := i, snd := b }.snd \u2227\n        \u2191(Embedding.sigmaMk { fst := i, snd := b }.fst) a_1 = { fst := i, snd := x }) \u2194\n    x \u2208 f a b\n[PROOFSTEP]\nrefine' \u27e8_, fun hx => \u27e8_, hx, rfl\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\nx : \u03b3 i\n\u22a2 (\u2203 a_1,\n      a_1 \u2208\n          f ((_ : { fst := i, snd := a }.fst = { fst := i, snd := a }.fst) \u25b8 { fst := i, snd := a }.snd)\n            { fst := i, snd := b }.snd \u2227\n        \u2191(Embedding.sigmaMk { fst := i, snd := b }.fst) a_1 = { fst := i, snd := x }) \u2192\n    x \u2208 f a b\n[PROOFSTEP]\nrintro \u27e8x, hx, _, rfl\u27e9\n[GOAL]\ncase intro.intro.refl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\ni : \u03b9\na : \u03b1 i\nb : \u03b2 i\nx : \u03b3 i\nhx :\n  x \u2208\n    f ((_ : { fst := i, snd := a }.fst = { fst := i, snd := a }.fst) \u25b8 { fst := i, snd := a }.snd)\n      { fst := i, snd := b }.snd\n\u22a2 x \u2208 f a b\n[PROOFSTEP]\nexact hx\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : Sigma \u03b1\nb : Sigma \u03b2\nx : Sigma \u03b3\nh : a.fst \u2260 x.fst\n\u22a2 \u00acx \u2208 sigmaLift f a b\n[PROOFSTEP]\nrw [mem_sigmaLift]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : Sigma \u03b1\nb : Sigma \u03b2\nx : Sigma \u03b3\nh : a.fst \u2260 x.fst\n\u22a2 \u00ac\u2203 ha hb, x.snd \u2208 f (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n[PROOFSTEP]\nexact fun H => h H.fst\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : Sigma \u03b1\nb : Sigma \u03b2\nx : Sigma \u03b3\nh : b.fst \u2260 x.fst\n\u22a2 \u00acx \u2208 sigmaLift f a b\n[PROOFSTEP]\nrw [mem_sigmaLift]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : Sigma \u03b1\nb : Sigma \u03b2\nx : Sigma \u03b3\nh : b.fst \u2260 x.fst\n\u22a2 \u00ac\u2203 ha hb, x.snd \u2208 f (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n[PROOFSTEP]\nexact fun H => h H.snd.fst\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 Finset.Nonempty (sigmaLift f a b) \u2194 \u2203 h, Finset.Nonempty (f (h \u25b8 a.snd) b.snd)\n[PROOFSTEP]\nsimp_rw [nonempty_iff_ne_empty, sigmaLift]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) \u2260 \u2205 \u2194\n    \u2203 h, f ((_ : a.fst = b.fst) \u25b8 a.snd) b.snd \u2260 \u2205\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nh : a.fst = b.fst\n\u22a2 map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) \u2260 \u2205 \u2194 \u2203 h, f ((_ : a.fst = b.fst) \u25b8 a.snd) b.snd \u2260 \u2205\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nh : \u00aca.fst = b.fst\n\u22a2 \u2205 \u2260 \u2205 \u2194 \u2203 h, f ((_ : a.fst = b.fst) \u25b8 a.snd) b.snd \u2260 \u2205\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 sigmaLift f a b = \u2205 \u2194 \u2200 (h : a.fst = b.fst), f (h \u25b8 a.snd) b.snd = \u2205\n[PROOFSTEP]\nsimp_rw [sigmaLift]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) = \u2205 \u2194\n    \u2200 (h : a.fst = b.fst), f (h \u25b8 a.snd) b.snd = \u2205\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nh : a.fst = b.fst\n\u22a2 map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) = \u2205 \u2194 \u2200 (h : a.fst = b.fst), f (h \u25b8 a.snd) b.snd = \u2205\n[PROOFSTEP]\nsimp [h, forall_prop_of_true h]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nh : \u00aca.fst = b.fst\n\u22a2 \u2205 = \u2205 \u2194 \u2200 (h : a.fst = b.fst), f (h \u25b8 a.snd) b.snd = \u2205\n[PROOFSTEP]\nsimp [h, forall_prop_of_false h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na\u271d : (i : \u03b9) \u00d7 \u03b1 i\nb\u271d : (i : \u03b9) \u00d7 \u03b2 i\nh : \u2200 \u2983i : \u03b9\u2984 \u2983a : \u03b1 i\u2984 \u2983b : \u03b2 i\u2984, f a b \u2286 g a b\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 sigmaLift f a b \u2286 sigmaLift g a b\n[PROOFSTEP]\nrintro x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na\u271d : (i : \u03b9) \u00d7 \u03b1 i\nb\u271d : (i : \u03b9) \u00d7 \u03b2 i\nh : \u2200 \u2983i : \u03b9\u2984 \u2983a : \u03b1 i\u2984 \u2983b : \u03b2 i\u2984, f a b \u2286 g a b\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nx : (i : \u03b9) \u00d7 \u03b3 i\nhx : x \u2208 sigmaLift f a b\n\u22a2 x \u2208 sigmaLift g a b\n[PROOFSTEP]\nrw [mem_sigmaLift] at hx \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na\u271d : (i : \u03b9) \u00d7 \u03b1 i\nb\u271d : (i : \u03b9) \u00d7 \u03b2 i\nh : \u2200 \u2983i : \u03b9\u2984 \u2983a : \u03b1 i\u2984 \u2983b : \u03b2 i\u2984, f a b \u2286 g a b\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nx : (i : \u03b9) \u00d7 \u03b3 i\nhx : \u2203 ha hb, x.snd \u2208 f (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n\u22a2 \u2203 ha hb, x.snd \u2208 g (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n[PROOFSTEP]\nobtain \u27e8ha, hb, hx\u27e9 := hx\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na\u271d : (i : \u03b9) \u00d7 \u03b1 i\nb\u271d : (i : \u03b9) \u00d7 \u03b2 i\nh : \u2200 \u2983i : \u03b9\u2984 \u2983a : \u03b1 i\u2984 \u2983b : \u03b2 i\u2984, f a b \u2286 g a b\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nx : (i : \u03b9) \u00d7 \u03b3 i\nha : a.fst = x.fst\nhb : b.fst = x.fst\nhx : x.snd \u2208 f (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n\u22a2 \u2203 ha hb, x.snd \u2208 g (ha \u25b8 a.snd) (hb \u25b8 b.snd)\n[PROOFSTEP]\nexact \u27e8ha, hb, h hx\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 card (sigmaLift f a b) = if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0\n[PROOFSTEP]\nsimp_rw [sigmaLift]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 card (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) =\n    if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nh : a.fst = b.fst\n\u22a2 card (map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd)) = card (f (h \u25b8 a.snd) b.snd)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\nh : \u00aca.fst = b.fst\n\u22a2 card \u2205 = 0\n[PROOFSTEP]\nsimp [h]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Sigma", "llama_tokens": 10566, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430478583168, "lm_q2_score": 0.6757646075489392, "lm_q1q2_score": 0.5616570555530046}}
{"text": "[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\ng1 g2 : G\np : P\nh : g1 +\u1d65 p = g2 +\u1d65 p\n\u22a2 g1 = g2\n[PROOFSTEP]\nrw [\u2190 vadd_vsub g1 p, h, vadd_vsub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\ng : G\np1 p2 : P\n\u22a2 g +\u1d65 p1 -\u1d65 p2 = g + (p1 -\u1d65 p2)\n[PROOFSTEP]\napply vadd_right_cancel p2\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\ng : G\np1 p2 : P\n\u22a2 g +\u1d65 p1 -\u1d65 p2 +\u1d65 p2 = g + (p1 -\u1d65 p2) +\u1d65 p2\n[PROOFSTEP]\nrw [vsub_vadd, add_vadd, vsub_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np : P\n\u22a2 p -\u1d65 p = 0\n[PROOFSTEP]\nrw [\u2190 zero_add (p -\u1d65 p), \u2190 vadd_vsub_assoc, vadd_vsub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 : P\nh : p1 -\u1d65 p2 = 0\n\u22a2 p1 = p2\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p1 p2, h, zero_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p2 + (p2 -\u1d65 p3) = p1 -\u1d65 p3\n[PROOFSTEP]\napply vadd_right_cancel p3\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p2 + (p2 -\u1d65 p3) +\u1d65 p3 = p1 -\u1d65 p3 +\u1d65 p3\n[PROOFSTEP]\nrw [add_vadd, vsub_vadd, vsub_vadd, vsub_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 : P\n\u22a2 -(p1 -\u1d65 p2) = p2 -\u1d65 p1\n[PROOFSTEP]\nrefine' neg_eq_of_add_eq_zero_right (vadd_right_cancel p1 _)\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 : P\n\u22a2 p1 -\u1d65 p2 + (p2 -\u1d65 p1) +\u1d65 p1 = 0 +\u1d65 p1\n[PROOFSTEP]\nrw [vsub_add_vsub_cancel, vsub_self]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\ng : G\np q : P\n\u22a2 g +\u1d65 p -\u1d65 q = g - (q -\u1d65 p)\n[PROOFSTEP]\nrw [vadd_vsub_assoc, sub_eq_add_neg, neg_vsub_eq_vsub_rev]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 : P\ng : G\n\u22a2 p1 -\u1d65 (g +\u1d65 p2) = p1 -\u1d65 p2 - g\n[PROOFSTEP]\nrw [\u2190 add_right_inj (p2 -\u1d65 p1 : G), vsub_add_vsub_cancel, \u2190 neg_vsub_eq_vsub_rev, vadd_vsub, \u2190 add_sub_assoc, \u2190\n  neg_vsub_eq_vsub_rev, neg_add_self, zero_sub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p3 - (p2 -\u1d65 p3) = p1 -\u1d65 p2\n[PROOFSTEP]\nrw [\u2190 vsub_vadd_eq_vsub_sub, vsub_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\nv\u2081 v\u2082 : G\np\u2081 p\u2082 : P\n\u22a2 v\u2081 +\u1d65 p\u2081 = v\u2082 +\u1d65 p\u2082 \u2194 -v\u2081 + v\u2082 = p\u2081 -\u1d65 p\u2082\n[PROOFSTEP]\nrw [eq_vadd_iff_vsub_eq, vadd_vsub_assoc, \u2190 add_right_inj (-v\u2081), neg_add_cancel_left, eq_comm]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np : P\n\u22a2 {p} -\u1d65 {p} = {0}\n[PROOFSTEP]\nrw [Set.singleton_vsub_singleton, vsub_self]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\nv\u2081 v\u2082 : G\np : P\n\u22a2 v\u2081 +\u1d65 p -\u1d65 (v\u2082 +\u1d65 p) = v\u2081 - v\u2082\n[PROOFSTEP]\nrw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, vsub_self, add_zero]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 p : P\nh : p1 -\u1d65 p = p2 -\u1d65 p\n\u22a2 p1 = p2\n[PROOFSTEP]\nrwa [\u2190 sub_eq_zero, vsub_sub_vsub_cancel_right, vsub_eq_zero_iff_eq] at h \n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 p : P\nh : p -\u1d65 p1 = p -\u1d65 p2\n\u22a2 p1 = p2\n[PROOFSTEP]\nrefine' vadd_left_cancel (p -\u1d65 p2) _\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d : AddGroup G\nT : AddTorsor G P\np1 p2 p : P\nh : p -\u1d65 p1 = p -\u1d65 p2\n\u22a2 p -\u1d65 p2 +\u1d65 p1 = p -\u1d65 p2 +\u1d65 p2\n[PROOFSTEP]\nrw [vsub_vadd, \u2190 h, vsub_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\np1 p2 p3 : P\n\u22a2 p3 -\u1d65 p2 - (p3 -\u1d65 p1) = p1 -\u1d65 p2\n[PROOFSTEP]\nrw [sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm, vsub_add_vsub_cancel]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\nv : G\np1 p2 : P\n\u22a2 v +\u1d65 p1 -\u1d65 (v +\u1d65 p2) = p1 -\u1d65 p2\n[PROOFSTEP]\nrw [vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, add_sub_cancel']\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p2 +\u1d65 p3 = p3 -\u1d65 p2 +\u1d65 p1\n[PROOFSTEP]\nrw [\u2190 @vsub_eq_zero_iff_eq G, vadd_vsub_assoc, vsub_vadd_eq_vsub_sub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p2 + (p3 -\u1d65 p1 - (p3 -\u1d65 p2)) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\nv\u2081 v\u2082 : G\np\u2081 p\u2082 : P\n\u22a2 v\u2081 +\u1d65 p\u2081 = v\u2082 +\u1d65 p\u2082 \u2194 v\u2082 - v\u2081 = p\u2081 -\u1d65 p\u2082\n[PROOFSTEP]\nrw [vadd_eq_vadd_iff_neg_add_eq_vsub, neg_add_eq_sub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 p\u2081 -\u1d65 p\u2082 - (p\u2083 -\u1d65 p\u2084) = p\u2081 -\u1d65 p\u2083 - (p\u2082 -\u1d65 p\u2084)\n[PROOFSTEP]\nrw [\u2190 vsub_vadd_eq_vsub_sub, vsub_vadd_comm, vsub_vadd_eq_vsub_sub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\np p' : P\n\u22a2 (fun v => -v +\u1d65 p) ((fun x x_1 => x -\u1d65 x_1) p p') = p'\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\np : P\nv : G\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p ((fun v => -v +\u1d65 p) v) = v\n[PROOFSTEP]\nsimp [vsub_vadd_eq_vsub_sub]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nv : G\np : P\n\u22a2 (fun x x_1 => x +\u1d65 x_1) (-v) ((fun x x_1 => x +\u1d65 x_1) v p) = p\n[PROOFSTEP]\nsimp [vadd_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nv : G\np : P\n\u22a2 (fun x x_1 => x +\u1d65 x_1) v ((fun x x_1 => x +\u1d65 x_1) (-v) p) = p\n[PROOFSTEP]\nsimp [vadd_vadd]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nx y : P\n\u22a2 -(x -\u1d65 \u2191(pointReflection x) y) = -(y -\u1d65 x)\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nx y : P\n\u22a2 \u2191(pointReflection x) y -\u1d65 y = 2 \u2022 (x -\u1d65 y)\n[PROOFSTEP]\nsimp [pointReflection, two_nsmul, vadd_vsub_assoc]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nx y : P\n\u22a2 -(y -\u1d65 \u2191(pointReflection x) y) = -(2 \u2022 (y -\u1d65 x))\n[PROOFSTEP]\nsimp [\u2190 neg_nsmul]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nx : P\n\u22a2 \u2200 (x_1 : P), \u2191(pointReflection x).symm x_1 = \u2191(pointReflection x) x_1\n[PROOFSTEP]\nsimp [pointReflection]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nx y : P\n\u22a2 \u2191(pointReflection x) y = \u2191(pointReflection x).symm y\n[PROOFSTEP]\nrw [pointReflection_symm]\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\nx y : P\nh : Injective bit0\n\u22a2 \u2191(pointReflection x) y = y \u2194 y = x\n[PROOFSTEP]\nrw [pointReflection_apply, eq_comm, eq_vadd_iff_vsub_eq, \u2190 neg_vsub_eq_vsub_rev, neg_eq_iff_add_eq_zero, \u2190 bit0, \u2190\n  bit0_zero, h.eq_iff, vsub_eq_zero_iff_eq, eq_comm]\n[GOAL]\nG\u271d : Type u_1\nP\u271d : Type u_2\ninst\u271d\u00b3 : AddGroup G\u271d\ninst\u271d\u00b2 : AddTorsor G\u271d P\u271d\nG : Type u_3\nP : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : AddTorsor G P\nh : Injective bit0\ny x\u2081 x\u2082 : P\nhy : \u2191(pointReflection x\u2081) y = \u2191(pointReflection x\u2082) y\n\u22a2 x\u2081 = x\u2082\n[PROOFSTEP]\nrwa [pointReflection_apply, pointReflection_apply, vadd_eq_vadd_iff_sub_eq_vsub, vsub_sub_vsub_cancel_right, \u2190\n  neg_vsub_eq_vsub_rev, neg_eq_iff_add_eq_zero, \u2190 bit0, \u2190 bit0_zero, h.eq_iff, vsub_eq_zero_iff_eq] at hy \n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\n\u22a2 Subsingleton G \u2194 Subsingleton P\n[PROOFSTEP]\ninhabit P\n[GOAL]\nG : Type u_1\nP : Type u_2\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : AddTorsor G P\ninhabited_h : Inhabited P\n\u22a2 Subsingleton G \u2194 Subsingleton P\n[PROOFSTEP]\nexact (Equiv.vaddConst default).subsingleton_congr\n", "meta": {"mathlib_filename": "Mathlib.Algebra.AddTorsor", "llama_tokens": 4135, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737869342623, "lm_q2_score": 0.6992544273261175, "lm_q1q2_score": 0.5616228264260668}}
{"text": "[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nh : x \u2260 y\n\u22a2 x (firstDiff x y) \u2260 y (firstDiff x y)\n[PROOFSTEP]\nrw [firstDiff_def, dif_pos h]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nh : x \u2260 y\n\u22a2 x (Nat.find (_ : \u2203 a, x a \u2260 y a)) \u2260 y (Nat.find (_ : \u2203 a, x a \u2260 y a))\n[PROOFSTEP]\nexact Nat.find_spec (ne_iff.1 h)\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n < firstDiff x y\n\u22a2 x n = y n\n[PROOFSTEP]\nrw [firstDiff_def] at hn \n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n < if h : x \u2260 y then Nat.find (_ : \u2203 a, x a \u2260 y a) else 0\n\u22a2 x n = y n\n[PROOFSTEP]\nsplit_ifs at hn  with h\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : x \u2260 y\nhn : n < Nat.find (_ : \u2203 a, x a \u2260 y a)\n\u22a2 x n = y n\n[PROOFSTEP]\nconvert Nat.find_min (ne_iff.1 h) hn\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : x \u2260 y\nhn : n < Nat.find (_ : \u2203 a, x a \u2260 y a)\n\u22a2 x n = y n \u2194 \u00acx n \u2260 y n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : \u00acx \u2260 y\nhn : n < 0\n\u22a2 x n = y n\n[PROOFSTEP]\nexact (not_lt_zero' hn).elim\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\n\u22a2 firstDiff x y = firstDiff y x\n[PROOFSTEP]\nsimp only [firstDiff_def, ne_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nh : x \u2260 z\n\u22a2 min (firstDiff x y) (firstDiff y z) \u2264 firstDiff x z\n[PROOFSTEP]\nby_contra' H\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nh : x \u2260 z\nH : firstDiff x z < min (firstDiff x y) (firstDiff y z)\n\u22a2 False\n[PROOFSTEP]\nrw [lt_min_iff] at H \n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nh : x \u2260 z\nH : firstDiff x z < firstDiff x y \u2227 firstDiff x z < firstDiff y z\n\u22a2 False\n[PROOFSTEP]\nrefine apply_firstDiff_ne h ?_\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nh : x \u2260 z\nH : firstDiff x z < firstDiff x y \u2227 firstDiff x z < firstDiff y z\n\u22a2 x (firstDiff x z) = z (firstDiff x z)\n[PROOFSTEP]\ncalc\n  x (firstDiff x z) = y (firstDiff x z) := apply_eq_of_lt_firstDiff H.1\n  _ = z (firstDiff x z) := apply_eq_of_lt_firstDiff H.2\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 cylinder x n = Set.pi \u2191(Finset.range n) fun i => {x i}\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\n\u22a2 y \u2208 cylinder x n \u2194 y \u2208 Set.pi \u2191(Finset.range n) fun i => {x i}\n[PROOFSTEP]\nsimp [cylinder]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\n\u22a2 cylinder x 0 = univ\n[PROOFSTEP]\nsimp [cylinder_eq_pi]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 x \u2208 cylinder x n\n[PROOFSTEP]\nsimp\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2194 cylinder y n = cylinder x n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2192 cylinder y n = cylinder x n\n[PROOFSTEP]\nintro hy\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder y n = cylinder x n\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder y n \u2286 cylinder x n\n[PROOFSTEP]\nintro z hz i hi\n[GOAL]\ncase mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder y n\ni : \u2115\nhi : i < n\n\u22a2 z i = x i\n[PROOFSTEP]\nrw [\u2190 hy i hi]\n[GOAL]\ncase mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder y n\ni : \u2115\nhi : i < n\n\u22a2 z i = y i\n[PROOFSTEP]\nexact hz i hi\n[GOAL]\ncase mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder x n \u2286 cylinder y n\n[PROOFSTEP]\nintro z hz i hi\n[GOAL]\ncase mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder x n\ni : \u2115\nhi : i < n\n\u22a2 z i = y i\n[PROOFSTEP]\nrw [hy i hi]\n[GOAL]\ncase mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder x n\ni : \u2115\nhi : i < n\n\u22a2 z i = x i\n[PROOFSTEP]\nexact hz i hi\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 cylinder y n = cylinder x n \u2192 y \u2208 cylinder x n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : cylinder y n = cylinder x n\n\u22a2 y \u2208 cylinder x n\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : cylinder y n = cylinder x n\n\u22a2 y \u2208 cylinder y n\n[PROOFSTEP]\nexact self_mem_cylinder _ _\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2194 x \u2208 cylinder y n\n[PROOFSTEP]\nsimp [mem_cylinder_iff_eq, eq_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\ni : \u2115\n\u22a2 x \u2208 cylinder y i \u2194 i \u2264 firstDiff x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\ni : \u2115\n\u22a2 x \u2208 cylinder y i \u2192 i \u2264 firstDiff x y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\ni : \u2115\nh : x \u2208 cylinder y i\n\u22a2 i \u2264 firstDiff x y\n[PROOFSTEP]\nby_contra'\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\ni : \u2115\nh : x \u2208 cylinder y i\nthis : firstDiff x y < i\n\u22a2 False\n[PROOFSTEP]\nexact apply_firstDiff_ne hne (h _ this)\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\ni : \u2115\n\u22a2 i \u2264 firstDiff x y \u2192 x \u2208 cylinder y i\n[PROOFSTEP]\nintro hi j hj\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\ni : \u2115\nhi : i \u2264 firstDiff x y\nj : \u2115\nhj : j < i\n\u22a2 x j = y j\n[PROOFSTEP]\nexact apply_eq_of_lt_firstDiff (hj.trans_le hi)\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\n\u22a2 cylinder x n = cylinder y n\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_eq]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\n\u22a2 x \u2208 cylinder y n\n[PROOFSTEP]\nintro i hi\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhn : n \u2264 firstDiff x y\ni : \u2115\nhi : i < n\n\u22a2 x i = y i\n[PROOFSTEP]\nexact apply_eq_of_lt_firstDiff (hi.trans_le hn)\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 \u22c3 (k : E n), cylinder (update x n k) (n + 1) = cylinder x n\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\n\u22a2 y \u2208 \u22c3 (k : E n), cylinder (update x n k) (n + 1) \u2194 y \u2208 cylinder x n\n[PROOFSTEP]\nsimp only [mem_cylinder_iff, mem_iUnion]\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\n\u22a2 (\u2203 i, \u2200 (i_1 : \u2115), i_1 < n + 1 \u2192 y i_1 = update x n i i_1) \u2194 \u2200 (i : \u2115), i < n \u2192 y i = x i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\n\u22a2 (\u2203 i, \u2200 (i_1 : \u2115), i_1 < n + 1 \u2192 y i_1 = update x n i i_1) \u2192 \u2200 (i : \u2115), i < n \u2192 y i = x i\n[PROOFSTEP]\nrintro \u27e8k, hk\u27e9 i hi\n[GOAL]\ncase h.mp.intro\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\nk : E n\nhk : \u2200 (i : \u2115), i < n + 1 \u2192 y i = update x n k i\ni : \u2115\nhi : i < n\n\u22a2 y i = x i\n[PROOFSTEP]\nsimpa [hi.ne] using hk i (Nat.lt_succ_of_lt hi)\n[GOAL]\ncase h.mpr\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\n\u22a2 (\u2200 (i : \u2115), i < n \u2192 y i = x i) \u2192 \u2203 i, \u2200 (i_1 : \u2115), i_1 < n + 1 \u2192 y i_1 = update x n i i_1\n[PROOFSTEP]\nintro H\n[GOAL]\ncase h.mpr\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\nH : \u2200 (i : \u2115), i < n \u2192 y i = x i\n\u22a2 \u2203 i, \u2200 (i_1 : \u2115), i_1 < n + 1 \u2192 y i_1 = update x n i i_1\n[PROOFSTEP]\nrefine' \u27e8y n, fun i hi => _\u27e9\n[GOAL]\ncase h.mpr\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\nH : \u2200 (i : \u2115), i < n \u2192 y i = x i\ni : \u2115\nhi : i < n + 1\n\u22a2 y i = update x n (y n) i\n[PROOFSTEP]\nrcases Nat.lt_succ_iff_lt_or_eq.1 hi with (h'i | rfl)\n[GOAL]\ncase h.mpr.inl\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : (n : \u2115) \u2192 E n\nH : \u2200 (i : \u2115), i < n \u2192 y i = x i\ni : \u2115\nhi : i < n + 1\nh'i : i < n\n\u22a2 y i = update x n (y n) i\n[PROOFSTEP]\nsimp [H i h'i, h'i.ne]\n[GOAL]\ncase h.mpr.inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\ni : \u2115\nH : \u2200 (i_1 : \u2115), i_1 < i \u2192 y i_1 = x i_1\nhi : i < i + 1\n\u22a2 y i = update x i (y i) i\n[PROOFSTEP]\nsimp\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn : \u2115\ny : E n\ni : \u2115\nhi : i < n\n\u22a2 update x n y i = x i\n[PROOFSTEP]\nsimp [hi.ne]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 length (res x n) = n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx : \u2115 \u2192 \u03b1\n\u22a2 length (res x Nat.zero) = Nat.zero\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nn_ih\u271d : length (res x n\u271d) = n\u271d\n\u22a2 length (res x (Nat.succ n\u271d)) = Nat.succ n\u271d\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 res x n = res y n \u2194 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 (\u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m) \u2192 res x n = res y n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\nh : res x n = res y n\n\u22a2 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\nh : \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\n\u22a2 res x n = res y n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase mp.zero\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\nh\u271d : res x n = res y n\nh : res x Nat.zero = res y Nat.zero\n\u22a2 \u2200 \u2983m : \u2115\u2984, m < Nat.zero \u2192 x m = y m\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nh : res x (Nat.succ n) = res y (Nat.succ n)\n\u22a2 \u2200 \u2983m : \u2115\u2984, m < Nat.succ n \u2192 x m = y m\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase mp.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nh : res x (Nat.succ n) = res y (Nat.succ n)\nm : \u2115\nhm : m < Nat.succ n\n\u22a2 x m = y m\n[PROOFSTEP]\nrw [Nat.lt_succ_iff_lt_or_eq] at hm \n[GOAL]\ncase mp.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nh : res x (Nat.succ n) = res y (Nat.succ n)\nm : \u2115\nhm : m < n \u2228 m = n\n\u22a2 x m = y m\n[PROOFSTEP]\nsimp only [res_succ, cons.injEq] at h \n[GOAL]\ncase mp.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nm : \u2115\nhm : m < n \u2228 m = n\nh : x n = y n \u2227 res x n = res y n\n\u22a2 x m = y m\n[PROOFSTEP]\ncases' hm with hm hm\n[GOAL]\ncase mp.succ.inl\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nm : \u2115\nh : x n = y n \u2227 res x n = res y n\nhm : m < n\n\u22a2 x m = y m\n[PROOFSTEP]\nexact ih h.2 hm\n[GOAL]\ncase mp.succ.inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nm : \u2115\nh : x n = y n \u2227 res x n = res y n\nhm : m = n\n\u22a2 x m = y m\n[PROOFSTEP]\nrw [hm]\n[GOAL]\ncase mp.succ.inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : res x n\u271d = res y n\u271d\nn : \u2115\nih : res x n = res y n \u2192 \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nm : \u2115\nh : x n = y n \u2227 res x n = res y n\nhm : m = n\n\u22a2 x n = y n\n[PROOFSTEP]\nexact h.1\n[GOAL]\ncase mpr.zero\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn : \u2115\nh\u271d : \u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m\nh : \u2200 \u2983m : \u2115\u2984, m < Nat.zero \u2192 x m = y m\n\u22a2 res x Nat.zero = res y Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : \u2200 \u2983m : \u2115\u2984, m < n\u271d \u2192 x m = y m\nn : \u2115\nih : (\u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m) \u2192 res x n = res y n\nh : \u2200 \u2983m : \u2115\u2984, m < Nat.succ n \u2192 x m = y m\n\u22a2 res x (Nat.succ n) = res y (Nat.succ n)\n[PROOFSTEP]\nsimp only [res_succ, cons.injEq]\n[GOAL]\ncase mpr.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : \u2200 \u2983m : \u2115\u2984, m < n\u271d \u2192 x m = y m\nn : \u2115\nih : (\u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m) \u2192 res x n = res y n\nh : \u2200 \u2983m : \u2115\u2984, m < Nat.succ n \u2192 x m = y m\n\u22a2 x n = y n \u2227 res x n = res y n\n[PROOFSTEP]\nrefine' \u27e8h (Nat.lt_succ_self _), ih fun m hm => _\u27e9\n[GOAL]\ncase mpr.succ\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nn\u271d : \u2115\nh\u271d : \u2200 \u2983m : \u2115\u2984, m < n\u271d \u2192 x m = y m\nn : \u2115\nih : (\u2200 \u2983m : \u2115\u2984, m < n \u2192 x m = y m) \u2192 res x n = res y n\nh : \u2200 \u2983m : \u2115\u2984, m < Nat.succ n \u2192 x m = y m\nm : \u2115\nhm : m < n\n\u22a2 x m = y m\n[PROOFSTEP]\nexact h (hm.trans (Nat.lt_succ_self _))\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\n\u22a2 Injective res\n[PROOFSTEP]\nintro x y h\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nh : res x = res y\n\u22a2 x = y\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nh : res x = res y\nn : \u2115\n\u22a2 x n = y n\n[PROOFSTEP]\napply res_eq_res.mp _ (Nat.lt_succ_self _)\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx y : \u2115 \u2192 \u03b1\nh : res x = res y\nn : \u2115\n\u22a2 res x (Nat.succ n) = res (fun n => y n) (Nat.succ n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 cylinder x n = {y | res y n = res x n}\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx : \u2115 \u2192 \u03b1\nn : \u2115\ny : \u2115 \u2192 \u03b1\n\u22a2 y \u2208 cylinder x n \u2194 y \u2208 {y | res y n = res x n}\n[PROOFSTEP]\ndsimp [cylinder]\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\nx : \u2115 \u2192 \u03b1\nn : \u2115\ny : \u2115 \u2192 \u03b1\n\u22a2 (\u2200 (i : \u2115), i < n \u2192 y i = x i) \u2194 res y n = res x n\n[PROOFSTEP]\nrw [res_eq_res]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nh : x \u2260 y\n\u22a2 dist x y = (1 / 2) ^ firstDiff x y\n[PROOFSTEP]\nsimp [dist, h]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\n\u22a2 dist x x = 0\n[PROOFSTEP]\nsimp [dist]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\n\u22a2 dist x y = dist y x\n[PROOFSTEP]\nsimp [dist, @eq_comm _ x y, firstDiff_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\n\u22a2 0 \u2264 dist x y\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | h)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\n\u22a2 0 \u2264 dist x x\n[PROOFSTEP]\nsimp [dist]\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nh : x \u2260 y\n\u22a2 0 \u2264 dist x y\n[PROOFSTEP]\nsimp [dist, h, zero_le_two]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\n\u22a2 dist x z \u2264 max (dist x y) (dist y z)\n[PROOFSTEP]\nrcases eq_or_ne x z with (rfl | hxz)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\n\u22a2 dist x x \u2264 max (dist x y) (dist y x)\n[PROOFSTEP]\nsimp [PiNat.dist_self x, PiNat.dist_nonneg]\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nhxz : x \u2260 z\n\u22a2 dist x z \u2264 max (dist x y) (dist y z)\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | hxy)\n[GOAL]\ncase inr.inl\nE : \u2115 \u2192 Type u_1\nx z : (n : \u2115) \u2192 E n\nhxz : x \u2260 z\n\u22a2 dist x z \u2264 max (dist x x) (dist x z)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nhxz : x \u2260 z\nhxy : x \u2260 y\n\u22a2 dist x z \u2264 max (dist x y) (dist y z)\n[PROOFSTEP]\nrcases eq_or_ne y z with (rfl | hyz)\n[GOAL]\ncase inr.inr.inl\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhxy hxz : x \u2260 y\n\u22a2 dist x y \u2264 max (dist x y) (dist y y)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.inr\nE : \u2115 \u2192 Type u_1\nx y z : (n : \u2115) \u2192 E n\nhxz : x \u2260 z\nhxy : x \u2260 y\nhyz : y \u2260 z\n\u22a2 dist x z \u2264 max (dist x y) (dist y z)\n[PROOFSTEP]\nsimp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_two, Ne.def, not_false_iff, le_max_iff,\n  pow_le_pow_iff, one_lt_two, pow_pos, min_le_iff.1 (min_firstDiff_le x y z hxz)]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhxy : dist x y = 0\n\u22a2 x = y\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | h)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nhxy : dist x x = 0\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nhxy : dist x y = 0\nh : x \u2260 y\n\u22a2 x = y\n[PROOFSTEP]\nsimp [dist_eq_of_ne h] at hxy \n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nh : x \u2260 y\nhxy : 2 ^ firstDiff x y = 0\n\u22a2 x = y\n[PROOFSTEP]\nexact (two_ne_zero (pow_eq_zero hxy)).elim\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2194 dist y x \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nrcases eq_or_ne y x with (rfl | hne)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ny : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder y n \u2194 dist y y \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nsimp [PiNat.dist_self]\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\n\u22a2 y \u2208 cylinder x n \u2194 dist y x \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nsuffices (\u2200 i : \u2115, i < n \u2192 y i = x i) \u2194 n \u2264 firstDiff y x by simpa [dist_eq_of_ne hne]\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\nthis : (\u2200 (i : \u2115), i < n \u2192 y i = x i) \u2194 n \u2264 firstDiff y x\n\u22a2 y \u2208 cylinder x n \u2194 dist y x \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nsimpa [dist_eq_of_ne hne]\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\n\u22a2 (\u2200 (i : \u2115), i < n \u2192 y i = x i) \u2194 n \u2264 firstDiff y x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\n\u22a2 (\u2200 (i : \u2115), i < n \u2192 y i = x i) \u2192 n \u2264 firstDiff y x\n[PROOFSTEP]\nintro hy\n[GOAL]\ncase inr.mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\nhy : \u2200 (i : \u2115), i < n \u2192 y i = x i\n\u22a2 n \u2264 firstDiff y x\n[PROOFSTEP]\nby_contra' H\n[GOAL]\ncase inr.mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\nhy : \u2200 (i : \u2115), i < n \u2192 y i = x i\nH : firstDiff y x < n\n\u22a2 False\n[PROOFSTEP]\nexact apply_firstDiff_ne hne (hy _ H)\n[GOAL]\ncase inr.mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\n\u22a2 n \u2264 firstDiff y x \u2192 \u2200 (i : \u2115), i < n \u2192 y i = x i\n[PROOFSTEP]\nintro h i hi\n[GOAL]\ncase inr.mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhne : y \u2260 x\nh : n \u2264 firstDiff y x\ni : \u2115\nhi : i < n\n\u22a2 y i = x i\n[PROOFSTEP]\nexact apply_eq_of_lt_firstDiff (hi.trans_le h)\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : dist x y < (1 / 2) ^ n\ni : \u2115\nhi : i \u2264 n\n\u22a2 x i = y i\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | hne)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\nx : (n : \u2115) \u2192 E n\nn i : \u2115\nhi : i \u2264 n\nh : dist x x < (1 / 2) ^ n\n\u22a2 x i = x i\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : dist x y < (1 / 2) ^ n\ni : \u2115\nhi : i \u2264 n\nhne : x \u2260 y\n\u22a2 x i = y i\n[PROOFSTEP]\nhave : n < firstDiff x y := by simpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff, one_lt_two] using h\n[GOAL]\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : dist x y < (1 / 2) ^ n\ni : \u2115\nhi : i \u2264 n\nhne : x \u2260 y\n\u22a2 n < firstDiff x y\n[PROOFSTEP]\nsimpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff, one_lt_two] using h\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : dist x y < (1 / 2) ^ n\ni : \u2115\nhi : i \u2264 n\nhne : x \u2260 y\nthis : n < firstDiff x y\n\u22a2 x i = y i\n[PROOFSTEP]\nexact apply_eq_of_lt_firstDiff (hi.trans_lt this)\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\n\u22a2 (\u2200 (x y : (n : \u2115) \u2192 E n), dist (f x) (f y) \u2264 dist x y) \u2194\n    \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\n\u22a2 (\u2200 (x y : (n : \u2115) \u2192 E n), dist (f x) (f y) \u2264 dist x y) \u2192\n    \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nintro H x y n hxy\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n), dist (f x) (f y) \u2264 dist x y\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhxy : y \u2208 cylinder x n\n\u22a2 dist (f x) (f y) \u2264 (1 / 2) ^ n\n[PROOFSTEP]\napply (H x y).trans\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n), dist (f x) (f y) \u2264 dist x y\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhxy : y \u2208 cylinder x n\n\u22a2 dist x y \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nrw [PiNat.dist_comm]\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n), dist (f x) (f y) \u2264 dist x y\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhxy : y \u2208 cylinder x n\n\u22a2 dist y x \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nexact mem_cylinder_iff_dist_le.1 hxy\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\n\u22a2 (\u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n) \u2192\n    \u2200 (x y : (n : \u2115) \u2192 E n), dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nintro H x y\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\nx y : (n : \u2115) \u2192 E n\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | hne)\n[GOAL]\ncase mpr.inl\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\nx : (n : \u2115) \u2192 E n\n\u22a2 dist (f x) (f x) \u2264 dist x x\n[PROOFSTEP]\nsimp [PiNat.dist_nonneg]\n[GOAL]\ncase mpr.inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nrw [dist_eq_of_ne hne]\n[GOAL]\ncase mpr.inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\n\u22a2 dist (f x) (f y) \u2264 (1 / 2) ^ firstDiff x y\n[PROOFSTEP]\napply H x y (firstDiff x y)\n[GOAL]\ncase mpr.inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\n\u22a2 y \u2208 cylinder x (firstDiff x y)\n[PROOFSTEP]\nrw [firstDiff_comm]\n[GOAL]\ncase mpr.inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nf : ((n : \u2115) \u2192 E n) \u2192 \u03b1\nH : \u2200 (x y : (n : \u2115) \u2192 E n) (n : \u2115), y \u2208 cylinder x n \u2192 dist (f x) (f y) \u2264 (1 / 2) ^ n\nx y : (n : \u2115) \u2192 E n\nhne : x \u2260 y\n\u22a2 y \u2208 cylinder x (firstDiff y x)\n[PROOFSTEP]\nexact mem_cylinder_firstDiff _ _\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 IsOpen (cylinder x n)\n[PROOFSTEP]\nrw [PiNat.cylinder_eq_pi]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 IsOpen (Set.pi \u2191(Finset.range n) fun i => {x i})\n[PROOFSTEP]\nexact isOpen_set_pi (Finset.range n).finite_toSet fun a _ => isOpen_discrete _\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 IsTopologicalBasis {s | \u2203 x n, s = cylinder x n}\n[PROOFSTEP]\napply isTopologicalBasis_of_open_of_nhds\n[GOAL]\ncase h_open\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2200 (u : Set ((n : \u2115) \u2192 E n)), u \u2208 {s | \u2203 x n, s = cylinder x n} \u2192 IsOpen u\n[PROOFSTEP]\nrintro u \u27e8x, n, rfl\u27e9\n[GOAL]\ncase h_open.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 IsOpen (cylinder x n)\n[PROOFSTEP]\napply isOpen_cylinder\n[GOAL]\ncase h_nhds\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2200 (a : (n : \u2115) \u2192 E n) (u : Set ((n : \u2115) \u2192 E n)),\n    a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 a \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nintro x u hx u_open\n[GOAL]\ncase h_nhds\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\n\u22a2 \u2203 v, v \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nobtain \u27e8v, \u27e8U, F, -, rfl\u27e9, xU, Uu\u27e9 :\n  \u2203\n    v \u2208\n      {S : Set (\u2200 i : \u2115, E i) |\n        \u2203 (U : \u2200 i : \u2115, Set (E i)) (F : Finset \u2115),\n          (\u2200 i : \u2115, i \u2208 F \u2192 U i \u2208 {s : Set (E i) | IsOpen s}) \u2227 S = (F : Set \u2115).pi U},\n    x \u2208 v \u2227 v \u2286 u :=\n  (isTopologicalBasis_pi fun n : \u2115 => isTopologicalBasis_opens).exists_subset_of_mem_open hx u_open\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\n\u22a2 \u2203 v, v \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrcases Finset.bddAbove F with \u27e8n, hn\u27e9\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\n\u22a2 \u2203 v, v \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 x \u2208 v \u2227 v \u2286 u\n[PROOFSTEP]\nrefine' \u27e8cylinder x (n + 1), \u27e8x, n + 1, rfl\u27e9, self_mem_cylinder _ _, Subset.trans _ Uu\u27e9\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\n\u22a2 cylinder x (n + 1) \u2286 Set.pi (\u2191F) U\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\n\u22a2 y \u2208 Set.pi (\u2191F) U\n[PROOFSTEP]\nsuffices \u2200 i : \u2115, i \u2208 F \u2192 y i \u2208 U i by simpa\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\nthis : \u2200 (i : \u2115), i \u2208 F \u2192 y i \u2208 U i\n\u22a2 y \u2208 Set.pi (\u2191F) U\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\n\u22a2 \u2200 (i : \u2115), i \u2208 F \u2192 y i \u2208 U i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\ni : \u2115\nhi : i \u2208 F\n\u22a2 y i \u2208 U i\n[PROOFSTEP]\nhave : y i = x i := mem_cylinder_iff.1 hy i ((hn hi).trans_lt (lt_add_one n))\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\ni : \u2115\nhi : i \u2208 F\nthis : y i = x i\n\u22a2 y i \u2208 U i\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nxU : x \u2208 Set.pi (\u2191F) U\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\ni : \u2115\nhi : i \u2208 F\nthis : y i = x i\n\u22a2 x i \u2208 U i\n[PROOFSTEP]\nsimp only [Set.mem_pi, Finset.mem_coe] at xU \n[GOAL]\ncase h_nhds.intro.intro.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nu : Set ((n : \u2115) \u2192 E n)\nhx : x \u2208 u\nu_open : IsOpen u\nU : (i : \u2115) \u2192 Set (E i)\nF : Finset \u2115\nUu : Set.pi (\u2191F) U \u2286 u\nn : \u2115\nhn : n \u2208 upperBounds \u2191F\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x (n + 1)\ni : \u2115\nhi : i \u2208 F\nthis : y i = x i\nxU : \u2200 (i : \u2115), i \u2208 F \u2192 x i \u2208 U i\n\u22a2 x i \u2208 U i\n[PROOFSTEP]\nexact xU i hi\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\n\u22a2 IsOpen s \u2194 \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\n\u22a2 IsOpen s \u2192 \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n[PROOFSTEP]\nintro hs x hx\n[GOAL]\ncase mp\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsOpen s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n[PROOFSTEP]\nobtain \u27e8v, \u27e8y, n, rfl\u27e9, h'x, h's\u27e9 : \u2203 v \u2208 {s | \u2203 (x : \u2200 n : \u2115, E n) (n : \u2115), s = cylinder x n}, x \u2208 v \u2227 v \u2286 s :=\n  (isTopologicalBasis_cylinders E).exists_subset_of_mem_open hx hs\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsOpen s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\ny : (n : \u2115) \u2192 E n\nn : \u2115\nh'x : x \u2208 cylinder y n\nh's : cylinder y n \u2286 s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_eq.1 h'x] at h's \n[GOAL]\ncase mp.intro.intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsOpen s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\ny : (n : \u2115) \u2192 E n\nn : \u2115\nh'x : x \u2208 cylinder y n\nh's : cylinder x n \u2286 s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n[PROOFSTEP]\nexact \u27e8(1 / 2 : \u211d) ^ n, by simp, fun y hy => h's fun i hi => (apply_eq_of_dist_lt hy hi.le).symm\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsOpen s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\ny : (n : \u2115) \u2192 E n\nn : \u2115\nh'x : x \u2208 cylinder y n\nh's : cylinder x n \u2286 s\n\u22a2 (1 / 2) ^ n > 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\n\u22a2 (\u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s) \u2192 IsOpen s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nh : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n\u22a2 IsOpen s\n[PROOFSTEP]\nrefine (isTopologicalBasis_cylinders E).isOpen_iff.2 fun x hx => ?_\n[GOAL]\ncase mpr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nh : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 \u2203 t, t \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 x \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nrcases h x hx with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n[GOAL]\ncase mpr.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nh : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\n\u22a2 \u2203 t, t \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 x \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2 : \u211d) ^ n < \u03b5 := exists_pow_lt_of_lt_one \u03b5pos one_half_lt_one\n[GOAL]\ncase mpr.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nh : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\n\u22a2 \u2203 t, t \u2208 {s | \u2203 x n, s = cylinder x n} \u2227 x \u2208 t \u2227 t \u2286 s\n[PROOFSTEP]\nrefine' \u27e8cylinder x n, \u27e8x, n, rfl\u27e9, self_mem_cylinder x n, fun y hy => h\u03b5 y _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nh : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x n\n\u22a2 dist x y < \u03b5\n[PROOFSTEP]\nrw [PiNat.dist_comm]\n[GOAL]\ncase mpr.intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nh : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : \u2200 (y : (n : \u2115) \u2192 E n), dist x y < \u03b5 \u2192 y \u2208 s\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 cylinder x n\n\u22a2 dist y x < \u03b5\n[PROOFSTEP]\nexact (mem_cylinder_iff_dist_le.1 hy).trans_lt hn\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nx\u271d\u00b9 x\u271d : (n : \u2115) \u2192 E n\n\u22a2 (fun x y => \u2191{ val := dist x y, property := (_ : 0 \u2264 dist x y) }) x\u271d\u00b9 x\u271d = ENNReal.ofReal (dist x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nexact ENNReal.coe_nnreal_eq _\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 uniformity ((n : \u2115) \u2192 E n) = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp [Pi.uniformity, comap_iInf, gt_iff_lt, preimage_setOf_eq, comap_principal, PseudoMetricSpace.uniformity_dist, h,\n  idRel]\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf {a | Prod.fst a i = Prod.snd a i} = \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2a05 (i : \u2115), \ud835\udcdf {a | Prod.fst a i = Prod.snd a i} \u2264 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [le_iInf_iff, le_principal_iff]\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 {p | dist p.fst p.snd < i} \u2208 \u2a05 (i : \u2115), \ud835\udcdf {a | Prod.fst a i = Prod.snd a i}\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 {p | dist p.fst p.snd < \u03b5} \u2208 \u2a05 (i : \u2115), \ud835\udcdf {a | Prod.fst a i = Prod.snd a i}\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, (1 / 2 : \u211d) ^ n < \u03b5 := exists_pow_lt_of_lt_one \u03b5pos (by norm_num)\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase a.intro\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\n\u22a2 {p | dist p.fst p.snd < \u03b5} \u2208 \u2a05 (i : \u2115), \ud835\udcdf {a | Prod.fst a i = Prod.snd a i}\n[PROOFSTEP]\napply\n  @mem_iInf_of_iInter _ _ _ _ _ (Finset.range n).finite_toSet fun i =>\n    {p : (\u2200 n : \u2115, E n) \u00d7 \u2200 n : \u2115, E n | p.fst i = p.snd i}\n[GOAL]\ncase a.intro.hV\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\n\u22a2 \u2200 (i : \u2191\u2191(Finset.range n)), {p | Prod.fst p \u2191i = Prod.snd p \u2191i} \u2208 \ud835\udcdf {a | Prod.fst a \u2191i = Prod.snd a \u2191i}\n[PROOFSTEP]\nsimp only [mem_principal, setOf_subset_setOf, imp_self, imp_true_iff]\n[GOAL]\ncase a.intro.hU\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\n\u22a2 \u22c2 (i : \u2191\u2191(Finset.range n)), {p | Prod.fst p \u2191i = Prod.snd p \u2191i} \u2286 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 hxy\n[GOAL]\ncase a.intro.hU.mk\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\nx y : (n : \u2115) \u2192 E n\nhxy : (x, y) \u2208 \u22c2 (i : \u2191\u2191(Finset.range n)), {p | Prod.fst p \u2191i = Prod.snd p \u2191i}\n\u22a2 (x, y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [Finset.mem_coe, Finset.mem_range, iInter_coe_set, mem_iInter, mem_setOf_eq] at hxy \n[GOAL]\ncase a.intro.hU.mk\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\nx y : (n : \u2115) \u2192 E n\nhxy : \u2200 (i : \u2115), i < n \u2192 x i = y i\n\u22a2 (x, y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\napply lt_of_le_of_lt _ hn\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\nx y : (n : \u2115) \u2192 E n\nhxy : \u2200 (i : \u2115), i < n \u2192 x i = y i\n\u22a2 dist (x, y).fst (x, y).snd \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_dist_le, mem_cylinder_iff]\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : (1 / 2) ^ n < \u03b5\nx y : (n : \u2115) \u2192 E n\nhxy : \u2200 (i : \u2115), i < n \u2192 x i = y i\n\u22a2 \u2200 (i : \u2115), i < n \u2192 Prod.fst (x, y) i = Prod.snd (x, y) i\n[PROOFSTEP]\nexact hxy\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5} \u2264 \u2a05 (i : \u2115), \ud835\udcdf {a | Prod.fst a i = Prod.snd a i}\n[PROOFSTEP]\nsimp only [le_iInf_iff, le_principal_iff]\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2200 (i : \u2115), {a | Prod.fst a i = Prod.snd a i} \u2208 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nintro n\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nn : \u2115\n\u22a2 {a | Prod.fst a n = Prod.snd a n} \u2208 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrefine' mem_iInf_of_mem ((1 / 2) ^ n : \u211d) _\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nn : \u2115\n\u22a2 {a | Prod.fst a n = Prod.snd a n} \u2208 \u2a05 (_ : 0 < (1 / 2) ^ n), \ud835\udcdf {p | dist p.fst p.snd < (1 / 2) ^ n}\n[PROOFSTEP]\nrefine' mem_iInf_of_mem (by positivity) _\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nn : \u2115\n\u22a2 0 < (1 / 2) ^ n\n[PROOFSTEP]\npositivity\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nn : \u2115\n\u22a2 {a | Prod.fst a n = Prod.snd a n} \u2208 \ud835\udcdf {p | dist p.fst p.snd < (1 / 2) ^ n}\n[PROOFSTEP]\nsimp only [mem_principal, setOf_subset_setOf, Prod.forall]\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nn : \u2115\n\u22a2 \u2200 (a b : (n : \u2115) \u2192 E n), dist a b < (1 / 2) ^ n \u2192 a n = b n\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase a\nE\u271d : \u2115 \u2192 Type u_1\ninst\u271d\u00b2 : (n : \u2115) \u2192 TopologicalSpace (E\u271d n)\ninst\u271d\u00b9 : \u2200 (n : \u2115), DiscreteTopology (E\u271d n)\nE : \u2115 \u2192 Type u_2\ninst\u271d : (n : \u2115) \u2192 UniformSpace (E n)\nh : \u2200 (n : \u2115), uniformity (E n) = \ud835\udcdf idRel\nthis : \u2200 (n : \u2115), DiscreteTopology (E n)\nn : \u2115\nx y : (n : \u2115) \u2192 E n\nhxy : dist x y < (1 / 2) ^ n\n\u22a2 x n = y n\n[PROOFSTEP]\nexact apply_eq_of_dist_lt hxy le_rfl\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 CompleteSpace ((n : \u2115) \u2192 E n)\n[PROOFSTEP]\nrefine' Metric.complete_of_convergent_controlled_sequences (fun n => (1 / 2) ^ n) (by simp) _\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2200 (n : \u2115), 0 < (fun n => (1 / 2) ^ n) n\n[PROOFSTEP]\nsimp\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\n\u22a2 \u2200 (u : \u2115 \u2192 (n : \u2115) \u2192 E n),\n    (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun n => (1 / 2) ^ n) N) \u2192 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nintro u hu\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nu : \u2115 \u2192 (n : \u2115) \u2192 E n\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun n => (1 / 2) ^ n) N\n\u22a2 \u2203 x, Tendsto u atTop (\ud835\udcdd x)\n[PROOFSTEP]\nrefine' \u27e8fun n => u n n, tendsto_pi_nhds.2 fun i => _\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nu : \u2115 \u2192 (n : \u2115) \u2192 E n\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun n => (1 / 2) ^ n) N\ni : \u2115\n\u22a2 Tendsto (fun i_1 => u i_1 i) atTop (\ud835\udcdd (u i i))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.congr' _\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nu : \u2115 \u2192 (n : \u2115) \u2192 E n\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun n => (1 / 2) ^ n) N\ni : \u2115\n\u22a2 (fun x => u i i) =\u1da0[atTop] fun i_1 => u i_1 i\n[PROOFSTEP]\nfilter_upwards [Filter.Ici_mem_atTop i] with n hn\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nu : \u2115 \u2192 (n : \u2115) \u2192 E n\nhu : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < (fun n => (1 / 2) ^ n) N\ni n : \u2115\nhn : n \u2208 Ici i\n\u22a2 u i i = u n i\n[PROOFSTEP]\nexact apply_eq_of_dist_lt (hu i i n le_rfl hn) le_rfl\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hne)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nhs : IsClosed \u2205\nhx : \u00acx \u2208 \u2205\n\u22a2 \u2203 n, Disjoint \u2205 (cylinder x n)\n[PROOFSTEP]\nexact \u27e80, by simp\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nhs : IsClosed \u2205\nhx : \u00acx \u2208 \u2205\n\u22a2 Disjoint \u2205 (cylinder x 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)\n[PROOFSTEP]\nhave A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, (1 / 2 : \u211d) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one\n[GOAL]\ncase inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)\n[PROOFSTEP]\nrefine' \u27e8n, disjoint_left.2 fun y ys hy => ?_\u27e9\n[GOAL]\ncase inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl (infDist x s)\n[GOAL]\ncase inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 infDist x s < infDist x s\n[PROOFSTEP]\ncalc\n  infDist x s \u2264 dist x y := infDist_le_dist_of_mem ys\n  _ \u2264 (1 / 2) ^ n := by\n    rw [mem_cylinder_comm] at hy \n    exact mem_cylinder_iff_dist_le.1 hy\n  _ < infDist x s := hn\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 dist x y \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nrw [mem_cylinder_comm] at hy \n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y n\n\u22a2 dist x y \u2264 (1 / 2) ^ n\n[PROOFSTEP]\nexact mem_cylinder_iff_dist_le.1 hy\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\n\u22a2 firstDiff x y < shortestPrefixDiff x s\n[PROOFSTEP]\nhave A := exists_disjoint_cylinder hs hx\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\n\u22a2 firstDiff x y < shortestPrefixDiff x s\n[PROOFSTEP]\nrw [shortestPrefixDiff, dif_pos A]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\n\u22a2 firstDiff x y < Nat.find A\n[PROOFSTEP]\nhave B := Nat.find_spec A\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Disjoint s (cylinder x (Nat.find A))\n\u22a2 firstDiff x y < Nat.find A\n[PROOFSTEP]\ncontrapose! B\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A \u2264 firstDiff x y\n\u22a2 \u00acDisjoint s (cylinder x (Nat.find A))\n[PROOFSTEP]\nrw [not_disjoint_iff_nonempty_inter]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A \u2264 firstDiff x y\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A))\n[PROOFSTEP]\nrefine' \u27e8y, hy, _\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A \u2264 firstDiff x y\n\u22a2 y \u2208 cylinder x (Nat.find A)\n[PROOFSTEP]\nrw [mem_cylinder_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A \u2264 firstDiff x y\n\u22a2 x \u2208 cylinder y (Nat.find A)\n[PROOFSTEP]\nexact cylinder_anti y B (mem_cylinder_firstDiff x y)\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\n\u22a2 0 < shortestPrefixDiff x s\n[PROOFSTEP]\nrcases hne with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 s\n\u22a2 0 < shortestPrefixDiff x s\n[PROOFSTEP]\nexact (zero_le _).trans_lt (firstDiff_lt_shortestPrefixDiff hs hx hy)\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\n\u22a2 firstDiff x y \u2264 longestPrefix x s\n[PROOFSTEP]\nrw [longestPrefix, le_tsub_iff_right]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\n\u22a2 firstDiff x y + 1 \u2264 shortestPrefixDiff x s\n[PROOFSTEP]\nexact firstDiff_lt_shortestPrefixDiff hs hx hy\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx y : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhy : y \u2208 s\n\u22a2 1 \u2264 shortestPrefixDiff x s\n[PROOFSTEP]\nexact shortestPrefixDiff_pos hs \u27e8y, hy\u27e9 hx\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n[PROOFSTEP]\nby_cases hx : x \u2208 s\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n[PROOFSTEP]\nexact \u27e8x, hx, self_mem_cylinder _ _\u27e9\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n[PROOFSTEP]\nhave A := exists_disjoint_cylinder hs hx\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n[PROOFSTEP]\nhave B : longestPrefix x s < shortestPrefixDiff x s := Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : longestPrefix x s < shortestPrefixDiff x s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n[PROOFSTEP]\nrw [longestPrefix, shortestPrefixDiff, dif_pos A] at B \u22a2\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A - 1))\n[PROOFSTEP]\nobtain \u27e8y, ys, hy\u27e9 : \u2203 y : \u2200 n : \u2115, E n, y \u2208 s \u2227 x \u2208 cylinder y (Nat.find A - 1) := by\n  simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n\u22a2 \u2203 y, y \u2208 s \u2227 x \u2208 cylinder y (Nat.find A - 1)\n[PROOFSTEP]\nsimpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B\n[GOAL]\ncase neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y (Nat.find A - 1)\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A - 1))\n[PROOFSTEP]\nrefine' \u27e8y, ys, _\u27e9\n[GOAL]\ncase neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y (Nat.find A - 1)\n\u22a2 y \u2208 cylinder x (Nat.find A - 1)\n[PROOFSTEP]\nrw [mem_cylinder_iff_eq] at hy \u22a2\n[GOAL]\ncase neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : cylinder x (Nat.find A - 1) = cylinder y (Nat.find A - 1)\n\u22a2 cylinder y (Nat.find A - 1) = cylinder x (Nat.find A - 1)\n[PROOFSTEP]\nrw [hy]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : longestPrefix x s < n\n\u22a2 Disjoint s (cylinder x n)\n[PROOFSTEP]\ncontrapose! hn\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\n\u22a2 n \u2264 longestPrefix x s\n[PROOFSTEP]\nrcases not_disjoint_iff_nonempty_inter.1 hn with \u27e8y, ys, hy\u27e9\n[GOAL]\ncase intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 n \u2264 longestPrefix x s\n[PROOFSTEP]\napply le_trans _ (firstDiff_le_longestPrefix hs hx ys)\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 n \u2264 firstDiff x y\n[PROOFSTEP]\napply (mem_cylinder_iff_le_firstDiff (ne_of_mem_of_not_mem ys hx).symm _).1\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 x \u2208 cylinder y n\n[PROOFSTEP]\nrwa [mem_cylinder_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\n\u22a2 cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n[PROOFSTEP]\nhave l_eq : longestPrefix y s = longestPrefix x s :=\n  by\n  rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)\n  \u00b7 have Ax : (s \u2229 cylinder x (longestPrefix x s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne x\n    have Z := disjoint_cylinder_of_longestPrefix_lt hs ys L\n    rw [firstDiff_comm] at H \n    rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z \n    exact (Ax.not_disjoint Z).elim\n  \u00b7 exact L\n  \u00b7 have Ay : (s \u2229 cylinder y (longestPrefix y s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne y\n    have A'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty :=\n      Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))\n    have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)\n    rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z \n    exact (A'y.not_disjoint Z).elim\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nrcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix y s < longestPrefix x s\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nhave Ax : (s \u2229 cylinder x (longestPrefix x s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne x\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix y s < longestPrefix x s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nhave Z := disjoint_cylinder_of_longestPrefix_lt hs ys L\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix y s < longestPrefix x s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nZ : Disjoint s (cylinder y (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nrw [firstDiff_comm] at H \n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff y x\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix y s < longestPrefix x s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nZ : Disjoint s (cylinder y (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nrw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z \n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff y x\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix y s < longestPrefix x s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nZ : Disjoint s (cylinder x (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nexact (Ax.not_disjoint Z).elim\n[GOAL]\ncase inr.inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix y s = longestPrefix x s\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nexact L\n[GOAL]\ncase inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix x s < longestPrefix y s\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nhave Ay : (s \u2229 cylinder y (longestPrefix y s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne y\n[GOAL]\ncase inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix x s < longestPrefix y s\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nhave A'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty :=\n  Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))\n[GOAL]\ncase inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix x s < longestPrefix y s\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nA'y : Set.Nonempty (s \u2229 cylinder y (Nat.succ (longestPrefix x s)))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nhave Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)\n[GOAL]\ncase inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix x s < longestPrefix y s\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nA'y : Set.Nonempty (s \u2229 cylinder y (Nat.succ (longestPrefix x s)))\nZ : Disjoint s (cylinder x (Nat.succ (longestPrefix x s)))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nrw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z \n[GOAL]\ncase inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nL : longestPrefix x s < longestPrefix y s\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nA'y : Set.Nonempty (s \u2229 cylinder y (Nat.succ (longestPrefix x s)))\nZ : Disjoint s (cylinder y (Nat.succ (longestPrefix x s)))\n\u22a2 longestPrefix y s = longestPrefix x s\n[PROOFSTEP]\nexact (A'y.not_disjoint Z).elim\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nl_eq : longestPrefix y s = longestPrefix x s\n\u22a2 cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n[PROOFSTEP]\nrw [l_eq, \u2190 mem_cylinder_iff_eq]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nH : longestPrefix x s < firstDiff x y\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nl_eq : longestPrefix y s = longestPrefix x s\n\u22a2 x \u2208 cylinder y (longestPrefix x s)\n[PROOFSTEP]\nexact cylinder_anti y H.le (mem_cylinder_firstDiff x y)\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\n\u22a2 \u2203 f, (\u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x) \u2227 range f = s \u2227 LipschitzWith 1 f\n[PROOFSTEP]\nset f := fun x => if x \u2208 s then x else (inter_cylinder_longestPrefix_nonempty hs hne x).some\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\n\u22a2 \u2203 f, (\u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x) \u2227 range f = s \u2227 LipschitzWith 1 f\n[PROOFSTEP]\nhave fs : \u2200 x \u2208 s, f x = x := fun x xs => by simp [xs]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nx : (n : \u2115) \u2192 E n\nxs : x \u2208 s\n\u22a2 f x = x\n[PROOFSTEP]\nsimp [xs]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\n\u22a2 \u2203 f, (\u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x) \u2227 range f = s \u2227 LipschitzWith 1 f\n[PROOFSTEP]\nrefine'\n  \u27e8f, fs, _, _\u27e9\n    -- check that the range of `f` is `s`.\n[GOAL]\ncase refine'_1\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\n\u22a2 range f = s\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase refine'_1.h\u2081\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\n\u22a2 range f \u2286 s\n[PROOFSTEP]\nrintro x \u27e8y, rfl\u27e9\n[GOAL]\ncase refine'_1.h\u2081.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\ny : (n : \u2115) \u2192 E n\n\u22a2 f y \u2208 s\n[PROOFSTEP]\nby_cases hy : y \u2208 s\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\ny : (n : \u2115) \u2192 E n\nhy : y \u2208 s\n\u22a2 f y \u2208 s\n[PROOFSTEP]\nrwa [fs y hy]\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\ny : (n : \u2115) \u2192 E n\nhy : \u00acy \u2208 s\n\u22a2 f y \u2208 s\n[PROOFSTEP]\nsimpa [if_neg hy] using (inter_cylinder_longestPrefix_nonempty hs hne y).choose_spec.1\n[GOAL]\ncase refine'_1.h\u2082\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\n\u22a2 s \u2286 range f\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_1.h\u2082\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nrw [\u2190 fs x hx]\n[GOAL]\ncase refine'_1.h\u2082\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 f x \u2208 range f\n[PROOFSTEP]\nexact\n  mem_range_self\n    _\n      -- check that `f` is `1`-Lipschitz, by a case analysis.\n[GOAL]\ncase refine'_2\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\n\u22a2 LipschitzWith 1 f\n[PROOFSTEP]\nrefine\n  LipschitzWith.mk_one fun x y =>\n    ?_\n      -- exclude the trivial cases where `x = y`, or `f x = f y`.\n[GOAL]\ncase refine'_2\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | hxy)\n[GOAL]\ncase refine'_2.inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx : (n : \u2115) \u2192 E n\n\u22a2 dist (f x) (f x) \u2264 dist x x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nrcases eq_or_ne (f x) (f y) with (h' | hfxfy)\n[GOAL]\ncase refine'_2.inr.inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nh' : f x = f y\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nsimp [h', dist_nonneg]\n[GOAL]\ncase refine'_2.inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nhave I2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y) :=\n  by\n  rw [\u2190 mem_cylinder_iff_eq]\n  apply mem_cylinder_firstDiff\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\n\u22a2 cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_eq]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\n\u22a2 x \u2208 cylinder y (firstDiff x y)\n[PROOFSTEP]\napply mem_cylinder_firstDiff\n[GOAL]\ncase refine'_2.inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nsuffices firstDiff x y \u2264 firstDiff (f x) (f y) by\n  simpa [dist_eq_of_ne hxy, dist_eq_of_ne hfxfy]\n    -- case where `x \u2208 s`\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nthis : firstDiff x y \u2264 firstDiff (f x) (f y)\n\u22a2 dist (f x) (f y) \u2264 dist x y\n[PROOFSTEP]\nsimpa [dist_eq_of_ne hxy, dist_eq_of_ne hfxfy]\n  -- case where `x \u2208 s`\n[GOAL]\ncase refine'_2.inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nby_cases xs : x \u2208 s\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nrw [fs x xs] at hfxfy \u22a2\n  -- case where `y \u2208 s`, trivial\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff x (f y)\n[PROOFSTEP]\nby_cases ys : y \u2208 s\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : y \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff x (f y)\n[PROOFSTEP]\nrw [fs y ys]\n  -- case where `y \u2209 s`\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff x (f y)\n[PROOFSTEP]\nhave A : (s \u2229 cylinder y (longestPrefix y s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne y\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\n\u22a2 firstDiff x y \u2264 firstDiff x (f y)\n[PROOFSTEP]\nhave fy : f y = A.some := by simp_rw [if_neg ys]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\n\u22a2 f y = Set.Nonempty.some A\n[PROOFSTEP]\nsimp_rw [if_neg ys]\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some A\n\u22a2 firstDiff x y \u2264 firstDiff x (f y)\n[PROOFSTEP]\nhave I : cylinder A.some (firstDiff x y) = cylinder y (firstDiff x y) :=\n  by\n  rw [\u2190 mem_cylinder_iff_eq, firstDiff_comm]\n  apply cylinder_anti y _ A.some_mem.2\n  exact firstDiff_le_longestPrefix hs ys xs\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some A\n\u22a2 cylinder (Set.Nonempty.some A) (firstDiff x y) = cylinder y (firstDiff x y)\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_eq, firstDiff_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some A\n\u22a2 Set.Nonempty.some A \u2208 cylinder y (firstDiff y x)\n[PROOFSTEP]\napply cylinder_anti y _ A.some_mem.2\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some A\n\u22a2 firstDiff y x \u2264 longestPrefix y s\n[PROOFSTEP]\nexact firstDiff_le_longestPrefix hs ys xs\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : x \u2208 s\nys : \u00acy \u2208 s\nA : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some A\nI : cylinder (Set.Nonempty.some A) (firstDiff x y) = cylinder y (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff x (f y)\n[PROOFSTEP]\nrwa [\u2190 fy, \u2190 I2, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff_le_firstDiff hfxfy.symm, firstDiff_comm _ x] at I \n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nby_cases ys : y \u2208 s\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave A : (s \u2229 cylinder x (longestPrefix x s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne x\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave fx : f x = A.some := by simp_rw [if_neg xs]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n\u22a2 f x = Set.Nonempty.some A\n[PROOFSTEP]\nsimp_rw [if_neg xs]\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some A\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave I : cylinder A.some (firstDiff x y) = cylinder x (firstDiff x y) :=\n  by\n  rw [\u2190 mem_cylinder_iff_eq]\n  apply cylinder_anti x _ A.some_mem.2\n  apply firstDiff_le_longestPrefix hs xs ys\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some A\n\u22a2 cylinder (Set.Nonempty.some A) (firstDiff x y) = cylinder x (firstDiff x y)\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_eq]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some A\n\u22a2 Set.Nonempty.some A \u2208 cylinder x (firstDiff x y)\n[PROOFSTEP]\napply cylinder_anti x _ A.some_mem.2\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some A\n\u22a2 firstDiff x y \u2264 longestPrefix x s\n[PROOFSTEP]\napply firstDiff_le_longestPrefix hs xs ys\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some A\nI : cylinder (Set.Nonempty.some A) (firstDiff x y) = cylinder x (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nrw [fs y ys] at hfxfy \u22a2\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : y \u2208 s\nA : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some A\nI : cylinder (Set.Nonempty.some A) (firstDiff x y) = cylinder x (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) y\n[PROOFSTEP]\nrwa [\u2190 fx, I2, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff_le_firstDiff hfxfy] at I \n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave Ax : (s \u2229 cylinder x (longestPrefix x s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne x\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave fx : f x = Ax.some := by simp_rw [if_neg xs]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n\u22a2 f x = Set.Nonempty.some Ax\n[PROOFSTEP]\nsimp_rw [if_neg xs]\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave Ay : (s \u2229 cylinder y (longestPrefix y s)).Nonempty := inter_cylinder_longestPrefix_nonempty hs hne y\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave fy : f y = Ay.some := by\n  simp_rw [if_neg ys]\n    -- case where the common prefix to `x` and `s`, or `y` and `s`, is shorter than the\n            -- common part to `x` and `y` -- then `f x = f y`.\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\n\u22a2 f y = Set.Nonempty.some Ay\n[PROOFSTEP]\nsimp_rw [if_neg ys]\n  -- case where the common prefix to `x` and `s`, or `y` and `s`, is shorter than the\n          -- common part to `x` and `y` -- then `f x = f y`.\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nby_cases H : longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave : cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s) :=\n  by\n  cases' H with H H\n  \u00b7 exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H xs ys\n  \u00b7 symm\n    rw [firstDiff_comm] at H \n    exact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y\n\u22a2 cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n[PROOFSTEP]\ncases' H with H H\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix x s < firstDiff x y\n\u22a2 cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n[PROOFSTEP]\nexact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H xs ys\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix y s < firstDiff x y\n\u22a2 cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix y s < firstDiff x y\n\u22a2 cylinder y (longestPrefix y s) = cylinder x (longestPrefix x s)\n[PROOFSTEP]\nrw [firstDiff_comm] at H \n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix y s < firstDiff y x\n\u22a2 cylinder y (longestPrefix y s) = cylinder x (longestPrefix x s)\n[PROOFSTEP]\nexact cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff hs hne H ys xs\n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y\nthis : cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nrw [fx, fy] at hfxfy \n[GOAL]\ncase pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nhfxfy : Set.Nonempty.some Ax \u2260 Set.Nonempty.some Ay\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y\nthis : cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\napply (hfxfy _).elim\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nhfxfy : Set.Nonempty.some Ax \u2260 Set.Nonempty.some Ay\nfy : f y = Set.Nonempty.some Ay\nH : longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y\nthis : cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)\n\u22a2 Set.Nonempty.some Ax = Set.Nonempty.some Ay\n[PROOFSTEP]\ncongr\n  -- case where the common prefix to `x` and `s` is long, as well as the common prefix to\n          -- `y` and `s`. Then all points remain in the same cylinders.\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : \u00ac(longestPrefix x s < firstDiff x y \u2228 longestPrefix y s < firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\npush_neg at H \n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave I1 : cylinder Ax.some (firstDiff x y) = cylinder x (firstDiff x y) :=\n  by\n  rw [\u2190 mem_cylinder_iff_eq]\n  exact cylinder_anti x H.1 Ax.some_mem.2\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\n\u22a2 cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\n[PROOFSTEP]\nrw [\u2190 mem_cylinder_iff_eq]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\n\u22a2 Set.Nonempty.some Ax \u2208 cylinder x (firstDiff x y)\n[PROOFSTEP]\nexact cylinder_anti x H.1 Ax.some_mem.2\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave I3 : cylinder y (firstDiff x y) = cylinder Ay.some (firstDiff x y) :=\n  by\n  rw [eq_comm, \u2190 mem_cylinder_iff_eq]\n  exact cylinder_anti y H.2 Ay.some_mem.2\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\n\u22a2 cylinder y (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\n[PROOFSTEP]\nrw [eq_comm, \u2190 mem_cylinder_iff_eq]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\n\u22a2 Set.Nonempty.some Ay \u2208 cylinder y (firstDiff x y)\n[PROOFSTEP]\nexact cylinder_anti y H.2 Ay.some_mem.2\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\nI3 : cylinder y (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nhave : cylinder Ax.some (firstDiff x y) = cylinder Ay.some (firstDiff x y) := by rw [I1, I2, I3]\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\nI3 : cylinder y (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\n\u22a2 cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\n[PROOFSTEP]\nrw [I1, I2, I3]\n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\nI3 : cylinder y (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\nthis : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nrw [\u2190 fx, \u2190 fy, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff_le_firstDiff hfxfy] at this \n[GOAL]\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n :=\n  fun x => if x \u2208 s then x else Set.Nonempty.some (_ : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s)))\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nx y : (n : \u2115) \u2192 E n\nhxy : x \u2260 y\nhfxfy : f x \u2260 f y\nI2 : cylinder x (firstDiff x y) = cylinder y (firstDiff x y)\nxs : \u00acx \u2208 s\nys : \u00acy \u2208 s\nAx : Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\nfx : f x = Set.Nonempty.some Ax\nAy : Set.Nonempty (s \u2229 cylinder y (longestPrefix y s))\nfy : f y = Set.Nonempty.some Ay\nH : firstDiff x y \u2264 longestPrefix x s \u2227 firstDiff x y \u2264 longestPrefix y s\nI1 : cylinder (Set.Nonempty.some Ax) (firstDiff x y) = cylinder x (firstDiff x y)\nI3 : cylinder y (firstDiff x y) = cylinder (Set.Nonempty.some Ay) (firstDiff x y)\nthis : firstDiff x y \u2264 firstDiff (f x) (f y)\n\u22a2 firstDiff x y \u2264 firstDiff (f x) (f y)\n[PROOFSTEP]\nexact this\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\n\u22a2 \u2203 f, (\u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x) \u2227 range f = s \u2227 Continuous f\n[PROOFSTEP]\nrcases exists_lipschitz_retraction_of_isClosed hs hne with \u27e8f, fs, frange, hf\u27e9\n[GOAL]\ncase intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x\nfrange : range f = s\nhf : LipschitzWith 1 f\n\u22a2 \u2203 f, (\u2200 (x : (n : \u2115) \u2192 E n), x \u2208 s \u2192 f x = x) \u2227 range f = s \u2227 Continuous f\n[PROOFSTEP]\nexact \u27e8f, fs, frange, hf.continuous\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\n\u22a2 \u2203 f, (\u2200 (x : \u2191s), f \u2191x = x) \u2227 Surjective f \u2227 Continuous f\n[PROOFSTEP]\nobtain \u27e8f, fs, rfl, f_cont\u27e9 : \u2203 f : (\u2200 n, E n) \u2192 \u2200 n, E n, (\u2200 x \u2208 s, f x = x) \u2227 range f = s \u2227 Continuous f :=\n  exists_retraction_of_isClosed hs hne\n[GOAL]\ncase intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n\nf_cont : Continuous f\nhs : IsClosed (range f)\nhne : Set.Nonempty (range f)\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 range f \u2192 f x = x\n\u22a2 \u2203 f_1, (\u2200 (x : \u2191(range f)), f_1 \u2191x = x) \u2227 Surjective f_1 \u2227 Continuous f_1\n[PROOFSTEP]\nhave A : \u2200 x : range f, rangeFactorization f x = x := fun x \u21a6 Subtype.eq <| fs x x.2\n[GOAL]\ncase intro.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nf : ((n : \u2115) \u2192 E n) \u2192 (n : \u2115) \u2192 E n\nf_cont : Continuous f\nhs : IsClosed (range f)\nhne : Set.Nonempty (range f)\nfs : \u2200 (x : (n : \u2115) \u2192 E n), x \u2208 range f \u2192 f x = x\nA : \u2200 (x : \u2191(range f)), rangeFactorization f \u2191x = x\n\u22a2 \u2203 f_1, (\u2200 (x : \u2191(range f)), f_1 \u2191x = x) \u2227 Surjective f_1 \u2227 Continuous f_1\n[PROOFSTEP]\nexact \u27e8rangeFactorization f, A, fun x => \u27e8x, A x\u27e9, f_cont.subtype_mk _\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nletI : MetricSpace (\u2115 \u2192 \u2115) := PiNat.metricSpaceNatNat\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nhave I0 : (0 : \u211d) < 1 / 2 := by norm_num\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nhave I1 : (1 / 2 : \u211d) < 1 := by norm_num\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nrcases exists_dense_seq \u03b1 with \u27e8u, hu\u27e9\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nlet s : Set (\u2115 \u2192 \u2115) := {x | (\u22c2 n : \u2115, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty}\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nlet g : s \u2192 \u03b1 := fun x => x.2.some\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nhave A : \u2200 (x : s) (n : \u2115), dist (g x) (u ((x : \u2115 \u2192 \u2115) n)) \u2264 (1 / 2) ^ n := fun x n => (mem_iInter.1 x.2.some_mem n : _)\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nhave g_cont : Continuous g := by\n  refine continuous_iff_continuousAt.2 fun y => ?_\n  refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_\n  rcases eq_or_ne x y with (rfl | hne)\n  \u00b7 simp\n  have hne' : x.1 \u2260 y.1 := Subtype.coe_injective.ne hne\n  have dist' : dist x y = dist x.1 y.1 := rfl\n  let n := firstDiff x.1 y.1 - 1\n  have diff_pos : 0 < firstDiff x.1 y.1 := by\n    by_contra' h\n    apply apply_firstDiff_ne hne'\n    rw [le_zero_iff.1 h]\n    apply apply_eq_of_dist_lt _ le_rfl\n    rw [pow_zero]\n    exact hxy\n  have hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm\n  rw [dist', dist_eq_of_ne hne', hn]\n  have B : x.1 n = y.1 n := mem_cylinder_firstDiff x.1 y.1 n (Nat.pred_lt diff_pos.ne')\n  calc\n    dist (g x) (g y) \u2264 dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) := dist_triangle_right _ _ _\n    _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [\u2190 B]\n    _ \u2264 (1 / 2) ^ n + (1 / 2) ^ n := (add_le_add (A x n) (A y n))\n    _ = 4 * (1 / 2) ^ (n + 1) := by ring\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\n\u22a2 Continuous g\n[PROOFSTEP]\nrefine continuous_iff_continuousAt.2 fun y => ?_\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny : \u2191s\n\u22a2 ContinuousAt g y\n[PROOFSTEP]\nrefine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nrcases eq_or_ne x y with (rfl | hne)\n[GOAL]\ncase inl\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\nx : \u2191s\nhxy : dist x x < 1\n\u22a2 dist (g x) (g x) \u2264 4 * dist x x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nhave hne' : x.1 \u2260 y.1 := Subtype.coe_injective.ne hne\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nhave dist' : dist x y = dist x.1 y.1 := rfl\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nlet n := firstDiff x.1 y.1 - 1\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nhave diff_pos : 0 < firstDiff x.1 y.1 := by\n  by_contra' h\n  apply apply_firstDiff_ne hne'\n  rw [le_zero_iff.1 h]\n  apply apply_eq_of_dist_lt _ le_rfl\n  rw [pow_zero]\n  exact hxy\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\n\u22a2 0 < firstDiff \u2191x \u2191y\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\nh : firstDiff \u2191x \u2191y \u2264 0\n\u22a2 False\n[PROOFSTEP]\napply apply_firstDiff_ne hne'\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\nh : firstDiff \u2191x \u2191y \u2264 0\n\u22a2 \u2191x (firstDiff \u2191x \u2191y) = \u2191y (firstDiff \u2191x \u2191y)\n[PROOFSTEP]\nrw [le_zero_iff.1 h]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\nh : firstDiff \u2191x \u2191y \u2264 0\n\u22a2 \u2191x 0 = \u2191y 0\n[PROOFSTEP]\napply apply_eq_of_dist_lt _ le_rfl\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\nh : firstDiff \u2191x \u2191y \u2264 0\n\u22a2 dist \u2191x \u2191y < (1 / 2) ^ 0\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\nh : firstDiff \u2191x \u2191y \u2264 0\n\u22a2 dist \u2191x \u2191y < 1\n[PROOFSTEP]\nexact hxy\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\ndiff_pos : 0 < firstDiff \u2191x \u2191y\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nhave hn : firstDiff x.1 y.1 = n + 1 := (Nat.succ_pred_eq_of_pos diff_pos).symm\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\ndiff_pos : 0 < firstDiff \u2191x \u2191y\nhn : firstDiff \u2191x \u2191y = n + 1\n\u22a2 dist (g x) (g y) \u2264 4 * dist x y\n[PROOFSTEP]\nrw [dist', dist_eq_of_ne hne', hn]\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\ndiff_pos : 0 < firstDiff \u2191x \u2191y\nhn : firstDiff \u2191x \u2191y = n + 1\n\u22a2 dist (g x) (g y) \u2264 4 * (1 / 2) ^ (n + 1)\n[PROOFSTEP]\nhave B : x.1 n = y.1 n := mem_cylinder_firstDiff x.1 y.1 n (Nat.pred_lt diff_pos.ne')\n[GOAL]\ncase inr\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\ndiff_pos : 0 < firstDiff \u2191x \u2191y\nhn : firstDiff \u2191x \u2191y = n + 1\nB : \u2191x n = \u2191y n\n\u22a2 dist (g x) (g y) \u2264 4 * (1 / 2) ^ (n + 1)\n[PROOFSTEP]\ncalc\n  dist (g x) (g y) \u2264 dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) := dist_triangle_right _ _ _\n  _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [\u2190 B]\n  _ \u2264 (1 / 2) ^ n + (1 / 2) ^ n := (add_le_add (A x n) (A y n))\n  _ = 4 * (1 / 2) ^ (n + 1) := by ring\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\ndiff_pos : 0 < firstDiff \u2191x \u2191y\nhn : firstDiff \u2191x \u2191y = n + 1\nB : \u2191x n = \u2191y n\n\u22a2 dist (g x) (u (\u2191x n)) + dist (g y) (u (\u2191x n)) = dist (g x) (u (\u2191x n)) + dist (g y) (u (\u2191y n))\n[PROOFSTEP]\nrw [\u2190 B]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ny x : \u2191s\nhxy : dist x y < 1\nhne : x \u2260 y\nhne' : \u2191x \u2260 \u2191y\ndist' : dist x y = dist \u2191x \u2191y\nn : \u2115 := firstDiff \u2191x \u2191y - 1\ndiff_pos : 0 < firstDiff \u2191x \u2191y\nhn : firstDiff \u2191x \u2191y = n + 1\nB : \u2191x n = \u2191y n\n\u22a2 (1 / 2) ^ n + (1 / 2) ^ n = 4 * (1 / 2) ^ (n + 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nhave g_surj : Surjective g := fun y \u21a6\n  by\n  have : \u2200 n : \u2115, \u2203 j, y \u2208 closedBall (u j) ((1 / 2) ^ n) := fun n \u21a6\n    by\n    rcases hu.exists_dist_lt y (by simp : (0 : \u211d) < (1 / 2) ^ n) with \u27e8j, hj\u27e9\n    exact \u27e8j, hj.le\u27e9\n  choose x hx using this\n  have I : (\u22c2 n : \u2115, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty := \u27e8y, mem_iInter.2 hx\u27e9\n  refine' \u27e8\u27e8x, I\u27e9, _\u27e9\n  refine' dist_le_zero.1 _\n  have J : \u2200 n : \u2115, dist (g \u27e8x, I\u27e9) y \u2264 (1 / 2) ^ n + (1 / 2) ^ n := fun n =>\n    calc\n      dist (g \u27e8x, I\u27e9) y \u2264 dist (g \u27e8x, I\u27e9) (u (x n)) + dist y (u (x n)) := dist_triangle_right _ _ _\n      _ \u2264 (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A \u27e8x, I\u27e9 n) (hx n)\n  have L : Tendsto (fun n : \u2115 => (1 / 2 : \u211d) ^ n + (1 / 2) ^ n) atTop (\ud835\udcdd (0 + 0)) :=\n    (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).add (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1)\n  rw [add_zero] at L \n  exact ge_of_tendsto' L J\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\n\u22a2 \u2203 a, g a = y\n[PROOFSTEP]\nhave : \u2200 n : \u2115, \u2203 j, y \u2208 closedBall (u j) ((1 / 2) ^ n) := fun n \u21a6\n  by\n  rcases hu.exists_dist_lt y (by simp : (0 : \u211d) < (1 / 2) ^ n) with \u27e8j, hj\u27e9\n  exact \u27e8j, hj.le\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nn : \u2115\n\u22a2 \u2203 j, y \u2208 closedBall (u j) ((1 / 2) ^ n)\n[PROOFSTEP]\nrcases hu.exists_dist_lt y (by simp : (0 : \u211d) < (1 / 2) ^ n) with \u27e8j, hj\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nn : \u2115\n\u22a2 0 < (1 / 2) ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nn j : \u2115\nhj : dist y (u j) < (1 / 2) ^ n\n\u22a2 \u2203 j, y \u2208 closedBall (u j) ((1 / 2) ^ n)\n[PROOFSTEP]\nexact \u27e8j, hj.le\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis\u271d : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nthis : \u2200 (n : \u2115), \u2203 j, y \u2208 closedBall (u j) ((1 / 2) ^ n)\n\u22a2 \u2203 a, g a = y\n[PROOFSTEP]\nchoose x hx using this\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\n\u22a2 \u2203 a, g a = y\n[PROOFSTEP]\nhave I : (\u22c2 n : \u2115, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty := \u27e8y, mem_iInter.2 hx\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\nI : Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))\n\u22a2 \u2203 a, g a = y\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, I\u27e9, _\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\nI : Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))\n\u22a2 g { val := x, property := I } = y\n[PROOFSTEP]\nrefine' dist_le_zero.1 _\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\nI : Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))\n\u22a2 dist (g { val := x, property := I }) y \u2264 0\n[PROOFSTEP]\nhave J : \u2200 n : \u2115, dist (g \u27e8x, I\u27e9) y \u2264 (1 / 2) ^ n + (1 / 2) ^ n := fun n =>\n  calc\n    dist (g \u27e8x, I\u27e9) y \u2264 dist (g \u27e8x, I\u27e9) (u (x n)) + dist y (u (x n)) := dist_triangle_right _ _ _\n    _ \u2264 (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A \u27e8x, I\u27e9 n) (hx n)\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\nI : Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))\nJ : \u2200 (n : \u2115), dist (g { val := x, property := I }) y \u2264 (1 / 2) ^ n + (1 / 2) ^ n\n\u22a2 dist (g { val := x, property := I }) y \u2264 0\n[PROOFSTEP]\nhave L : Tendsto (fun n : \u2115 => (1 / 2 : \u211d) ^ n + (1 / 2) ^ n) atTop (\ud835\udcdd (0 + 0)) :=\n  (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).add (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1)\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\nI : Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))\nJ : \u2200 (n : \u2115), dist (g { val := x, property := I }) y \u2264 (1 / 2) ^ n + (1 / 2) ^ n\nL : Tendsto (fun n => (1 / 2) ^ n + (1 / 2) ^ n) atTop (\ud835\udcdd (0 + 0))\n\u22a2 dist (g { val := x, property := I }) y \u2264 0\n[PROOFSTEP]\nrw [add_zero] at L \n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ny : \u03b1\nx : \u2115 \u2192 \u2115\nhx : \u2200 (n : \u2115), y \u2208 closedBall (u (x n)) ((1 / 2) ^ n)\nI : Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))\nJ : \u2200 (n : \u2115), dist (g { val := x, property := I }) y \u2264 (1 / 2) ^ n + (1 / 2) ^ n\nL : Tendsto (fun n => (1 / 2) ^ n + (1 / 2) ^ n) atTop (\ud835\udcdd 0)\n\u22a2 dist (g { val := x, property := I }) y \u2264 0\n[PROOFSTEP]\nexact ge_of_tendsto' L J\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nhave s_closed : IsClosed s := by\n  refine isClosed_iff_clusterPt.mpr fun x hx \u21a6 ?_\n  have L : Tendsto (fun n : \u2115 => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0) :=\n    by\n    have : Tendsto (fun n : \u2115 => (2 : \u211d) * (1 / 2) ^ n) atTop (\ud835\udcdd (2 * 0)) :=\n      (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _\n    rw [mul_zero] at this \n    exact squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this\n  refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_ball) (fun n => bounded_closedBall) (fun N \u21a6 ?_) L\n  obtain \u27e8y, hxy, ys\u27e9 : \u2203 y, y \u2208 ball x ((1 / 2) ^ N) \u2229 s :=\n    clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))\n  have E :\n    \u22c2 (n : \u2115) (H : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n) =\n      \u22c2 (n : \u2115) (H : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n) :=\n    by\n    refine iInter_congr fun n \u21a6 iInter_congr fun hn \u21a6 ?_\n    have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn\n    rw [this]\n  rw [E]\n  apply Nonempty.mono _ ys\n  apply iInter_subset_iInter\u2082\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\n\u22a2 IsClosed s\n[PROOFSTEP]\nrefine isClosed_iff_clusterPt.mpr fun x hx \u21a6 ?_\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\n\u22a2 x \u2208 s\n[PROOFSTEP]\nhave L : Tendsto (fun n : \u2115 => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0) :=\n  by\n  have : Tendsto (fun n : \u2115 => (2 : \u211d) * (1 / 2) ^ n) atTop (\ud835\udcdd (2 * 0)) :=\n    (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _\n  rw [mul_zero] at this \n  exact squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\n\u22a2 Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun n : \u2115 => (2 : \u211d) * (1 / 2) ^ n) atTop (\ud835\udcdd (2 * 0)) :=\n  (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis\u271d : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nthis : Tendsto (fun n => 2 * (1 / 2) ^ n) atTop (\ud835\udcdd (2 * 0))\n\u22a2 Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [mul_zero] at this \n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis\u271d : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nthis : Tendsto (fun n => 2 * (1 / 2) ^ n) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrefine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_ball) (fun n => bounded_closedBall) (fun N \u21a6 ?_) L\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n))\n[PROOFSTEP]\nobtain \u27e8y, hxy, ys\u27e9 : \u2203 y, y \u2208 ball x ((1 / 2) ^ N) \u2229 s :=\n  clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))\n[GOAL]\ncase intro.intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n))\n[PROOFSTEP]\nhave E :\n  \u22c2 (n : \u2115) (H : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n) =\n    \u22c2 (n : \u2115) (H : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n) :=\n  by\n  refine iInter_congr fun n \u21a6 iInter_congr fun hn \u21a6 ?_\n  have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn\n  rw [this]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\n\u22a2 \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n) = \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n)\n[PROOFSTEP]\nrefine iInter_congr fun n \u21a6 iInter_congr fun hn \u21a6 ?_\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\nn : \u2115\nhn : n \u2264 N\n\u22a2 closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)\n[PROOFSTEP]\nhave : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis\u271d : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\nn : \u2115\nhn : n \u2264 N\nthis : x n = y n\n\u22a2 closedBall (u (x n)) ((1 / 2) ^ n) = closedBall (u (y n)) ((1 / 2) ^ n)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro\nE\u271d : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\nE :\n  \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n) = \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n)\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n))\n[PROOFSTEP]\nrw [E]\n[GOAL]\ncase intro.intro\nE\u271d : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\nE :\n  \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n) = \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n)\n\u22a2 Set.Nonempty (\u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n))\n[PROOFSTEP]\napply Nonempty.mono _ ys\n[GOAL]\nE\u271d : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\nx : \u2115 \u2192 \u2115\nhx : ClusterPt x (\ud835\udcdf s)\nL : Tendsto (fun n => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (\ud835\udcdd 0)\nN : \u2115\ny : \u2115 \u2192 \u2115\nhxy : y \u2208 ball x ((1 / 2) ^ N)\nys : y \u2208 s\nE :\n  \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (x n)) ((1 / 2) ^ n) = \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n)\n\u22a2 \u22c2 (n : \u2115), closedBall (u (y n)) ((1 / 2) ^ n) \u2286 \u22c2 (n : \u2115) (_ : n \u2264 N), closedBall (u (y n)) ((1 / 2) ^ n)\n[PROOFSTEP]\napply iInter_subset_iInter\u2082\n[GOAL]\ncase intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\ns_closed : IsClosed s\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nobtain \u27e8f, -, f_surj, f_cont\u27e9 : \u2203 f : (\u2115 \u2192 \u2115) \u2192 s, (\u2200 x : s, f x = x) \u2227 Surjective f \u2227 Continuous f :=\n  by\n  apply exists_retraction_subtype_of_isClosed s_closed\n  simpa only [nonempty_coe_sort] using g_surj.nonempty\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\ns_closed : IsClosed s\n\u22a2 \u2203 f, (\u2200 (x : \u2191s), f \u2191x = x) \u2227 Surjective f \u2227 Continuous f\n[PROOFSTEP]\napply exists_retraction_subtype_of_isClosed s_closed\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\ns_closed : IsClosed s\n\u22a2 Set.Nonempty s\n[PROOFSTEP]\nsimpa only [nonempty_coe_sort] using g_surj.nonempty\n[GOAL]\ncase intro.intro.intro.intro\nE : \u2115 \u2192 Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : MetricSpace \u03b1\ninst\u271d\u00b2 : CompleteSpace \u03b1\ninst\u271d\u00b9 : SecondCountableTopology \u03b1\ninst\u271d : Nonempty \u03b1\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : \u2115 \u2192 \u03b1\nhu : DenseRange u\ns : Set (\u2115 \u2192 \u2115) := {x | Set.Nonempty (\u22c2 (n : \u2115), closedBall (u (x n)) ((1 / 2) ^ n))}\ng : \u2191s \u2192 \u03b1 := fun x => Set.Nonempty.some (_ : \u2191x \u2208 s)\nA : \u2200 (x : \u2191s) (n : \u2115), dist (g x) (u (\u2191x n)) \u2264 (1 / 2) ^ n\ng_cont : Continuous g\ng_surj : Surjective g\ns_closed : IsClosed s\nf : (\u2115 \u2192 \u2115) \u2192 \u2191s\nf_surj : Surjective f\nf_cont : Continuous f\n\u22a2 \u2203 f, Continuous f \u2227 Surjective f\n[PROOFSTEP]\nexact \u27e8g \u2218 f, g_cont.comp f_cont, g_surj.comp f_surj\u27e9\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\n\u22a2 Summable fun i => min ((1 / 2) ^ encode i) (dist (x i) (y i))\n[PROOFSTEP]\nrefine summable_of_nonneg_of_le (fun i => ?_) (fun i => min_le_left _ _) summable_geometric_two_encode\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni : \u03b9\n\u22a2 0 \u2264 min ((1 / 2) ^ encode i) (dist (x i) (y i))\n[PROOFSTEP]\nexact le_min (pow_nonneg (by norm_num) _) dist_nonneg\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni : \u03b9\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni j : \u03b9\nx\u271d : j \u2260 i\n\u22a2 0 \u2264 (1 / 2) ^ encode j\n[PROOFSTEP]\nsimp\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni : \u03b9\nh : dist x y < (1 / 2) ^ encode i\n\u22a2 dist (x i) (y i) \u2264 dist x y\n[PROOFSTEP]\nsimpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx : (i : \u03b9) \u2192 F i\n\u22a2 dist x x = 0\n[PROOFSTEP]\nsimp [dist_eq_tsum]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\n\u22a2 dist x y = dist y x\n[PROOFSTEP]\nsimp [dist_eq_tsum, dist_comm]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y z : (i : \u03b9) \u2192 F i\ni : \u03b9\n\u22a2 min ((1 / 2) ^ encode i) (dist (x i) (y i) + dist (y i) (z i)) =\n    min ((1 / 2) ^ encode i) (min ((1 / 2) ^ encode i) (dist (x i) (y i)) + min ((1 / 2) ^ encode i) (dist (y i) (z i)))\n[PROOFSTEP]\nconvert congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d) (min_add_distrib ((1 / 2 : \u211d\u22650) ^ encode i) (nndist (x i) (y i)) (nndist (y i) (z i)))\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx\u271d\u00b9 x\u271d : (i : \u03b9) \u2192 F i\n\u22a2 (fun x y => \u2191{ val := dist x y, property := (_ : 0 \u2264 dist x y) }) x\u271d\u00b9 x\u271d = ENNReal.ofReal (dist x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nexact ENNReal.coe_nnreal_eq _\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u22a2 uniformity ((i : \u03b9) \u2192 F i) = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [Pi.uniformity, comap_iInf, gt_iff_lt, preimage_setOf_eq, comap_principal, PseudoMetricSpace.uniformity_dist]\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u22a2 \u2a05 (i : \u03b9) (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_1} =\n    \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u22a2 \u2a05 (i : \u03b9) (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_1} \u2264\n    \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [le_iInf_iff, le_principal_iff]\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u22a2 \u2200 (i : \u211d),\n    0 < i \u2192\n      {p | dist p.fst p.snd < i} \u2208 \u2a05 (i : \u03b9) (i_2 : \u211d) (_ : 0 < i_2), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_2}\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 {p | dist p.fst p.snd < \u03b5} \u2208 \u2a05 (i : \u03b9) (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_1}\n[PROOFSTEP]\nobtain \u27e8K, hK\u27e9 : \u2203 K : Finset \u03b9, (\u2211' i : { j // j \u2209 K }, (1 / 2 : \u211d) ^ encode (i : \u03b9)) < \u03b5 / 2 :=\n  ((tendsto_order.1 (tendsto_tsum_compl_atTop_zero fun i : \u03b9 => (1 / 2 : \u211d) ^ encode i)).2 _ (half_pos \u03b5pos)).exists\n[GOAL]\ncase a.intro\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u22a2 {p | dist p.fst p.snd < \u03b5} \u2208 \u2a05 (i : \u03b9) (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_1}\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 : \u211d, 0 < \u03b4 \u2227 (K.card : \u211d) * \u03b4 < \u03b5 / 2 := exists_pos_mul_lt (half_pos \u03b5pos) _\n[GOAL]\ncase a.intro.intro.intro\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\n\u22a2 {p | dist p.fst p.snd < \u03b5} \u2208 \u2a05 (i : \u03b9) (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_1}\n[PROOFSTEP]\napply\n  @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>\n    {p : (\u2200 i : \u03b9, F i) \u00d7 \u2200 i : \u03b9, F i | dist (p.fst i) (p.snd i) < \u03b4}\n[GOAL]\ncase a.intro.intro.intro.hV\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\n\u22a2 \u2200 (i : \u2191\u2191K),\n    {p | dist (Prod.fst p \u2191i) (Prod.snd p \u2191i) < \u03b4} \u2208\n      \u2a05 (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a \u2191i) (Prod.snd a \u2191i) < i_1}\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9\n[GOAL]\ncase a.intro.intro.intro.hV.mk\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\ni : \u03b9\nhi : i \u2208 \u2191K\n\u22a2 {p | dist (Prod.fst p \u2191{ val := i, property := hi }) (Prod.snd p \u2191{ val := i, property := hi }) < \u03b4} \u2208\n    \u2a05 (i_1 : \u211d) (_ : 0 < i_1),\n      \ud835\udcdf {a | dist (Prod.fst a \u2191{ val := i, property := hi }) (Prod.snd a \u2191{ val := i, property := hi }) < i_1}\n[PROOFSTEP]\nrefine' mem_iInf_of_mem \u03b4 (mem_iInf_of_mem \u03b4pos _)\n[GOAL]\ncase a.intro.intro.intro.hV.mk\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\ni : \u03b9\nhi : i \u2208 \u2191K\n\u22a2 {p | dist (Prod.fst p \u2191{ val := i, property := hi }) (Prod.snd p \u2191{ val := i, property := hi }) < \u03b4} \u2208\n    \ud835\udcdf {a | dist (Prod.fst a \u2191{ val := i, property := hi }) (Prod.snd a \u2191{ val := i, property := hi }) < \u03b4}\n[PROOFSTEP]\nsimp only [Prod.forall, imp_self, mem_principal, Subset.rfl]\n[GOAL]\ncase a.intro.intro.intro.hU\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\n\u22a2 \u22c2 (i : \u2191\u2191K), {p | dist (Prod.fst p \u2191i) (Prod.snd p \u2191i) < \u03b4} \u2286 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 hxy\n[GOAL]\ncase a.intro.intro.intro.hU.mk\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : (x, y) \u2208 \u22c2 (i : \u2191\u2191K), {p | dist (Prod.fst p \u2191i) (Prod.snd p \u2191i) < \u03b4}\n\u22a2 (x, y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy \n[GOAL]\ncase a.intro.intro.intro.hU.mk\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 (x, y) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\ncalc\n  dist x y = \u2211' i : \u03b9, min ((1 / 2) ^ encode i : \u211d) (dist (x i) (y i)) := rfl\n  _ =\n      (\u2211 i in K, min ((1 / 2) ^ encode i : \u211d) (dist (x i) (y i))) +\n        \u2211' i : \u2191(K : Set \u03b9)\u1d9c, min ((1 / 2) ^ encode (i : \u03b9) : \u211d) (dist (x i) (y i)) :=\n    (sum_add_tsum_compl (dist_summable _ _)).symm\n  _ \u2264 (\u2211 i in K, dist (x i) (y i)) + \u2211' i : \u2191(K : Set \u03b9)\u1d9c, ((1 / 2) ^ encode (i : \u03b9) : \u211d) :=\n    by\n    refine' add_le_add (Finset.sum_le_sum fun i _ => min_le_right _ _) _\n    refine' tsum_le_tsum (fun i => min_le_left _ _) _ _\n    \u00b7 apply Summable.subtype (dist_summable x y) (\u2191K : Set \u03b9)\u1d9c\n    \u00b7 apply Summable.subtype summable_geometric_two_encode (\u2191K : Set \u03b9)\u1d9c\n  _ < (\u2211 _i in K, \u03b4) + \u03b5 / 2 := by\n    apply add_lt_add_of_le_of_lt _ hK\n    refine Finset.sum_le_sum fun i hi => (hxy i ?_).le\n    simpa using hi\n  _ \u2264 \u03b5 / 2 + \u03b5 / 2 := (add_le_add_right (by simpa only [Finset.sum_const, nsmul_eq_mul] using h\u03b4.le) _)\n  _ = \u03b5 := add_halves _\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 \u2211 i in K, min ((1 / 2) ^ encode i) (dist (x i) (y i)) +\n      \u2211' (i : \u2191(\u2191K)\u1d9c), min ((1 / 2) ^ encode \u2191i) (dist (x \u2191i) (y \u2191i)) \u2264\n    \u2211 i in K, dist (x i) (y i) + \u2211' (i : \u2191(\u2191K)\u1d9c), (1 / 2) ^ encode \u2191i\n[PROOFSTEP]\nrefine' add_le_add (Finset.sum_le_sum fun i _ => min_le_right _ _) _\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 \u2211' (i : \u2191(\u2191K)\u1d9c), min ((1 / 2) ^ encode \u2191i) (dist (x \u2191i) (y \u2191i)) \u2264 \u2211' (i : \u2191(\u2191K)\u1d9c), (1 / 2) ^ encode \u2191i\n[PROOFSTEP]\nrefine' tsum_le_tsum (fun i => min_le_left _ _) _ _\n[GOAL]\ncase refine'_1\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 Summable fun i => min ((1 / 2) ^ encode \u2191i) (dist (x \u2191i) (y \u2191i))\n[PROOFSTEP]\napply Summable.subtype (dist_summable x y) (\u2191K : Set \u03b9)\u1d9c\n[GOAL]\ncase refine'_2\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 Summable fun i => (1 / 2) ^ encode \u2191i\n[PROOFSTEP]\napply Summable.subtype summable_geometric_two_encode (\u2191K : Set \u03b9)\u1d9c\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 \u2211 i in K, dist (x i) (y i) + \u2211' (i : \u2191(\u2191K)\u1d9c), (1 / 2) ^ encode \u2191i < \u2211 _i in K, \u03b4 + \u03b5 / 2\n[PROOFSTEP]\napply add_lt_add_of_le_of_lt _ hK\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 \u2211 i in K, dist (x i) (y i) \u2264 \u2211 _i in K, \u03b4\n[PROOFSTEP]\nrefine Finset.sum_le_sum fun i hi => (hxy i ?_).le\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\ni : \u03b9\nhi : i \u2208 K\n\u22a2 i \u2208 K\n[PROOFSTEP]\nsimpa using hi\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nK : Finset \u03b9\nhK : \u2211' (i : { j // \u00acj \u2208 K }), (1 / 2) ^ encode \u2191i < \u03b5 / 2\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2191(Finset.card K) * \u03b4 < \u03b5 / 2\nx y : (i : \u03b9) \u2192 F i\nhxy : \u2200 (x_1 : \u03b9), x_1 \u2208 K \u2192 dist (x x_1) (y x_1) < \u03b4\n\u22a2 \u2211 _i in K, \u03b4 \u2264 \u03b5 / 2\n[PROOFSTEP]\nsimpa only [Finset.sum_const, nsmul_eq_mul] using h\u03b4.le\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u22a2 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5} \u2264\n    \u2a05 (i : \u03b9) (i_1 : \u211d) (_ : 0 < i_1), \ud835\udcdf {a | dist (Prod.fst a i) (Prod.snd a i) < i_1}\n[PROOFSTEP]\nsimp only [le_iInf_iff, le_principal_iff]\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\n\u22a2 \u2200 (i : \u03b9) (i_1 : \u211d),\n    0 < i_1 \u2192 {a | dist (Prod.fst a i) (Prod.snd a i) < i_1} \u2208 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nintro i \u03b5 \u03b5pos\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 {a | dist (Prod.fst a i) (Prod.snd a i) < \u03b5} \u2208 \u2a05 (\u03b5 : \u211d) (_ : 0 < \u03b5), \ud835\udcdf {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrefine' mem_iInf_of_mem (min ((1 / 2) ^ encode i : \u211d) \u03b5) _\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 {a | dist (Prod.fst a i) (Prod.snd a i) < \u03b5} \u2208\n    \u2a05 (_ : 0 < min ((1 / 2) ^ encode i) \u03b5), \ud835\udcdf {p | dist p.fst p.snd < min ((1 / 2) ^ encode i) \u03b5}\n[PROOFSTEP]\nhave : 0 < min ((1 / 2) ^ encode i : \u211d) \u03b5 := lt_min (by simp) \u03b5pos\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 0 < (1 / 2) ^ encode i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nthis : 0 < min ((1 / 2) ^ encode i) \u03b5\n\u22a2 {a | dist (Prod.fst a i) (Prod.snd a i) < \u03b5} \u2208\n    \u2a05 (_ : 0 < min ((1 / 2) ^ encode i) \u03b5), \ud835\udcdf {p | dist p.fst p.snd < min ((1 / 2) ^ encode i) \u03b5}\n[PROOFSTEP]\nrefine' mem_iInf_of_mem this _\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nthis : 0 < min ((1 / 2) ^ encode i) \u03b5\n\u22a2 {a | dist (Prod.fst a i) (Prod.snd a i) < \u03b5} \u2208 \ud835\udcdf {p | dist p.fst p.snd < min ((1 / 2) ^ encode i) \u03b5}\n[PROOFSTEP]\nsimp only [and_imp, Prod.forall, setOf_subset_setOf, lt_min_iff, mem_principal]\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nthis : 0 < min ((1 / 2) ^ encode i) \u03b5\n\u22a2 \u2200 (a b : (i : \u03b9) \u2192 F i), dist a b < (1 / 2) ^ encode i \u2192 dist a b < \u03b5 \u2192 dist (a i) (b i) < \u03b5\n[PROOFSTEP]\nintro x y hn h\u03b5\n[GOAL]\ncase a\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\ni : \u03b9\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nthis : 0 < min ((1 / 2) ^ encode i) \u03b5\nx y : (i : \u03b9) \u2192 F i\nhn : dist x y < (1 / 2) ^ encode i\nh\u03b5 : dist x y < \u03b5\n\u22a2 dist (x i) (y i) < \u03b5\n[PROOFSTEP]\ncalc\n  dist (x i) (y i) \u2264 dist x y := dist_le_dist_pi_of_dist_lt hn\n  _ < \u03b5 := h\u03b5\n[GOAL]\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx\u271d y\u271d : (i : \u03b9) \u2192 F i\nhxy : dist x\u271d y\u271d = 0\n\u22a2 x\u271d = y\u271d\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx\u271d y\u271d : (i : \u03b9) \u2192 F i\nhxy : dist x\u271d y\u271d = 0\nn : \u03b9\n\u22a2 x\u271d n = y\u271d n\n[PROOFSTEP]\nrw [\u2190 dist_le_zero, \u2190 hxy]\n[GOAL]\ncase h\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx\u271d y\u271d : (i : \u03b9) \u2192 F i\nhxy : dist x\u271d y\u271d = 0\nn : \u03b9\n\u22a2 dist (x\u271d n) (y\u271d n) \u2264 dist x\u271d y\u271d\n[PROOFSTEP]\napply dist_le_dist_pi_of_dist_lt\n[GOAL]\ncase h.h\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx\u271d y\u271d : (i : \u03b9) \u2192 F i\nhxy : dist x\u271d y\u271d = 0\nn : \u03b9\n\u22a2 (dist x\u271d fun n => y\u271d n) < (1 / 2) ^ encode n\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\ncase h.h\nE : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx\u271d y\u271d : (i : \u03b9) \u2192 F i\nhxy : dist x\u271d y\u271d = 0\nn : \u03b9\n\u22a2 0 < (1 / 2) ^ encode n\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.PiNat", "llama_tokens": 92683, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.5612047158316272}}
{"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\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : OrderAddMonoidHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\nha : 0 \u2264 a\n\u22a2 0 \u2264 \u2191f a\n[PROOFSTEP]\nrw [\u2190 map_zero f]\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\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : OrderAddMonoidHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u2191f 0 \u2264 \u2191f a\n[PROOFSTEP]\nexact OrderHomClass.mono _ ha\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\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : OrderAddMonoidHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\nha : a \u2264 0\n\u22a2 \u2191f a \u2264 0\n[PROOFSTEP]\nrw [\u2190 map_zero f]\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\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : OrderAddMonoidHomClass F \u03b1 \u03b2\nf : F\na : \u03b1\nha : a \u2264 0\n\u22a2 \u2191f a \u2264 \u2191f 0\n[PROOFSTEP]\nexact OrderHomClass.mono _ ha\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\u00b2 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : AddMonoidHomClass F \u03b1 \u03b2\nf : F\nh : Monotone \u2191f\na : \u03b1\n\u22a2 0 \u2264 a \u2192 0 \u2264 \u2191f a\n[PROOFSTEP]\nrw [\u2190 map_zero f]\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\u00b2 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : AddMonoidHomClass F \u03b1 \u03b2\nf : F\nh : Monotone \u2191f\na : \u03b1\n\u22a2 0 \u2264 a \u2192 \u2191f 0 \u2264 \u2191f a\n[PROOFSTEP]\napply h\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\u00b2 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : AddMonoidHomClass F \u03b1 \u03b2\nf : F\nh : \u2200 (a : \u03b1), 0 \u2264 a \u2192 0 \u2264 \u2191f a\na b : \u03b1\nhl : a \u2264 b\n\u22a2 \u2191f a \u2264 \u2191f b\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel b a, map_add f]\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\u00b2 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b9 : OrderedAddCommMonoid \u03b2\ninst\u271d : AddMonoidHomClass F \u03b1 \u03b2\nf : F\nh : \u2200 (a : \u03b1), 0 \u2264 a \u2192 0 \u2264 \u2191f a\na b : \u03b1\nhl : a \u2264 b\n\u22a2 \u2191f a \u2264 \u2191f (b - a) + \u2191f a\n[PROOFSTEP]\nexact le_add_of_nonneg_left (h _ <| sub_nonneg.2 hl)\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\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n\u22a2 StrictMono \u2191f \u2194 \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\n[PROOFSTEP]\nrefine \u27e8fun h a => ?_, fun h a b hl => ?_\u27e9\n[GOAL]\ncase refine_1\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\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : StrictMono \u2191f\na : \u03b1\n\u22a2 0 < a \u2192 0 < \u2191f a\n[PROOFSTEP]\nrw [\u2190 map_zero f]\n[GOAL]\ncase refine_1\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\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : StrictMono \u2191f\na : \u03b1\n\u22a2 0 < a \u2192 \u2191f 0 < \u2191f a\n[PROOFSTEP]\napply h\n[GOAL]\ncase refine_2\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\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\na b : \u03b1\nhl : a < b\n\u22a2 \u2191f a < \u2191f b\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel b a, map_add f]\n[GOAL]\ncase refine_2\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\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\na b : \u03b1\nhl : a < b\n\u22a2 \u2191f a < \u2191f (b - a) + \u2191f a\n[PROOFSTEP]\nexact lt_add_of_pos_left _ (h _ <| sub_pos.2 hl)\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf\u271d g\u271d f g : \u03b1 \u2192*o \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := f\n[GOAL]\ncase 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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf g\u271d g : \u03b1 \u2192*o \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_one'\u271d : toFun\u271d 1 = 1\nmap_mul'\u271d :\n  \u2200 (x y : \u03b1),\n    OneHom.toFun { toFun := toFun\u271d, map_one' := map_one'\u271d } (x * y) =\n      OneHom.toFun { toFun := toFun\u271d, map_one' := map_one'\u271d } x *\n        OneHom.toFun { toFun := toFun\u271d, map_one' := map_one'\u271d } y\nmonotone'\u271d : Monotone (\u2191{ toOneHom := { toFun := toFun\u271d, map_one' := map_one'\u271d }, map_mul' := map_mul'\u271d }).toFun\nh :\n  (fun f => f.toFun)\n      { toMonoidHom := { toOneHom := { toFun := toFun\u271d, map_one' := map_one'\u271d }, map_mul' := map_mul'\u271d },\n        monotone' := monotone'\u271d } =\n    (fun f => f.toFun) g\n\u22a2 { toMonoidHom := { toOneHom := { toFun := toFun\u271d, map_one' := map_one'\u271d }, map_mul' := map_mul'\u271d },\n      monotone' := monotone'\u271d } =\n    g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf g : \u03b1 \u2192*o \u03b2\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_one'\u271d\u00b9 : toFun\u271d\u00b9 1 = 1\nmap_mul'\u271d\u00b9 :\n  \u2200 (x y : \u03b1),\n    OneHom.toFun { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 } (x * y) =\n      OneHom.toFun { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 } x *\n        OneHom.toFun { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 } y\nmonotone'\u271d\u00b9 : Monotone (\u2191{ toOneHom := { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 }, map_mul' := map_mul'\u271d\u00b9 }).toFun\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_one'\u271d : toFun\u271d 1 = 1\nmap_mul'\u271d :\n  \u2200 (x y : \u03b1),\n    OneHom.toFun { toFun := toFun\u271d, map_one' := map_one'\u271d } (x * y) =\n      OneHom.toFun { toFun := toFun\u271d, map_one' := map_one'\u271d } x *\n        OneHom.toFun { toFun := toFun\u271d, map_one' := map_one'\u271d } y\nmonotone'\u271d : Monotone (\u2191{ toOneHom := { toFun := toFun\u271d, map_one' := map_one'\u271d }, map_mul' := map_mul'\u271d }).toFun\nh :\n  (fun f => f.toFun)\n      { toMonoidHom := { toOneHom := { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 }, map_mul' := map_mul'\u271d\u00b9 },\n        monotone' := monotone'\u271d\u00b9 } =\n    (fun f => f.toFun)\n      { toMonoidHom := { toOneHom := { toFun := toFun\u271d, map_one' := map_one'\u271d }, map_mul' := map_mul'\u271d },\n        monotone' := monotone'\u271d }\n\u22a2 { toMonoidHom := { toOneHom := { toFun := toFun\u271d\u00b9, map_one' := map_one'\u271d\u00b9 }, map_mul' := map_mul'\u271d\u00b9 },\n      monotone' := monotone'\u271d\u00b9 } =\n    { toMonoidHom := { toOneHom := { toFun := toFun\u271d, map_one' := map_one'\u271d }, map_mul' := map_mul'\u271d },\n      monotone' := monotone'\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf\u271d g f : \u03b1 \u2192*o \u03b2\nh : Monotone (\u2191\u2191f).toFun\n\u22a2 { toMonoidHom := \u2191f, monotone' := h } = f\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\n\u03b4 : Type u_5\ninst\u271d\u2077 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf\u271d g f : \u03b1 \u2192*o \u03b2\nh : Monotone (\u2191\u2191f).toFun\na\u271d : \u03b1\n\u22a2 \u2191{ toMonoidHom := \u2191f, monotone' := h } a\u271d = \u2191f a\u271d\n[PROOFSTEP]\nrfl\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf\u271d g\u271d f g : \u03b1 \u2192*o \u03b2\nh : f.toMonoidHom = g.toMonoidHom\n\u22a2 \u2200 (a : \u03b1), \u2191f a = \u2191g a\n[PROOFSTEP]\nconvert FunLike.ext_iff.1 h using 0\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf\u271d g\u271d f g : \u03b1 \u2192*o \u03b2\nh : toOrderHom f = toOrderHom g\n\u22a2 \u2200 (a : \u03b1), \u2191f a = \u2191g a\n[PROOFSTEP]\nconvert FunLike.ext_iff.1 h using 0\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf\u271d g : \u03b1 \u2192*o \u03b2\ng\u2081 g\u2082 : \u03b2 \u2192*o \u03b3\nf : \u03b1 \u2192*o \u03b2\nhf : Surjective \u2191f\nx\u271d : g\u2081 = g\u2082\n\u22a2 comp g\u2081 f = comp g\u2082 f\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulOneClass \u03b1\ninst\u271d\u00b2 : MulOneClass \u03b2\ninst\u271d\u00b9 : MulOneClass \u03b3\ninst\u271d : MulOneClass \u03b4\nf g\u271d : \u03b1 \u2192*o \u03b2\ng : \u03b2 \u2192*o \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192*o \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[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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf\u271d g\u271d f g : \u03b1 \u2192*\u2080o \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := f\n[GOAL]\ncase 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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf g\u271d g : \u03b1 \u2192*\u2080o \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_zero'\u271d : toFun\u271d 0 = 0\nmap_one'\u271d : ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } 1 = 1\nmap_mul'\u271d :\n  \u2200 (x y : \u03b1),\n    ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } (x * y) =\n      ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } x *\n        ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } y\nmonotone'\u271d :\n  Monotone\n    (\u2191{ toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d }).toFun\nh :\n  (fun f => f.toFun)\n      {\n        toMonoidWithZeroHom :=\n          { toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d },\n        monotone' := monotone'\u271d } =\n    (fun f => f.toFun) g\n\u22a2 {\n      toMonoidWithZeroHom :=\n        { toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d },\n      monotone' := monotone'\u271d } =\n    g\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf g : \u03b1 \u2192*\u2080o \u03b2\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmap_zero'\u271d\u00b9 : toFun\u271d\u00b9 0 = 0\nmap_one'\u271d\u00b9 : ZeroHom.toFun { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } 1 = 1\nmap_mul'\u271d\u00b9 :\n  \u2200 (x y : \u03b1),\n    ZeroHom.toFun { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } (x * y) =\n      ZeroHom.toFun { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } x *\n        ZeroHom.toFun { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } y\nmonotone'\u271d\u00b9 :\n  Monotone\n    (\u2191{ toZeroHom := { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 }, map_one' := map_one'\u271d\u00b9,\n          map_mul' := map_mul'\u271d\u00b9 }).toFun\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmap_zero'\u271d : toFun\u271d 0 = 0\nmap_one'\u271d : ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } 1 = 1\nmap_mul'\u271d :\n  \u2200 (x y : \u03b1),\n    ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } (x * y) =\n      ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } x *\n        ZeroHom.toFun { toFun := toFun\u271d, map_zero' := map_zero'\u271d } y\nmonotone'\u271d :\n  Monotone\n    (\u2191{ toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d }).toFun\nh :\n  (fun f => f.toFun)\n      {\n        toMonoidWithZeroHom :=\n          { toZeroHom := { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 }, map_one' := map_one'\u271d\u00b9,\n            map_mul' := map_mul'\u271d\u00b9 },\n        monotone' := monotone'\u271d\u00b9 } =\n    (fun f => f.toFun)\n      {\n        toMonoidWithZeroHom :=\n          { toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d },\n        monotone' := monotone'\u271d }\n\u22a2 {\n      toMonoidWithZeroHom :=\n        { toZeroHom := { toFun := toFun\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 }, map_one' := map_one'\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 },\n      monotone' := monotone'\u271d\u00b9 } =\n    {\n      toMonoidWithZeroHom :=\n        { toZeroHom := { toFun := toFun\u271d, map_zero' := map_zero'\u271d }, map_one' := map_one'\u271d, map_mul' := map_mul'\u271d },\n      monotone' := monotone'\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf\u271d g\u271d f g : \u03b1 \u2192*\u2080o \u03b2\nh : toOrderMonoidHom f = toOrderMonoidHom g\n\u22a2 \u2200 (a : \u03b1), \u2191f a = \u2191g a\n[PROOFSTEP]\nconvert FunLike.ext_iff.1 h using 0\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf\u271d g\u271d f g : \u03b1 \u2192*\u2080o \u03b2\nh : f.toMonoidWithZeroHom = g.toMonoidWithZeroHom\n\u22a2 \u2200 (a : \u03b1), \u2191f a = \u2191g a\n[PROOFSTEP]\nconvert FunLike.ext_iff.1 h using 0\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf\u271d g : \u03b1 \u2192*\u2080o \u03b2\ng\u2081 g\u2082 : \u03b2 \u2192*\u2080o \u03b3\nf : \u03b1 \u2192*\u2080o \u03b2\nhf : Surjective \u2191f\nx\u271d : g\u2081 = g\u2082\n\u22a2 comp g\u2081 f = comp g\u2082 f\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 : Preorder \u03b1\ninst\u271d\u2076 : Preorder \u03b2\ninst\u271d\u2075 : Preorder \u03b3\ninst\u271d\u2074 : Preorder \u03b4\ninst\u271d\u00b3 : MulZeroOneClass \u03b1\ninst\u271d\u00b2 : MulZeroOneClass \u03b2\ninst\u271d\u00b9 : MulZeroOneClass \u03b3\ninst\u271d : MulZeroOneClass \u03b4\nf g\u271d : \u03b1 \u2192*\u2080o \u03b2\ng : \u03b2 \u2192*\u2080o \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192*\u2080o \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[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\nh\u03b1 : Preorder \u03b1\nh\u03b1' : MulZeroOneClass \u03b1\nh\u03b2 : Preorder \u03b2\nh\u03b2' : MulZeroOneClass \u03b2\nf : \u03b1 \u2192*\u2080o \u03b2\n\u22a2 f.toMonoidWithZeroHom = \u2191f\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Hom.Monoid", "llama_tokens": 7840, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314828740729, "lm_q2_score": 0.63341027059799, "lm_q1q2_score": 0.5609680772173657}}
{"text": "[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : Module R A\ne\u2081 : m \u2243 m'\ne\u2082 : n \u2243 n'\ne\u2081' : m' \u2243 m''\ne\u2082' : n' \u2243 n''\n\u22a2 LinearEquiv.trans (reindexLinearEquiv R A e\u2081 e\u2082) (reindexLinearEquiv R A e\u2081' e\u2082') =\n    reindexLinearEquiv R A (e\u2081.trans e\u2081') (e\u2082.trans e\u2082')\n[PROOFSTEP]\next\n[GOAL]\ncase h.a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : Module R A\ne\u2081 : m \u2243 m'\ne\u2082 : n \u2243 n'\ne\u2081' : m' \u2243 m''\ne\u2082' : n' \u2243 n''\nx\u271d\u00b9 : Matrix m n A\ni\u271d : m''\nx\u271d : n''\n\u22a2 \u2191(LinearEquiv.trans (reindexLinearEquiv R A e\u2081 e\u2082) (reindexLinearEquiv R A e\u2081' e\u2082')) x\u271d\u00b9 i\u271d x\u271d =\n    \u2191(reindexLinearEquiv R A (e\u2081.trans e\u2081') (e\u2082.trans e\u2082')) x\u271d\u00b9 i\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : Module R A\ne\u2081 : m \u2243 m'\ne\u2082 : n \u2243 n'\ne\u2081' : m' \u2243 m''\ne\u2082' : n' \u2243 n''\n\u22a2 \u2191(reindexLinearEquiv R A e\u2081' e\u2082') \u2218 \u2191(reindexLinearEquiv R A e\u2081 e\u2082) =\n    \u2191(reindexLinearEquiv R A (e\u2081.trans e\u2081') (e\u2082.trans e\u2082'))\n[PROOFSTEP]\nrw [\u2190 reindexLinearEquiv_trans]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid A\ninst\u271d : Module R A\ne\u2081 : m \u2243 m'\ne\u2082 : n \u2243 n'\ne\u2081' : m' \u2243 m''\ne\u2082' : n' \u2243 n''\n\u22a2 \u2191(reindexLinearEquiv R A e\u2081' e\u2082') \u2218 \u2191(reindexLinearEquiv R A e\u2081 e\u2082) =\n    \u2191(LinearEquiv.trans (reindexLinearEquiv R A e\u2081 e\u2082) (reindexLinearEquiv R A e\u2081' e\u2082'))\n[PROOFSTEP]\nrfl\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nl' : Type u_5\nm' : Type u_6\nn' : Type u_7\no' : Type u_8\nm'' : Type u_9\nn'' : Type u_10\nR : Type u_11\nA : Type u_12\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\ne : m \u2243 n\nsrc\u271d : Matrix m m R \u2243\u2097[R] Matrix n n R := reindexLinearEquiv R R e e\nr : R\n\u22a2 toFun\n      { toFun := \u2191(reindex e e), 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 (Matrix m m R)) r) =\n    \u2191(algebraMap R (Matrix n n R)) r\n[PROOFSTEP]\nsimp [algebraMap, Algebra.toRingHom, submatrix_smul _ 1]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.Reindex", "llama_tokens": 1388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696748, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.5605011266346394}}
{"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\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 => \u2220 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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\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 => \u2220 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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\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 => \u2220 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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\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 => \u2220 y.fst y.snd.fst y.snd.snd) x\n[PROOFSTEP]\nexact\n  (InnerProductGeometry.continuousAt_angle 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\u2077 : NormedAddCommGroup V\ninst\u271d\u2076 : InnerProductSpace \u211d V\ninst\u271d\u2075 : MetricSpace P\ninst\u271d\u2074 : NormedAddTorsor V P\nV\u2082 : Type u_3\nP\u2082 : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup V\u2082\ninst\u271d\u00b2 : InnerProductSpace \u211d V\u2082\ninst\u271d\u00b9 : MetricSpace P\u2082\ninst\u271d : NormedAddTorsor V\u2082 P\u2082\nf : P \u2192\u1d43\u2071[\u211d] P\u2082\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2220 (\u2191f p\u2081) (\u2191f p\u2082) (\u2191f p\u2083) = \u2220 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nsimp_rw [angle, \u2190 AffineIsometry.map_vsub, LinearIsometry.angle_map]\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\nv\u2081 v\u2082 v\u2083 v : V\n\u22a2 \u2220 (v\u2081 - v) (v\u2082 - v) (v\u2083 - v) = \u2220 v\u2081 v\u2082 v\u2083\n[PROOFSTEP]\nsimpa only [vsub_eq_sub] using angle_vsub_const v\u2081 v\u2082 v\u2083 v\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\nv v\u2081 v\u2082 v\u2083 : V\n\u22a2 \u2220 (v - v\u2081) (v - v\u2082) (v - v\u2083) = \u2220 v\u2081 v\u2082 v\u2083\n[PROOFSTEP]\nsimpa only [vsub_eq_sub] using angle_const_vsub v v\u2081 v\u2082 v\u2083\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\nv\u2081 v\u2082 v\u2083 : V\n\u22a2 \u2220 (-v\u2081) (-v\u2082) (-v\u2083) = \u2220 v\u2081 v\u2082 v\u2083\n[PROOFSTEP]\nsimpa only [zero_sub] using angle_const_sub 0 v\u2081 v\u2082 v\u2083\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 : P\n\u22a2 \u2220 p1 p1 p2 = \u03c0 / 2\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 : P\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p1) (p2 -\u1d65 p1) = \u03c0 / 2\n[PROOFSTEP]\nrw [vsub_self]\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 : P\n\u22a2 InnerProductGeometry.angle 0 (p2 -\u1d65 p1) = \u03c0 / 2\n[PROOFSTEP]\nexact angle_zero_left _\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 : P\n\u22a2 \u2220 p1 p2 p2 = \u03c0 / 2\n[PROOFSTEP]\nrw [angle_comm, angle_eq_left]\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 = \u03c0\n\u22a2 \u2220 p2 p1 p3 = 0\n[PROOFSTEP]\nunfold angle 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 : InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = \u03c0\n\u22a2 \u2220 p2 p1 p3 = 0\n[PROOFSTEP]\nrw [angle_eq_pi_iff] 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 : p1 -\u1d65 p2 \u2260 0 \u2227 \u2203 r, r < 0 \u2227 p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 \u2220 p2 p1 p3 = 0\n[PROOFSTEP]\nrcases h with \u27e8hp1p2, \u27e8r, \u27e8hr, hpr\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\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\nhp1p2 : p1 -\u1d65 p2 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 \u2220 p2 p1 p3 = 0\n[PROOFSTEP]\nunfold angle\n[GOAL]\ncase intro.intro.intro\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\nhp1p2 : p1 -\u1d65 p2 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 InnerProductGeometry.angle (p2 -\u1d65 p1) (p3 -\u1d65 p1) = 0\n[PROOFSTEP]\nrw [angle_eq_zero_iff]\n[GOAL]\ncase intro.intro.intro\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\nhp1p2 : p1 -\u1d65 p2 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 p2 -\u1d65 p1 \u2260 0 \u2227 \u2203 r, 0 < r \u2227 p3 -\u1d65 p1 = r \u2022 (p2 -\u1d65 p1)\n[PROOFSTEP]\nrw [\u2190 neg_vsub_eq_vsub_rev, neg_ne_zero] at hp1p2 \n[GOAL]\ncase intro.intro.intro\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\nhp1p2 : p2 -\u1d65 p1 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 p2 -\u1d65 p1 \u2260 0 \u2227 \u2203 r, 0 < r \u2227 p3 -\u1d65 p1 = r \u2022 (p2 -\u1d65 p1)\n[PROOFSTEP]\nuse hp1p2, -r + 1, add_pos (neg_pos_of_neg hr) zero_lt_one\n[GOAL]\ncase right\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\nhp1p2 : p2 -\u1d65 p1 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 p3 -\u1d65 p1 = (-r + 1) \u2022 (p2 -\u1d65 p1)\n[PROOFSTEP]\nrw [add_smul, \u2190 neg_vsub_eq_vsub_rev p1 p2, smul_neg]\n[GOAL]\ncase right\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\nhp1p2 : p2 -\u1d65 p1 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p3 -\u1d65 p2 = r \u2022 (p1 -\u1d65 p2)\n\u22a2 p3 -\u1d65 p1 = -(-r \u2022 (p1 -\u1d65 p2)) + 1 \u2022 -(p1 -\u1d65 p2)\n[PROOFSTEP]\nsimp [\u2190 hpr]\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 = \u03c0\n\u22a2 \u2220 p2 p3 p1 = 0\n[PROOFSTEP]\nrw [angle_comm] 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 : \u2220 p3 p2 p1 = \u03c0\n\u22a2 \u2220 p2 p3 p1 = 0\n[PROOFSTEP]\nexact angle_eq_zero_of_angle_eq_pi_left 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 p4 : P\nh : \u2220 p2 p3 p4 = \u03c0\n\u22a2 \u2220 p1 p2 p3 = \u2220 p1 p2 p4\n[PROOFSTEP]\nunfold angle at *\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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p2) (p4 -\u1d65 p2)\n[PROOFSTEP]\nrcases angle_eq_pi_iff.1 h with \u27e8_, \u27e8r, \u27e8hr, hpr\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\nleft\u271d : p2 -\u1d65 p3 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p4 -\u1d65 p3 = r \u2022 (p2 -\u1d65 p3)\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p2) (p4 -\u1d65 p2)\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase intro.intro.intro\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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\nleft\u271d : p2 -\u1d65 p3 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p4 -\u1d65 p3 = r \u2022 (p2 -\u1d65 p3)\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p2) (p4 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2)\n[PROOFSTEP]\nconvert angle_smul_right_of_pos (p1 -\u1d65 p2) (p3 -\u1d65 p2) (add_pos (neg_pos_of_neg hr) zero_lt_one)\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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\nleft\u271d : p2 -\u1d65 p3 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p4 -\u1d65 p3 = r \u2022 (p2 -\u1d65 p3)\n\u22a2 p4 -\u1d65 p2 = (-r + 1) \u2022 (p3 -\u1d65 p2)\n[PROOFSTEP]\nrw [add_smul, \u2190 neg_vsub_eq_vsub_rev p2 p3, smul_neg, neg_smul, \u2190 hpr]\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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\nleft\u271d : p2 -\u1d65 p3 \u2260 0\nr : \u211d\nhr : r < 0\nhpr : p4 -\u1d65 p3 = r \u2022 (p2 -\u1d65 p3)\n\u22a2 p4 -\u1d65 p2 = - -(p4 -\u1d65 p3) + 1 \u2022 -(p2 -\u1d65 p3)\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\np1 p2 p3 p4 : P\nh : \u2220 p2 p3 p4 = \u03c0\n\u22a2 \u2220 p1 p3 p2 + \u2220 p1 p3 p4 = \u03c0\n[PROOFSTEP]\nunfold angle 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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\n\u22a2 \u2220 p1 p3 p2 + \u2220 p1 p3 p4 = \u03c0\n[PROOFSTEP]\nrw [angle_comm p1 p3 p2, angle_comm p1 p3 p4]\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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\n\u22a2 \u2220 p2 p3 p1 + \u2220 p4 p3 p1 = \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 p4 : P\nh : InnerProductGeometry.angle (p2 -\u1d65 p3) (p4 -\u1d65 p3) = \u03c0\n\u22a2 InnerProductGeometry.angle (p2 -\u1d65 p3) (p1 -\u1d65 p3) + InnerProductGeometry.angle (p4 -\u1d65 p3) (p1 -\u1d65 p3) = \u03c0\n[PROOFSTEP]\nexact angle_add_angle_eq_pi_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\np1 p2 p3 p4 p5 : P\nhapc : \u2220 p1 p5 p3 = \u03c0\nhbpd : \u2220 p2 p5 p4 = \u03c0\n\u22a2 \u2220 p1 p5 p2 = \u2220 p3 p5 p4\n[PROOFSTEP]\nlinarith [angle_add_angle_eq_pi_of_angle_eq_pi p1 hbpd, angle_comm p4 p5 p1,\n  angle_add_angle_eq_pi_of_angle_eq_pi p4 hapc, angle_comm p4 p5 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 : \u2220 p1 p2 p3 = \u03c0\n\u22a2 dist p1 p2 \u2260 0\n[PROOFSTEP]\nby_contra heq\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 = \u03c0\nheq : dist p1 p2 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [dist_eq_zero] at heq \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 = \u03c0\nheq : p1 = p2\n\u22a2 False\n[PROOFSTEP]\nrw [heq, angle_eq_left] 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 : \u03c0 / 2 = \u03c0\nheq : p1 = p2\n\u22a2 False\n[PROOFSTEP]\nexact Real.pi_ne_zero (by linarith)\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 : \u03c0 / 2 = \u03c0\nheq : p1 = p2\n\u22a2 \u03c0 = 0\n[PROOFSTEP]\nlinarith\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 = \u03c0\n\u22a2 dist p1 p3 = dist p1 p2 + dist p3 p2\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, \u2190 vsub_sub_vsub_cancel_right]\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 = \u03c0\n\u22a2 \u2016p1 -\u1d65 ?p3 - (p3 -\u1d65 ?p3)\u2016 = \u2016p1 -\u1d65 p2\u2016 + \u2016p3 -\u1d65 p2\u2016\ncase p3\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 = \u03c0\n\u22a2 P\n[PROOFSTEP]\nexact norm_sub_eq_add_norm_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\np1 p2 p3 : P\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 dist p1 p3 = dist p1 p2 + dist p3 p2 \u2194 \u2220 p1 p2 p3 = \u03c0\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, \u2190 vsub_sub_vsub_cancel_right]\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\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 \u2016p1 -\u1d65 ?p3 - (p3 -\u1d65 ?p3)\u2016 = \u2016p1 -\u1d65 p2\u2016 + \u2016p3 -\u1d65 p2\u2016 \u2194 \u2220 p1 p2 p3 = \u03c0\ncase p3\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\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 P\n[PROOFSTEP]\nexact\n  norm_sub_eq_add_norm_iff_angle_eq_pi (fun he => hp1p2 (vsub_eq_zero_iff_eq.1 he)) fun he =>\n    hp3p2 (vsub_eq_zero_iff_eq.1 he)\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 = 0\n\u22a2 dist p1 p3 = |dist p1 p2 - dist p3 p2|\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, \u2190 vsub_sub_vsub_cancel_right]\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 = 0\n\u22a2 \u2016p1 -\u1d65 ?p3 - (p3 -\u1d65 ?p3)\u2016 = |\u2016p1 -\u1d65 p2\u2016 - \u2016p3 -\u1d65 p2\u2016|\ncase p3\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 = 0\n\u22a2 P\n[PROOFSTEP]\nexact norm_sub_eq_abs_sub_norm_of_angle_eq_zero 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\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 dist p1 p3 = |dist p1 p2 - dist p3 p2| \u2194 \u2220 p1 p2 p3 = 0\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, \u2190 vsub_sub_vsub_cancel_right]\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\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 \u2016p1 -\u1d65 ?p3 - (p3 -\u1d65 ?p3)\u2016 = |\u2016p1 -\u1d65 p2\u2016 - \u2016p3 -\u1d65 p2\u2016| \u2194 \u2220 p1 p2 p3 = 0\ncase p3\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\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 P\n[PROOFSTEP]\nexact\n  norm_sub_eq_abs_sub_norm_iff_angle_eq_zero (fun he => hp1p2 (vsub_eq_zero_iff_eq.1 he)) fun he =>\n    hp3p2 (vsub_eq_zero_iff_eq.1 he)\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 : P\nhp1p2 : p1 \u2260 p2\n\u22a2 \u2220 p1 (midpoint \u211d p1 p2) p2 = \u03c0\n[PROOFSTEP]\nsimp [angle, hp1p2, -zero_lt_one]\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 : P\nhp1p2 : p1 \u2260 p2\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p2) (p2 -\u1d65 p1) = \u03c0\n[PROOFSTEP]\nrw [\u2190 neg_vsub_eq_vsub_rev p1 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 : P\nhp1p2 : p1 \u2260 p2\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p2) (-(p1 -\u1d65 p2)) = \u03c0\n[PROOFSTEP]\napply angle_self_neg_of_nonzero\n[GOAL]\ncase 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\np1 p2 : P\nhp1p2 : p1 \u2260 p2\n\u22a2 p1 -\u1d65 p2 \u2260 0\n[PROOFSTEP]\nsimpa only [ne_eq, vsub_eq_zero_iff_eq]\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 p3 p1 = dist p3 p2\n\u22a2 \u2220 p3 (midpoint \u211d p1 p2) p1 = \u03c0 / 2\n[PROOFSTEP]\nlet m : P := midpoint \u211d p1 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\nh : dist p3 p1 = dist p3 p2\nm : P := midpoint \u211d p1 p2\n\u22a2 \u2220 p3 (midpoint \u211d p1 p2) p1 = \u03c0 / 2\n[PROOFSTEP]\nhave h1 : p3 -\u1d65 p1 = p3 -\u1d65 m - (p1 -\u1d65 m) := (vsub_sub_vsub_cancel_right p3 p1 m).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 p3 p1 = dist p3 p2\nm : P := midpoint \u211d p1 p2\nh1 : p3 -\u1d65 p1 = p3 -\u1d65 m - (p1 -\u1d65 m)\n\u22a2 \u2220 p3 (midpoint \u211d p1 p2) p1 = \u03c0 / 2\n[PROOFSTEP]\nhave h2 : p3 -\u1d65 p2 = p3 -\u1d65 m + (p1 -\u1d65 m) := by rw [left_vsub_midpoint, \u2190 midpoint_vsub_right, vsub_add_vsub_cancel]\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 p3 p1 = dist p3 p2\nm : P := midpoint \u211d p1 p2\nh1 : p3 -\u1d65 p1 = p3 -\u1d65 m - (p1 -\u1d65 m)\n\u22a2 p3 -\u1d65 p2 = p3 -\u1d65 m + (p1 -\u1d65 m)\n[PROOFSTEP]\nrw [left_vsub_midpoint, \u2190 midpoint_vsub_right, vsub_add_vsub_cancel]\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 p3 p1 = dist p3 p2\nm : P := midpoint \u211d p1 p2\nh1 : p3 -\u1d65 p1 = p3 -\u1d65 m - (p1 -\u1d65 m)\nh2 : p3 -\u1d65 p2 = p3 -\u1d65 m + (p1 -\u1d65 m)\n\u22a2 \u2220 p3 (midpoint \u211d p1 p2) p1 = \u03c0 / 2\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V p3 p1, dist_eq_norm_vsub V p3 p2, h1, h2] 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\nm : P := midpoint \u211d p1 p2\nh : \u2016p3 -\u1d65 m - (p1 -\u1d65 m)\u2016 = \u2016p3 -\u1d65 m + (p1 -\u1d65 m)\u2016\nh1 : p3 -\u1d65 p1 = p3 -\u1d65 m - (p1 -\u1d65 m)\nh2 : p3 -\u1d65 p2 = p3 -\u1d65 m + (p1 -\u1d65 m)\n\u22a2 \u2220 p3 (midpoint \u211d p1 p2) p1 = \u03c0 / 2\n[PROOFSTEP]\nexact (norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (p3 -\u1d65 m) (p1 -\u1d65 m)).mp h.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 p3 p1 = dist p3 p2\n\u22a2 \u2220 p3 (midpoint \u211d p1 p2) p2 = \u03c0 / 2\n[PROOFSTEP]\nrw [midpoint_comm p1 p2, angle_left_midpoint_eq_pi_div_two_of_dist_eq h.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\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nrw [angle, angle_eq_pi_iff]\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\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 p\u2081 -\u1d65 p\u2082 \u2260 0 \u2227 \u2203 r, r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nrcases h with \u27e8\u27e8r, \u27e8hr0, hr1\u27e9, hp\u2082\u27e9, hp\u2082p\u2081, hp\u2082p\u2083\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 p\u2081 -\u1d65 p\u2082 \u2260 0 \u2227 \u2203 r, r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nrefine' \u27e8vsub_ne_zero.2 hp\u2082p\u2081.symm, -(1 - r) / r, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 -(1 - r) / r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = (-(1 - r) / r) \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nhave hr0' : r \u2260 0 := by\n  rintro rfl\n  rw [\u2190 hp\u2082] at hp\u2082p\u2081 \n  simp at hp\u2082p\u2081 \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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 r \u2260 0\n[PROOFSTEP]\nrintro rfl\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 0 = p\u2082\nhr0 : 0 \u2264 0\nhr1 : 0 \u2264 1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 hp\u2082] at hp\u2082p\u2081 \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\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 0 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 0 = p\u2082\nhr0 : 0 \u2264 0\nhr1 : 0 \u2264 1\n\u22a2 False\n[PROOFSTEP]\nsimp at hp\u2082p\u2081 \n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhr0' : r \u2260 0\n\u22a2 -(1 - r) / r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = (-(1 - r) / r) \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nhave hr1' : r \u2260 1 := by\n  rintro rfl\n  rw [\u2190 hp\u2082] at hp\u2082p\u2083 \n  simp at hp\u2082p\u2083 \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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhr0' : r \u2260 0\n\u22a2 r \u2260 1\n[PROOFSTEP]\nrintro rfl\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 1 = p\u2082\nhr0 : 0 \u2264 1\nhr1 : 1 \u2264 1\nhr0' : 1 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 hp\u2082] at hp\u2082p\u2083 \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\np\u2081 p\u2082 p\u2083 : P\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 1 \u2260 p\u2083\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 1 = p\u2082\nhr0 : 0 \u2264 1\nhr1 : 1 \u2264 1\nhr0' : 1 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp at hp\u2082p\u2083 \n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhr0' : r \u2260 0\nhr1' : r \u2260 1\n\u22a2 -(1 - r) / r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = (-(1 - r) / r) \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nreplace hr0 := hr0.lt_of_ne hr0'.symm\n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhr1 : r \u2264 1\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhr0' : r \u2260 0\nhr1' : r \u2260 1\nhr0 : 0 < r\n\u22a2 -(1 - r) / r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = (-(1 - r) / r) \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nreplace hr1 := hr1.lt_of_ne hr1'\n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhr0' : r \u2260 0\nhr1' : r \u2260 1\nhr0 : 0 < r\nhr1 : r < 1\n\u22a2 -(1 - r) / r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = (-(1 - r) / r) \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nrefine' \u27e8div_neg_of_neg_of_pos (Left.neg_neg_iff.2 (sub_pos.2 hr1)) hr0, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nr : \u211d\nhp\u2082 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r = p\u2082\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhr0' : r \u2260 0\nhr1' : r \u2260 1\nhr0 : 0 < r\nhr1 : r < 1\n\u22a2 p\u2083 -\u1d65 p\u2082 = (-(1 - r) / r) \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [\u2190 hp\u2082, AffineMap.lineMap_apply, vsub_vadd_eq_vsub_sub, vsub_vadd_eq_vsub_sub, vsub_self, zero_sub, smul_neg,\n  smul_smul, div_mul_cancel _ hr0', neg_smul, neg_neg, sub_eq_iff_eq_add, \u2190 add_smul, sub_add_cancel, one_smul]\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\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2220 p\u2083 p\u2082 p\u2081 = \u03c0\n[PROOFSTEP]\nrw [\u2190 h.angle\u2081\u2082\u2083_eq_pi, angle_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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 \u2194 Sbtw \u211d p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrefine' \u27e8_, fun h => h.angle\u2081\u2082\u2083_eq_pi\u27e9\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 \u2192 Sbtw \u211d p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [angle, angle_eq_pi_iff]\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 (p\u2081 -\u1d65 p\u2082 \u2260 0 \u2227 \u2203 r, r < 0 \u2227 p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)) \u2192 Sbtw \u211d p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrintro \u27e8hp\u2081p\u2082, r, hr, hp\u2083p\u2082\u27e9\n[GOAL]\ncase intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\n\u22a2 Sbtw \u211d p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrefine'\n  \u27e8\u27e81 / (1 - r),\n      \u27e8div_nonneg zero_le_one (sub_nonneg.2 (hr.le.trans zero_le_one)),\n        (div_le_one (sub_pos.2 (hr.trans zero_lt_one))).2 ((le_sub_self_iff 1).2 hr.le)\u27e9,\n      _\u27e9,\n    (vsub_ne_zero.1 hp\u2081p\u2082).symm, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\n\u22a2 \u2191(AffineMap.lineMap p\u2081 p\u2083) (1 / (1 - r)) = p\u2082\n[PROOFSTEP]\nrw [\u2190 eq_vadd_iff_vsub_eq] at hp\u2083p\u2082 \n[GOAL]\ncase intro.intro.intro.refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 = r \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2082\n\u22a2 \u2191(AffineMap.lineMap p\u2081 p\u2083) (1 / (1 - r)) = p\u2082\n[PROOFSTEP]\nrw [AffineMap.lineMap_apply, hp\u2083p\u2082, vadd_vsub_assoc, \u2190 neg_vsub_eq_vsub_rev p\u2082 p\u2081, smul_neg, \u2190 neg_smul, smul_add,\n  smul_smul, \u2190 add_smul, eq_comm, eq_vadd_iff_vsub_eq]\n[GOAL]\ncase intro.intro.intro.refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 = r \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2082\n\u22a2 p\u2082 -\u1d65 p\u2081 = (1 / (1 - r) * -r + 1 / (1 - r)) \u2022 (p\u2082 -\u1d65 p\u2081)\n[PROOFSTEP]\nconvert (one_smul \u211d (p\u2082 -\u1d65 p\u2081)).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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 = r \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2082\n\u22a2 1 / (1 - r) * -r + 1 / (1 - r) = 1\n[PROOFSTEP]\nfield_simp [(sub_pos.2 (hr.trans zero_lt_one)).ne.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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 = r \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2082\n\u22a2 -(r * (1 - r)) + (1 - r) = (1 - r) * (1 - r)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro.intro.refine'_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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\n\u22a2 p\u2082 \u2260 p\u2083\n[PROOFSTEP]\nrw [ne_comm, \u2190 @vsub_ne_zero V, hp\u2083p\u2082, smul_ne_zero_iff]\n[GOAL]\ncase intro.intro.intro.refine'_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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr : r < 0\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\n\u22a2 r \u2260 0 \u2227 p\u2081 -\u1d65 p\u2082 \u2260 0\n[PROOFSTEP]\nexact \u27e8hr.ne, hp\u2081p\u2082\u27e9\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\n\u22a2 \u2220 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nrw [angle, angle_eq_zero_iff]\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\n\u22a2 p\u2082 -\u1d65 p\u2081 \u2260 0 \u2227 \u2203 r, 0 < r \u2227 p\u2083 -\u1d65 p\u2081 = r \u2022 (p\u2082 -\u1d65 p\u2081)\n[PROOFSTEP]\nrcases h with \u27e8r, \u27e8hr0, hr1\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\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\np\u2081 p\u2083 : P\nr : \u211d\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r \u2260 p\u2081\n\u22a2 \u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081 \u2260 0 \u2227 \u2203 r_1, 0 < r_1 \u2227 p\u2083 -\u1d65 p\u2081 = r_1 \u2022 (\u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081)\n[PROOFSTEP]\nhave hr0' : r \u2260 0 := by\n  rintro rfl\n  simp at hp\u2082p\u2081 \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\np\u2081 p\u2083 : P\nr : \u211d\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r \u2260 p\u2081\n\u22a2 r \u2260 0\n[PROOFSTEP]\nrintro rfl\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\np\u2081 p\u2083 : P\nhr0 : 0 \u2264 0\nhr1 : 0 \u2264 1\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) 0 \u2260 p\u2081\n\u22a2 False\n[PROOFSTEP]\nsimp at hp\u2082p\u2081 \n[GOAL]\ncase intro.intro.intro\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\np\u2081 p\u2083 : P\nr : \u211d\nhr0 : 0 \u2264 r\nhr1 : r \u2264 1\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r \u2260 p\u2081\nhr0' : r \u2260 0\n\u22a2 \u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081 \u2260 0 \u2227 \u2203 r_1, 0 < r_1 \u2227 p\u2083 -\u1d65 p\u2081 = r_1 \u2022 (\u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081)\n[PROOFSTEP]\nreplace hr0 := hr0.lt_of_ne hr0'.symm\n[GOAL]\ncase intro.intro.intro\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\np\u2081 p\u2083 : P\nr : \u211d\nhr1 : r \u2264 1\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r \u2260 p\u2081\nhr0' : r \u2260 0\nhr0 : 0 < r\n\u22a2 \u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081 \u2260 0 \u2227 \u2203 r_1, 0 < r_1 \u2227 p\u2083 -\u1d65 p\u2081 = r_1 \u2022 (\u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081)\n[PROOFSTEP]\nrefine' \u27e8vsub_ne_zero.2 hp\u2082p\u2081, r\u207b\u00b9, inv_pos.2 hr0, _\u27e9\n[GOAL]\ncase intro.intro.intro\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\np\u2081 p\u2083 : P\nr : \u211d\nhr1 : r \u2264 1\nhp\u2082p\u2081 : \u2191(AffineMap.lineMap p\u2081 p\u2083) r \u2260 p\u2081\nhr0' : r \u2260 0\nhr0 : 0 < r\n\u22a2 p\u2083 -\u1d65 p\u2081 = r\u207b\u00b9 \u2022 (\u2191(AffineMap.lineMap p\u2081 p\u2083) r -\u1d65 p\u2081)\n[PROOFSTEP]\nrw [AffineMap.lineMap_apply, vadd_vsub_assoc, vsub_self, add_zero, smul_smul, inv_mul_cancel hr0', one_smul]\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\n\u22a2 \u2220 p\u2083 p\u2081 p\u2082 = 0\n[PROOFSTEP]\nrw [\u2190 h.angle\u2082\u2081\u2083_eq_zero_of_ne hp\u2082p\u2081, angle_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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2194 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\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2192 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\n[PROOFSTEP]\nrw [angle, angle_eq_zero_iff]\n[GOAL]\ncase mp\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 (p\u2081 -\u1d65 p\u2082 \u2260 0 \u2227 \u2203 r, 0 < r \u2227 p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)) \u2192 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\n[PROOFSTEP]\nrintro \u27e8hp\u2081p\u2082, r, hr0, hp\u2083p\u2082\u27e9\n[GOAL]\ncase mp.intro.intro.intro\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 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\n[PROOFSTEP]\nrcases le_or_lt 1 r with (hr1 | hr1)\n[GOAL]\ncase mp.intro.intro.intro.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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\nhr1 : 1 \u2264 r\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\n[PROOFSTEP]\nrefine' Or.inl \u27e8vsub_ne_zero.1 hp\u2081p\u2082, r\u207b\u00b9, \u27e8(inv_pos.2 hr0).le, inv_le_one hr1\u27e9, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\nhr1 : 1 \u2264 r\n\u22a2 \u2191(AffineMap.lineMap p\u2082 p\u2083) r\u207b\u00b9 = p\u2081\n[PROOFSTEP]\nrw [AffineMap.lineMap_apply, hp\u2083p\u2082, smul_smul, inv_mul_cancel hr0.ne.symm, one_smul, vsub_vadd]\n[GOAL]\ncase mp.intro.intro.intro.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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\nhr1 : r < 1\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\n[PROOFSTEP]\nrefine' Or.inr \u27e8_, r, \u27e8hr0.le, hr1.le\u27e9, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.inr.refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\nhr1 : r < 1\n\u22a2 p\u2083 \u2260 p\u2082\n[PROOFSTEP]\nrw [\u2190 @vsub_ne_zero V, hp\u2083p\u2082, smul_ne_zero_iff]\n[GOAL]\ncase mp.intro.intro.intro.inr.refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\nhr1 : r < 1\n\u22a2 r \u2260 0 \u2227 p\u2081 -\u1d65 p\u2082 \u2260 0\n[PROOFSTEP]\nexact \u27e8hr0.ne.symm, hp\u2081p\u2082\u27e9\n[GOAL]\ncase mp.intro.intro.intro.inr.refine'_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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 -\u1d65 p\u2082 \u2260 0\nr : \u211d\nhr0 : 0 < r\nhp\u2083p\u2082 : p\u2083 -\u1d65 p\u2082 = r \u2022 (p\u2081 -\u1d65 p\u2082)\nhr1 : r < 1\n\u22a2 \u2191(AffineMap.lineMap p\u2082 p\u2081) r = p\u2083\n[PROOFSTEP]\nrw [AffineMap.lineMap_apply, \u2190 hp\u2083p\u2082, vsub_vadd]\n[GOAL]\ncase mpr\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\np\u2081 p\u2082 p\u2083 : P\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 \u2192 \u2220 p\u2081 p\u2082 p\u2083 = 0\n[PROOFSTEP]\nrintro (\u27e8hp\u2081p\u2082, h\u27e9 | \u27e8hp\u2083p\u2082, h\u27e9)\n[GOAL]\ncase mpr.inl.intro\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0\n[PROOFSTEP]\nexact h.angle\u2082\u2081\u2083_eq_zero_of_ne hp\u2081p\u2082\n[GOAL]\ncase mpr.inr.intro\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2083p\u2082 : p\u2083 \u2260 p\u2082\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0\n[PROOFSTEP]\nexact h.angle\u2083\u2081\u2082_eq_zero_of_ne hp\u2083p\u2082\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2194 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nrw [angle_eq_zero_iff_ne_and_wbtw]\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\np\u2081 p\u2082 p\u2083 : P\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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 = 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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = 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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nby_cases hp\u2081p\u2083 : p\u2081 = p\u2083\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2081p\u2083 : p\u2081 = p\u2083\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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nsimp [hp\u2081p\u2083]\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\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2081p\u2083 : \u00acp\u2081 = p\u2083\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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2081p\u2083 : \u00acp\u2081 = p\u2083\nhp\u2083p\u2082 : p\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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \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\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2081p\u2083 : \u00acp\u2081 = p\u2083\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 p\u2081 = p\u2083 \u2227 p\u2081 \u2260 p\u2082 \u2228 Sbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Sbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nsimp [hp\u2081p\u2082, hp\u2081p\u2083, Ne.symm hp\u2081p\u2083, Sbtw, hp\u2083p\u2082]\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} \u2194 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2083}\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nreplace h := h.wbtw_or_wbtw_or_wbtw\n[GOAL]\ncase refine'_1\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nby_cases h\u2081\u2082 : p\u2081 = p\u2082\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : p\u2081 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nexact Or.inl h\u2081\u2082\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nby_cases h\u2083\u2082 : p\u2083 = p\u2082\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : p\u2083 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nexact Or.inr (Or.inl h\u2083\u2082)\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nrw [or_iff_right h\u2081\u2082, or_iff_right h\u2083\u2082]\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nrcases h with (h | h | h)\n[GOAL]\ncase neg.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\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nexact Or.inr (angle_eq_pi_iff_sbtw.2 \u27e8h, Ne.symm h\u2081\u2082, Ne.symm h\u2083\u2082\u27e9)\n[GOAL]\ncase neg.inr.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\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nexact Or.inl (h.angle\u2083\u2081\u2082_eq_zero_of_ne h\u2083\u2082)\n[GOAL]\ncase neg.inr.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\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2083 p\u2081 p\u2082\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n[PROOFSTEP]\nexact Or.inl (h.angle\u2082\u2083\u2081_eq_zero_of_ne h\u2081\u2082)\n[GOAL]\ncase refine'_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\np\u2081 p\u2082 p\u2083 : P\nh : p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrcases h with (rfl | rfl | h | h)\n[GOAL]\ncase refine'_2.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\np\u2081 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2081, p\u2083}\n[PROOFSTEP]\nsimpa using collinear_pair \u211d p\u2081 p\u2083\n[GOAL]\ncase refine'_2.inr.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\np\u2081 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2083, p\u2083}\n[PROOFSTEP]\nsimpa using collinear_pair \u211d p\u2081 p\u2083\n[GOAL]\ncase refine'_2.inr.inr.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\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = 0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrw [angle_eq_zero_iff_ne_and_wbtw] at h \n[GOAL]\ncase refine'_2.inr.inr.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\np\u2081 p\u2082 p\u2083 : P\nh : 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\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrcases h with (\u27e8-, h\u27e9 | \u27e8-, h\u27e9)\n[GOAL]\ncase refine'_2.inr.inr.inl.inl.intro\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrw [Set.insert_comm]\n[GOAL]\ncase refine'_2.inr.inr.inl.inl.intro\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 Collinear \u211d {p\u2082, p\u2081, p\u2083}\n[PROOFSTEP]\nexact h.collinear\n[GOAL]\ncase refine'_2.inr.inr.inl.inr.intro\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrw [Set.insert_comm, Set.pair_comm]\n[GOAL]\ncase refine'_2.inr.inr.inl.inr.intro\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\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2082, p\u2083, p\u2081}\n[PROOFSTEP]\nexact h.collinear\n[GOAL]\ncase refine'_2.inr.inr.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\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrw [angle_eq_pi_iff_sbtw] at h \n[GOAL]\ncase refine'_2.inr.inr.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\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nexact h.wbtw.collinear\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} \u2194 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 sin (\u2220 p\u2081 p\u2082 p\u2083) = 0\n[PROOFSTEP]\nrw [sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi, collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi]\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\np\u2081 p\u2082 p\u2083 : P\nh : sin (\u2220 p\u2081 p\u2082 p\u2083) = 0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrevert 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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 sin (\u2220 p\u2081 p\u2082 p\u2083) = 0 \u2192 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\ncontrapose\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\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u00acCollinear \u211d {p\u2081, p\u2082, p\u2083} \u2192 \u00acsin (\u2220 p\u2081 p\u2082 p\u2083) = 0\n[PROOFSTEP]\nexact sin_ne_zero_of_not_collinear\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine", "llama_tokens": 27059, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424373085145, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.5599486621903119}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\n\u22a2 z \u2022 r \u2208 zmultiples p \u2194 \u2203 k, r - \u2191k \u2022 (p / \u2191z) \u2208 zmultiples p\n[PROOFSTEP]\nrw [AddSubgroup.mem_zmultiples_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\n\u22a2 (\u2203 k, k \u2022 p = z \u2022 r) \u2194 \u2203 k, r - \u2191k \u2022 (p / \u2191z) \u2208 zmultiples p\n[PROOFSTEP]\nsimp_rw [AddSubgroup.mem_zmultiples_iff, div_eq_mul_inv, \u2190 smul_mul_assoc, eq_sub_iff_add_eq]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\n\u22a2 (\u2203 k, k \u2022 p = z \u2022 r) \u2194 \u2203 k k_1, k_1 \u2022 p + \u2191k \u2022 p * (\u2191z)\u207b\u00b9 = r\n[PROOFSTEP]\nhave hz' : (z : R) \u2260 0 := Int.cast_ne_zero.mpr hz\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n\u22a2 (\u2203 k, k \u2022 p = z \u2022 r) \u2194 \u2203 k k_1, k_1 \u2022 p + \u2191k \u2022 p * (\u2191z)\u207b\u00b9 = r\n[PROOFSTEP]\nconv_rhs => simp (config := { singlePass := true }) only [\u2190 (mul_right_injective\u2080 hz').eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n| \u2203 k k_1, k_1 \u2022 p + \u2191k \u2022 p * (\u2191z)\u207b\u00b9 = r\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 (mul_right_injective\u2080 hz').eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n| \u2203 k k_1, k_1 \u2022 p + \u2191k \u2022 p * (\u2191z)\u207b\u00b9 = r\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 (mul_right_injective\u2080 hz').eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n| \u2203 k k_1, k_1 \u2022 p + \u2191k \u2022 p * (\u2191z)\u207b\u00b9 = r\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 (mul_right_injective\u2080 hz').eq_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n\u22a2 (\u2203 k, k \u2022 p = z \u2022 r) \u2194\n    \u2203 k k_1, (fun x x_1 => x * x_1) (\u2191z) (k_1 \u2022 p + \u2191k \u2022 p * (\u2191z)\u207b\u00b9) = (fun x x_1 => x * x_1) (\u2191z) r\n[PROOFSTEP]\nsimp_rw [\u2190 zsmul_eq_mul, smul_add, \u2190 mul_smul_comm, zsmul_eq_mul (z : R)\u207b\u00b9, mul_inv_cancel hz', mul_one, \u2190\n  coe_nat_zsmul, smul_smul, \u2190 add_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n\u22a2 (\u2203 k, k \u2022 p = z \u2022 r) \u2194 \u2203 k k_1, (z * k_1 + \u2191\u2191k) \u2022 p = z \u2022 r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n\u22a2 (\u2203 k, k \u2022 p = z \u2022 r) \u2192 \u2203 k k_1, (z * k_1 + \u2191\u2191k) \u2022 p = z \u2022 r\n[PROOFSTEP]\nrintro \u27e8k, h\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 \u2203 k k_1, (z * k_1 + \u2191\u2191k) \u2022 p = z \u2022 r\n[PROOFSTEP]\nsimp_rw [\u2190 h]\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 \u2203 k_1 k_2, (z * k_2 + \u2191\u2191k_1) \u2022 p = k \u2022 p\n[PROOFSTEP]\nrefine' \u27e8\u27e8(k % z).toNat, _\u27e9, k / z, _\u27e9\n[GOAL]\ncase mp.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 Int.toNat (k % z) < Int.natAbs z\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_lt, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n[GOAL]\ncase mp.intro.refine'_1\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 k % z < \u2191(Int.natAbs z)\n[PROOFSTEP]\nexact (Int.emod_lt _ hz).trans_eq (Int.abs_eq_natAbs _)\n[GOAL]\ncase mp.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 (z * (k / z) + \u2191\u2191{ val := Int.toNat (k % z), isLt := (_ : Int.toNat (k % z) < Int.natAbs z) }) \u2022 p = k \u2022 p\n[PROOFSTEP]\nrw [Fin.val_mk, Int.toNat_of_nonneg (Int.emod_nonneg _ hz)]\n[GOAL]\ncase mp.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 (z * (k / z) + k % z) \u2022 p = k \u2022 p\n[PROOFSTEP]\nnth_rewrite 3 [\u2190 Int.div_add_mod k z]\n[GOAL]\ncase mp.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 (z * (k / z) + k % z) \u2022 p = (z * Int.div k z + Int.mod k z) \u2022 p\n[PROOFSTEP]\nrw [Int.mod_def, \u2190 Int.div_def', Int.emod_def]\n[GOAL]\ncase mp.intro.refine'_2\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : \u2124\nh : k \u2022 p = z \u2022 r\n\u22a2 (z * (k / z) + (k - z * (k / z))) \u2022 p = (z * (k / z) + (k - z * (k / z))) \u2022 p\n[PROOFSTEP]\nsimp only [add_sub_cancel'_right, zsmul_eq_mul, Int.div_def']\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\n\u22a2 (\u2203 k k_1, (z * k_1 + \u2191\u2191k) \u2022 p = z \u2022 r) \u2192 \u2203 k, k \u2022 p = z \u2022 r\n[PROOFSTEP]\nrintro \u27e8k, n, h\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nz : \u2124\nhz : z \u2260 0\nhz' : \u2191z \u2260 0\nk : Fin (Int.natAbs z)\nn : \u2124\nh : (z * n + \u2191\u2191k) \u2022 p = z \u2022 r\n\u22a2 \u2203 k, k \u2022 p = z \u2022 r\n[PROOFSTEP]\nexact \u27e8_, h\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nn : \u2115\nhn : n \u2260 0\n\u22a2 n \u2022 r \u2208 zmultiples p \u2194 \u2203 k, r - \u2191k \u2022 (p / \u2191n) \u2208 zmultiples p\n[PROOFSTEP]\nrw [\u2190 coe_nat_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.coe_nat_ne_zero.mpr hn), Int.cast_ofNat]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np r : R\nn : \u2115\nhn : n \u2260 0\n\u22a2 (\u2203 k, r - \u2191k \u2022 (p / \u2191n) \u2208 zmultiples p) \u2194 \u2203 k, r - \u2191k \u2022 (p / \u2191n) \u2208 zmultiples p\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\n\u03c8 \u03b8 : R \u29f8 AddSubgroup.zmultiples p\nz : \u2124\nhz : z \u2260 0\n\u22a2 z \u2022 \u03c8 = z \u2022 \u03b8 \u2194 \u2203 k, \u03c8 = \u03b8 + \u2191(\u2191k \u2022 (p / \u2191z))\n[PROOFSTEP]\ninduction \u03c8 using Quotient.inductionOn'\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\n\u03b8 : R \u29f8 AddSubgroup.zmultiples p\nz : \u2124\nhz : z \u2260 0\na\u271d : R\n\u22a2 z \u2022 Quotient.mk'' a\u271d = z \u2022 \u03b8 \u2194 \u2203 k, Quotient.mk'' a\u271d = \u03b8 + \u2191(\u2191k \u2022 (p / \u2191z))\n[PROOFSTEP]\ninduction \u03b8 using Quotient.inductionOn'\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\nz : \u2124\nhz : z \u2260 0\na\u271d\u00b9 a\u271d : R\n\u22a2 z \u2022 Quotient.mk'' a\u271d\u00b9 = z \u2022 Quotient.mk'' a\u271d \u2194 \u2203 k, Quotient.mk'' a\u271d\u00b9 = Quotient.mk'' a\u271d + \u2191(\u2191k \u2022 (p / \u2191z))\n[PROOFSTEP]\nlet Zp := AddSubgroup.zmultiples p\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\nz : \u2124\nhz : z \u2260 0\na\u271d\u00b9 a\u271d : R\nZp : AddSubgroup R := AddSubgroup.zmultiples p\n\u22a2 z \u2022 Quotient.mk'' a\u271d\u00b9 = z \u2022 Quotient.mk'' a\u271d \u2194 \u2203 k, Quotient.mk'' a\u271d\u00b9 = Quotient.mk'' a\u271d + \u2191(\u2191k \u2022 (p / \u2191z))\n[PROOFSTEP]\nhave : (Quotient.mk'' : R \u2192 R \u29f8 Zp) = ((\u2191) : R \u2192 R \u29f8 Zp) := rfl\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\nz : \u2124\nhz : z \u2260 0\na\u271d\u00b9 a\u271d : R\nZp : AddSubgroup R := AddSubgroup.zmultiples p\nthis : Quotient.mk'' = mk\n\u22a2 z \u2022 Quotient.mk'' a\u271d\u00b9 = z \u2022 Quotient.mk'' a\u271d \u2194 \u2203 k, Quotient.mk'' a\u271d\u00b9 = Quotient.mk'' a\u271d + \u2191(\u2191k \u2022 (p / \u2191z))\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\nz : \u2124\nhz : z \u2260 0\na\u271d\u00b9 a\u271d : R\nZp : AddSubgroup R := AddSubgroup.zmultiples p\nthis : Quotient.mk'' = mk\n\u22a2 z \u2022 \u2191a\u271d\u00b9 = z \u2022 \u2191a\u271d \u2194 \u2203 k, \u2191a\u271d\u00b9 = \u2191a\u271d + \u2191(\u2191k \u2022 (p / \u2191z))\n[PROOFSTEP]\nsimp_rw [\u2190 QuotientAddGroup.mk_zsmul, \u2190 QuotientAddGroup.mk_add, QuotientAddGroup.eq_iff_sub_mem, \u2190 smul_sub, \u2190 sub_sub]\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\nz : \u2124\nhz : z \u2260 0\na\u271d\u00b9 a\u271d : R\nZp : AddSubgroup R := AddSubgroup.zmultiples p\nthis : Quotient.mk'' = mk\n\u22a2 z \u2022 (a\u271d\u00b9 - a\u271d) \u2208 AddSubgroup.zmultiples p \u2194 \u2203 k, a\u271d\u00b9 - a\u271d - \u2191k \u2022 (p / \u2191z) \u2208 AddSubgroup.zmultiples p\n[PROOFSTEP]\nexact AddSubgroup.zsmul_mem_zmultiples_iff_exists_sub_div hz\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\n\u03c8 \u03b8 : R \u29f8 AddSubgroup.zmultiples p\nn : \u2115\nhz : n \u2260 0\n\u22a2 n \u2022 \u03c8 = n \u2022 \u03b8 \u2194 \u2203 k, \u03c8 = \u03b8 + \u2191(\u2191k \u2022 (p / \u2191n))\n[PROOFSTEP]\nrw [\u2190 coe_nat_zsmul \u03c8, \u2190 coe_nat_zsmul \u03b8, zmultiples_zsmul_eq_zsmul_iff (Int.coe_nat_ne_zero.mpr hz), Int.cast_ofNat]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\np : R\n\u03c8 \u03b8 : R \u29f8 AddSubgroup.zmultiples p\nn : \u2115\nhz : n \u2260 0\n\u22a2 (\u2203 k, \u03c8 = \u03b8 + \u2191(\u2191k \u2022 (p / \u2191n))) \u2194 \u2203 k, \u03c8 = \u03b8 + \u2191(\u2191k \u2022 (p / \u2191n))\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharZero.Quotient", "llama_tokens": 4617, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.819893335913536, "lm_q2_score": 0.6825737214979745, "lm_q1q2_score": 0.5596376455258911}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\n\u22a2 annIdealGenerator \ud835\udd5c a = 0 \u2194 annIdeal \ud835\udd5c a = \u22a5\n[PROOFSTEP]\nsimp only [annIdealGenerator, mul_eq_zero, IsPrincipal.eq_bot_iff_generator_eq_zero, Polynomial.C_eq_zero, inv_eq_zero,\n  Polynomial.leadingCoeff_eq_zero, or_self_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\n\u22a2 Ideal.span {annIdealGenerator \ud835\udd5c a} = annIdeal \ud835\udd5c a\n[PROOFSTEP]\nby_cases h : annIdealGenerator \ud835\udd5c a = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : annIdealGenerator \ud835\udd5c a = 0\n\u22a2 Ideal.span {annIdealGenerator \ud835\udd5c a} = annIdeal \ud835\udd5c a\n[PROOFSTEP]\nrw [h, annIdealGenerator_eq_zero_iff.mp h, Set.singleton_zero, Ideal.span_zero]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 Ideal.span {annIdealGenerator \ud835\udd5c a} = annIdeal \ud835\udd5c a\n[PROOFSTEP]\nrw [annIdealGenerator, Ideal.span_singleton_mul_right_unit, Ideal.span_singleton_generator]\n[GOAL]\ncase neg.h2\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 IsUnit (\u2191C (leadingCoeff (IsPrincipal.generator (annIdeal \ud835\udd5c a)))\u207b\u00b9)\n[PROOFSTEP]\napply Polynomial.isUnit_C.mpr\n[GOAL]\ncase neg.h2\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 IsUnit (leadingCoeff (IsPrincipal.generator (annIdeal \ud835\udd5c a)))\u207b\u00b9\n[PROOFSTEP]\napply IsUnit.mk0\n[GOAL]\ncase neg.h2.hx\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 (leadingCoeff (IsPrincipal.generator (annIdeal \ud835\udd5c a)))\u207b\u00b9 \u2260 0\n[PROOFSTEP]\napply inv_eq_zero.not.mpr\n[GOAL]\ncase neg.h2.hx\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 \u00acleadingCoeff (IsPrincipal.generator (annIdeal \ud835\udd5c a)) = 0\n[PROOFSTEP]\napply Polynomial.leadingCoeff_eq_zero.not.mpr\n[GOAL]\ncase neg.h2.hx\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 \u00acIsPrincipal.generator (annIdeal \ud835\udd5c a) = 0\n[PROOFSTEP]\napply (mul_ne_zero_iff.mp h).1\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\np : \ud835\udd5c[X]\na : A\n\u22a2 p \u2208 annIdeal \ud835\udd5c a \u2194 \u2203 s, p = s \u2022 annIdealGenerator \ud835\udd5c a\n[PROOFSTEP]\nsimp_rw [@eq_comm _ p, \u2190 mem_span_singleton, \u2190 span_singleton_annIdealGenerator \ud835\udd5c a, Ideal.span]\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\np : \ud835\udd5c[X]\na : A\n\u22a2 p \u2208 annIdeal \ud835\udd5c a \u2194 annIdealGenerator \ud835\udd5c a \u2223 p\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_span_singleton, span_singleton_annIdealGenerator]\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\n\u22a2 annIdealGenerator \ud835\udd5c a = minpoly \ud835\udd5c a\n[PROOFSTEP]\nby_cases h : annIdealGenerator \ud835\udd5c a = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : annIdealGenerator \ud835\udd5c a = 0\n\u22a2 annIdealGenerator \ud835\udd5c a = minpoly \ud835\udd5c a\n[PROOFSTEP]\nrw [h, minpoly.eq_zero]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : annIdealGenerator \ud835\udd5c a = 0\n\u22a2 \u00acIsIntegral \ud835\udd5c a\n[PROOFSTEP]\nrintro \u27e8p, p_monic, hp : aeval a p = 0\u27e9\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : annIdealGenerator \ud835\udd5c a = 0\np : \ud835\udd5c[X]\np_monic : Monic p\nhp : \u2191(aeval a) p = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' p_monic.ne_zero (Ideal.mem_bot.mp _)\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : annIdealGenerator \ud835\udd5c a = 0\np : \ud835\udd5c[X]\np_monic : Monic p\nhp : \u2191(aeval a) p = 0\n\u22a2 p \u2208 \u22a5\n[PROOFSTEP]\nsimpa only [annIdealGenerator_eq_zero_iff.mp h] using mem_annIdeal_iff_aeval_eq_zero.mpr hp\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nh : \u00acannIdealGenerator \ud835\udd5c a = 0\n\u22a2 annIdealGenerator \ud835\udd5c a = minpoly \ud835\udd5c a\n[PROOFSTEP]\nexact\n  minpoly.unique _ _ (monic_annIdealGenerator _ _ h) (annIdealGenerator_aeval_eq_zero _ _) fun q q_monic hq =>\n    degree_annIdealGenerator_le_of_mem a q (mem_annIdeal_iff_aeval_eq_zero.mpr hq) q_monic.ne_zero\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\np : \ud835\udd5c[X]\np_monic : Monic p\np_gen : Ideal.span {p} = annIdeal \ud835\udd5c a\n\u22a2 annIdealGenerator \ud835\udd5c a = p\n[PROOFSTEP]\nby_cases h : p = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\np : \ud835\udd5c[X]\np_monic : Monic p\np_gen : Ideal.span {p} = annIdeal \ud835\udd5c a\nh : p = 0\n\u22a2 annIdealGenerator \ud835\udd5c a = p\n[PROOFSTEP]\nrwa [h, annIdealGenerator_eq_zero_iff, \u2190 p_gen, Ideal.span_singleton_eq_bot.mpr]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\np : \ud835\udd5c[X]\np_monic : Monic p\np_gen : Ideal.span {p} = annIdeal \ud835\udd5c a\nh : \u00acp = 0\n\u22a2 annIdealGenerator \ud835\udd5c a = p\n[PROOFSTEP]\nrw [\u2190 span_singleton_annIdealGenerator, Ideal.span_singleton_eq_span_singleton] at p_gen \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\np : \ud835\udd5c[X]\np_monic : Monic p\np_gen : Associated p (annIdealGenerator \ud835\udd5c a)\nh : \u00acp = 0\n\u22a2 annIdealGenerator \ud835\udd5c a = p\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\np : \ud835\udd5c[X]\np_monic : Monic p\np_gen : Associated p (annIdealGenerator \ud835\udd5c a)\nh : \u00acp = 0\n\u22a2 p = annIdealGenerator \ud835\udd5c a\n[PROOFSTEP]\napply eq_of_monic_of_associated p_monic _ p_gen\n[GOAL]\n\ud835\udd5c : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\np : \ud835\udd5c[X]\np_monic : Monic p\np_gen : Associated p (annIdealGenerator \ud835\udd5c a)\nh : \u00acp = 0\n\u22a2 Monic (annIdealGenerator \ud835\udd5c a)\n[PROOFSTEP]\napply monic_annIdealGenerator _ _ ((Associated.ne_zero_iff p_gen).mp h)\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AnnihilatingPolynomial", "llama_tokens": 3037, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737869342624, "lm_q2_score": 0.6959583376458152, "lm_q1q2_score": 0.5589754935954634}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\na : \u03b1\nl : List (Sigma \u03b2)\nh\u2081 : \u00aca \u2208 keys l\ns : Sigma \u03b2\nh\u2082 : s \u2208 l\ne : a = s.fst\n\u22a2 \u00acs.fst \u2208 keys l\n[PROOFSTEP]\nrwa [e] at h\u2081 \n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ns : Sigma \u03b2\nl : List (Sigma \u03b2)\n\u22a2 NodupKeys (s :: l) \u2194 \u00acs.fst \u2208 keys l \u2227 NodupKeys l\n[PROOFSTEP]\nsimp [keys, NodupKeys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\na : \u03b1\nb b' : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\nh' : { fst := a, snd := b' } \u2208 l\n\u22a2 b = b'\n[PROOFSTEP]\ncases nd.eq_of_fst_eq h h' rfl\n[GOAL]\ncase refl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh h' : { fst := a, snd := b } \u2208 l\n\u22a2 b = b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\nL : List (List (Sigma \u03b2))\n\u22a2 NodupKeys (join L) \u2194 (\u2200 (l : List (Sigma \u03b2)), l \u2208 L \u2192 NodupKeys l) \u2227 Pairwise Disjoint (map keys L)\n[PROOFSTEP]\nrw [nodupKeys_iff_pairwise, pairwise_join, pairwise_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\nL : List (List (Sigma \u03b2))\n\u22a2 (\u2200 (l : List (Sigma \u03b2)), l \u2208 L \u2192 Pairwise (fun s s' => s.fst \u2260 s'.fst) l) \u2227\n      Pairwise (fun l\u2081 l\u2082 => \u2200 (x : Sigma \u03b2), x \u2208 l\u2081 \u2192 \u2200 (y : Sigma \u03b2), y \u2208 l\u2082 \u2192 x.fst \u2260 y.fst) L \u2194\n    (\u2200 (l : List (Sigma \u03b2)), l \u2208 L \u2192 NodupKeys l) \u2227 Pairwise (fun a b => Disjoint (keys a) (keys b)) L\n[PROOFSTEP]\nrefine' and_congr (ball_congr fun l _ => by simp [nodupKeys_iff_pairwise]) _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\nL : List (List (Sigma \u03b2))\nl : List (Sigma \u03b2)\nx\u271d : l \u2208 L\n\u22a2 Pairwise (fun s s' => s.fst \u2260 s'.fst) l \u2194 NodupKeys l\n[PROOFSTEP]\nsimp [nodupKeys_iff_pairwise]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\nL : List (List (Sigma \u03b2))\n\u22a2 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : Sigma \u03b2), x \u2208 l\u2081 \u2192 \u2200 (y : Sigma \u03b2), y \u2208 l\u2082 \u2192 x.fst \u2260 y.fst) L \u2194\n    Pairwise (fun a b => Disjoint (keys a) (keys b)) L\n[PROOFSTEP]\napply iff_of_eq\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\nL : List (List (Sigma \u03b2))\n\u22a2 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : Sigma \u03b2), x \u2208 l\u2081 \u2192 \u2200 (y : Sigma \u03b2), y \u2208 l\u2082 \u2192 x.fst \u2260 y.fst) L =\n    Pairwise (fun a b => Disjoint (keys a) (keys b)) L\n[PROOFSTEP]\ncongr with (l\u2081 l\u2082)\n[GOAL]\ncase a.e_R.h.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\nL : List (List (Sigma \u03b2))\nl\u2081 l\u2082 : List (Sigma \u03b2)\n\u22a2 (\u2200 (x : Sigma \u03b2), x \u2208 l\u2081 \u2192 \u2200 (y : Sigma \u03b2), y \u2208 l\u2082 \u2192 x.fst \u2260 y.fst) \u2194 Disjoint (keys l\u2081) (keys l\u2082)\n[PROOFSTEP]\nsimp [keys, disjoint_iff_ne]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\nl : List \u03b1\n\u22a2 Nodup (map Prod.fst (enum l))\n[PROOFSTEP]\nsimp [List.nodup_range]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 Option.isSome (dlookup a []) = true \u2194 a \u2208 keys []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb : \u03b2 a\n\u22a2 Option.isSome (dlookup a ({ fst := a, snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a, snd := b } :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)\n[PROOFSTEP]\nsimp [h, dlookup_isSome]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\n\u22a2 dlookup a l = none \u2194 \u00aca \u2208 keys l\n[PROOFSTEP]\nsimp [\u2190 dlookup_isSome, Option.isNone_iff_eq_none]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nH : b \u2208 dlookup a ({ fst := a', snd := b' } :: l)\n\u22a2 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nH : b \u2208 dlookup a ({ fst := a', snd := b' } :: l)\nh : a = a'\n\u22a2 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nb' : \u03b2 a\nH : b \u2208 dlookup a ({ fst := a, snd := b' } :: l)\n\u22a2 { fst := a, snd := b } \u2208 { fst := a, snd := b' } :: l\n[PROOFSTEP]\nsimp at H \n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nb' : \u03b2 a\nH : b' = b\n\u22a2 { fst := a, snd := b } \u2208 { fst := a, snd := b' } :: l\n[PROOFSTEP]\nsimp [H]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nH : b \u2208 dlookup a ({ fst := a', snd := b' } :: l)\nh : \u00aca = a'\n\u22a2 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsimp only [ne_eq, h, not_false_iff, dlookup_cons_ne] at H \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\nH : b \u2208 dlookup a l\n\u22a2 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsimp [of_mem_dlookup H]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\n\u22a2 b \u2208 dlookup a l\n[PROOFSTEP]\ncases' Option.isSome_iff_exists.mp (dlookup_isSome.mpr (mem_keys_of_mem h)) with b' h'\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\nb' : \u03b2 { fst := a, snd := b }.fst\nh' : dlookup { fst := a, snd := b }.fst l = some b'\n\u22a2 b \u2208 dlookup a l\n[PROOFSTEP]\ncases nd.eq_of_mk_mem h (of_mem_dlookup h')\n[GOAL]\ncase intro.refl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\nh' : dlookup { fst := a, snd := b }.fst l = some b\n\u22a2 b \u2208 dlookup a l\n[PROOFSTEP]\nexact h'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 Option.map (Sigma.mk a) (dlookup a ({ fst := a', snd := b' } :: l)) =\n    find? (fun s => decide (a = s.fst)) ({ fst := a', snd := b' } :: l)\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 Option.map (Sigma.mk a) (dlookup a ({ fst := a', snd := b' } :: l)) =\n    find? (fun s => decide (a = s.fst)) ({ fst := a', snd := b' } :: l)\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb' : \u03b2 a\n\u22a2 Option.map (Sigma.mk a) (dlookup a ({ fst := a, snd := b' } :: l)) =\n    find? (fun s => decide (a = s.fst)) ({ fst := a, snd := b' } :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 Option.map (Sigma.mk a) (dlookup a ({ fst := a', snd := b' } :: l)) =\n    find? (fun s => decide (a = s.fst)) ({ fst := a', snd := b' } :: l)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 Option.map (Sigma.mk a) (dlookup a l) = find? (fun s => decide (a = s.fst)) l\n[PROOFSTEP]\nexact map_dlookup_eq_find a l\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys l\u2081\nnd\u2082 : NodupKeys l\u2082\np : l\u2081 ~ l\u2082\n\u22a2 dlookup a l\u2081 = dlookup a l\u2082\n[PROOFSTEP]\next b\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys l\u2081\nnd\u2082 : NodupKeys l\u2082\np : l\u2081 ~ l\u2082\nb : \u03b2 a\n\u22a2 b \u2208 dlookup a l\u2081 \u2194 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nsimp only [mem_dlookup_iff nd\u2081, mem_dlookup_iff nd\u2082]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys l\u2081\nnd\u2082 : NodupKeys l\u2082\np : l\u2081 ~ l\u2082\nb : \u03b2 a\n\u22a2 { fst := a, snd := b } \u2208 l\u2081 \u2194 { fst := a, snd := b } \u2208 l\u2082\n[PROOFSTEP]\nexact p.mem_iff\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2080 l\u2081 : List (Sigma \u03b2)\nnd\u2080 : NodupKeys l\u2080\nnd\u2081 : NodupKeys l\u2081\nh : \u2200 (x : \u03b1) (y : \u03b2 x), y \u2208 dlookup x l\u2080 \u2194 y \u2208 dlookup x l\u2081\nx\u271d : Sigma \u03b2\na : \u03b1\nb : \u03b2 a\n\u22a2 { fst := a, snd := b } \u2208 l\u2080 \u2194 { fst := a, snd := b } \u2208 l\u2081\n[PROOFSTEP]\nrw [\u2190 mem_dlookup_iff, \u2190 mem_dlookup_iff, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2080 l\u2081 : List (Sigma \u03b2)\nnd\u2080 : NodupKeys l\u2080\nnd\u2081 : NodupKeys l\u2081\nh : \u2200 (x : \u03b1) (y : \u03b2 x), y \u2208 dlookup x l\u2080 \u2194 y \u2208 dlookup x l\u2081\nx\u271d : Sigma \u03b2\na : \u03b1\nb : \u03b2 a\n\u22a2 NodupKeys l\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2080 l\u2081 : List (Sigma \u03b2)\nnd\u2080 : NodupKeys l\u2080\nnd\u2081 : NodupKeys l\u2081\nh : \u2200 (x : \u03b1) (y : \u03b2 x), y \u2208 dlookup x l\u2080 \u2194 y \u2208 dlookup x l\u2081\nx\u271d : Sigma \u03b2\na : \u03b1\nb : \u03b2 a\n\u22a2 NodupKeys l\u2080\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 lookupAll a [] = [] \u2194 \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 lookupAll a ({ fst := a', snd := b } :: l) = [] \u2194\n    \u2200 (b_1 : \u03b2 a), \u00ac{ fst := a, snd := b_1 } \u2208 { fst := a', snd := b } :: l\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 lookupAll a ({ fst := a', snd := b } :: l) = [] \u2194\n    \u2200 (b_1 : \u03b2 a), \u00ac{ fst := a, snd := b_1 } \u2208 { fst := a', snd := b } :: l\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb : \u03b2 a\n\u22a2 lookupAll a ({ fst := a, snd := b } :: l) = [] \u2194\n    \u2200 (b_1 : \u03b2 a), \u00ac{ fst := a, snd := b_1 } \u2208 { fst := a, snd := b } :: l\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 lookupAll a ({ fst := a', snd := b } :: l) = [] \u2194\n    \u2200 (b_1 : \u03b2 a), \u00ac{ fst := a, snd := b_1 } \u2208 { fst := a', snd := b } :: l\n[PROOFSTEP]\nsimp [h, lookupAll_eq_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 head? (lookupAll a []) = dlookup a []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 head? (lookupAll a ({ fst := a', snd := b } :: l)) = dlookup a ({ fst := a', snd := b } :: l)\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 head? (lookupAll a ({ fst := a', snd := b } :: l)) = dlookup a ({ fst := a', snd := b } :: l)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb : \u03b2 a\n\u22a2 head? (lookupAll a ({ fst := a, snd := b } :: l)) = dlookup a ({ fst := a, snd := b } :: l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 head? (lookupAll a ({ fst := a', snd := b } :: l)) = dlookup a ({ fst := a', snd := b } :: l)\n[PROOFSTEP]\nrw [lookupAll_cons_ne, dlookup_cons_ne, head?_lookupAll a l]\n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 a \u2260 { fst := a', snd := b }.fst\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 a \u2260 { fst := a', snd := b }.fst\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\n\u22a2 b \u2208 lookupAll a [] \u2194 { fst := a, snd := b } \u2208 []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 b \u2208 lookupAll a ({ fst := a', snd := b' } :: l) \u2194 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 b \u2208 lookupAll a ({ fst := a', snd := b' } :: l) \u2194 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nb' : \u03b2 a\n\u22a2 b \u2208 lookupAll a ({ fst := a, snd := b' } :: l) \u2194 { fst := a, snd := b } \u2208 { fst := a, snd := b' } :: l\n[PROOFSTEP]\nsimp [*, mem_lookupAll]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 b \u2208 lookupAll a ({ fst := a', snd := b' } :: l) \u2194 { fst := a, snd := b } \u2208 { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsimp [*, mem_lookupAll]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 map (Sigma.mk a) (lookupAll a []) <+ []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb' : \u03b2 a\n\u22a2 map (Sigma.mk a) (lookupAll a ({ fst := a, snd := b' } :: l)) <+ { fst := a, snd := b' } :: l\n[PROOFSTEP]\nsimp only [ne_eq, not_true, lookupAll_cons_eq, List.map]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb' : \u03b2 a\n\u22a2 { fst := a, snd := b' } :: map (Sigma.mk a) (lookupAll a l) <+ { fst := a, snd := b' } :: l\n[PROOFSTEP]\nexact (lookupAll_sublist a l).cons\u2082 _\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l\n[PROOFSTEP]\nsimp only [ne_eq, h, not_false_iff, lookupAll_cons_ne]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 map (Sigma.mk a) (lookupAll a l) <+ { fst := a', snd := b' } :: l\n[PROOFSTEP]\nexact (lookupAll_sublist a l).cons _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 length (lookupAll a l) \u2264 1\n[PROOFSTEP]\nhave := Nodup.sublist ((lookupAll_sublist a l).map _) h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nthis : Nodup (map Sigma.fst (map (Sigma.mk a) (lookupAll a l)))\n\u22a2 length (lookupAll a l) \u2264 1\n[PROOFSTEP]\nrw [map_map] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nthis : Nodup (map (Sigma.fst \u2218 Sigma.mk a) (lookupAll a l))\n\u22a2 length (lookupAll a l) \u2264 1\n[PROOFSTEP]\nrwa [\u2190 nodup_replicate, \u2190 map_const]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 lookupAll a l = Option.toList (dlookup a l)\n[PROOFSTEP]\nrw [\u2190 head?_lookupAll]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 lookupAll a l = Option.toList (head? (lookupAll a l))\n[PROOFSTEP]\nhave h1 := lookupAll_length_le_one a h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nh1 : length (lookupAll a l) \u2264 1\n\u22a2 lookupAll a l = Option.toList (head? (lookupAll a l))\n[PROOFSTEP]\nrevert h1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 length (lookupAll a l) \u2264 1 \u2192 lookupAll a l = Option.toList (head? (lookupAll a l))\n[PROOFSTEP]\nrcases lookupAll a l with (_ | \u27e8b, _ | \u27e8c, l\u27e9\u27e9)\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 length [] \u2264 1 \u2192 [] = Option.toList (head? [])\n[PROOFSTEP]\nintro h1\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nb : \u03b2 a\n\u22a2 length [b] \u2264 1 \u2192 [b] = Option.toList (head? [b])\n[PROOFSTEP]\nintro h1\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u271d : List (Sigma \u03b2)\nh : NodupKeys l\u271d\nb c : \u03b2 a\nl : List (\u03b2 a)\n\u22a2 length (b :: c :: l) \u2264 1 \u2192 b :: c :: l = Option.toList (head? (b :: c :: l))\n[PROOFSTEP]\nintro h1\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nh1 : length [] \u2264 1\n\u22a2 [] = Option.toList (head? [])\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nh1 : length [] \u2264 1\n\u22a2 [] = Option.toList (head? [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nb : \u03b2 a\nh1 : length [b] \u2264 1\n\u22a2 [b] = Option.toList (head? [b])\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase cons.nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\nb : \u03b2 a\nh1 : length [b] \u2264 1\n\u22a2 [b] = Option.toList (head? [b])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u271d : List (Sigma \u03b2)\nh : NodupKeys l\u271d\nb c : \u03b2 a\nl : List (\u03b2 a)\nh1 : length (b :: c :: l) \u2264 1\n\u22a2 b :: c :: l = Option.toList (head? (b :: c :: l))\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u271d : List (Sigma \u03b2)\nh : NodupKeys l\u271d\nb c : \u03b2 a\nl : List (\u03b2 a)\nh1 : length (b :: c :: l) \u2264 1\n\u22a2 b :: c :: l = Option.toList (head? (b :: c :: l))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u271d : List (Sigma \u03b2)\nh : NodupKeys l\u271d\nb c : \u03b2 a\nl : List (\u03b2 a)\nh1 : length (b :: c :: l) \u2264 1\n\u22a2 b :: c :: l = Option.toList (head? (b :: c :: l))\n[PROOFSTEP]\nexact absurd h1 (by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u271d : List (Sigma \u03b2)\nh : NodupKeys l\u271d\nb c : \u03b2 a\nl : List (\u03b2 a)\nh1 : length (b :: c :: l) \u2264 1\n\u22a2 \u00aclength (b :: c :: l) \u2264 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 Nodup (lookupAll a l)\n[PROOFSTEP]\nrw [lookupAll_eq_dlookup a h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : NodupKeys l\n\u22a2 Nodup (Option.toList (dlookup a l))\n[PROOFSTEP]\napply Option.toList_nodup\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys l\u2081\nnd\u2082 : NodupKeys l\u2082\np : l\u2081 ~ l\u2082\n\u22a2 lookupAll a l\u2081 = lookupAll a l\u2082\n[PROOFSTEP]\nsimp [lookupAll_eq_dlookup, nd\u2081, nd\u2082, perm_dlookup a nd\u2081 nd\u2082 p]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nH : \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 l\n\u22a2 \u2200 (a_1 : Sigma \u03b2), a_1 \u2208 l \u2192 (if a = a_1.fst then some { fst := a, snd := b } else none) = none\n[PROOFSTEP]\nrintro \u27e8a', b'\u27e9 h\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nH : \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh : { fst := a', snd := b' } \u2208 l\n\u22a2 (if a = { fst := a', snd := b' }.fst then some { fst := a, snd := b } else none) = none\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nH : \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh : { fst := a', snd := b' } \u2208 l\n\u22a2 (if a = a' then some { fst := a, snd := b } else none) = none\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nH : \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh : { fst := a', snd := b' } \u2208 l\nh\u271d : a = a'\n\u22a2 False\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nH : \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 l\nb' : \u03b2 a\nh : { fst := a, snd := b' } \u2208 l\n\u22a2 False\n[PROOFSTEP]\nexact H _ h\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nH : \u2200 (b : \u03b2 a), \u00ac{ fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh : { fst := a', snd := b' } \u2208 l\nh\u271d : \u00aca = a'\n\u22a2 none = none\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\n\u22a2 kreplace a b l = l\n[PROOFSTEP]\nrefine' (lookmap_congr _).trans (lookmap_id' (Option.guard fun (s : Sigma \u03b2) => a = s.1) _ _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\n\u22a2 \u2200 (a_1 : Sigma \u03b2),\n    a_1 \u2208 l \u2192 (if a = a_1.fst then some { fst := a, snd := b } else none) = Option.guard (fun s => a = s.fst) a_1\n[PROOFSTEP]\nrintro \u27e8a', b'\u27e9 h'\n[GOAL]\ncase refine'_1.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh' : { fst := a', snd := b' } \u2208 l\n\u22a2 (if a = { fst := a', snd := b' }.fst then some { fst := a, snd := b } else none) =\n    Option.guard (fun s => a = s.fst) { fst := a', snd := b' }\n[PROOFSTEP]\ndsimp [Option.guard]\n[GOAL]\ncase refine'_1.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh' : { fst := a', snd := b' } \u2208 l\n\u22a2 (if a = a' then some { fst := a, snd := b } else none) = if a = a' then some { fst := a', snd := b' } else none\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh' : { fst := a', snd := b' } \u2208 l\nh\u271d : a = a'\n\u22a2 some { fst := a, snd := b } = some { fst := a', snd := b' }\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\nb' : \u03b2 a\nh' : { fst := a, snd := b' } \u2208 l\n\u22a2 some { fst := a, snd := b } = some { fst := a, snd := b' }\n[PROOFSTEP]\nsimp [nd.eq_of_mk_mem h h']\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na' : \u03b1\nb' : \u03b2 a'\nh' : { fst := a', snd := b' } \u2208 l\nh\u271d : \u00aca = a'\n\u22a2 none = none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\n\u22a2 \u2200 (a_1 b : Sigma \u03b2), b \u2208 Option.guard (fun s => a = s.fst) a_1 \u2192 a_1 = b\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9\n[GOAL]\ncase refine'_2.mk.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\n\u22a2 { fst := a\u2082, snd := b\u2082 } \u2208 Option.guard (fun s => a = s.fst) { fst := a\u2081, snd := b\u2081 } \u2192\n    { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n[PROOFSTEP]\ndsimp [Option.guard]\n[GOAL]\ncase refine'_2.mk.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\n\u22a2 ({ fst := a\u2082, snd := b\u2082 } \u2208 if a = a\u2081 then some { fst := a\u2081, snd := b\u2081 } else none) \u2192\n    { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\nh\u271d : a = a\u2081\n\u22a2 { fst := a\u2082, snd := b\u2082 } \u2208 some { fst := a\u2081, snd := b\u2081 } \u2192 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\nnd : NodupKeys l\nh : { fst := a, snd := b } \u2208 l\na\u2081 : \u03b1\nb\u2081 : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\nh\u271d : \u00aca = a\u2081\n\u22a2 { fst := a\u2082, snd := b\u2082 } \u2208 none \u2192 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\n\u22a2 \u2200 (a_1 b_1 : Sigma \u03b2), (b_1 \u2208 if a = a_1.fst then some { fst := a, snd := b } else none) \u2192 a_1.fst = b_1.fst\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2082\u27e9 \u27e8a\u2082, b\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na\u2081 : \u03b1\nb\u2082\u271d : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\n\u22a2 ({ fst := a\u2082, snd := b\u2082 } \u2208 if a = { fst := a\u2081, snd := b\u2082\u271d }.fst then some { fst := a, snd := b } else none) \u2192\n    { fst := a\u2081, snd := b\u2082\u271d }.fst = { fst := a\u2082, snd := b\u2082 }.fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na\u2081 : \u03b1\nb\u2082\u271d : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\n\u22a2 ({ fst := a\u2082, snd := b\u2082 } \u2208 if a = a\u2081 then some { fst := a, snd := b } else none) \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na\u2081 : \u03b1\nb\u2082\u271d : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\nh : a = a\u2081\n\u22a2 { fst := a\u2082, snd := b\u2082 } \u2208 some { fst := a, snd := b } \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nsimp (config := { contextual := true }) [h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\na\u2081 : \u03b1\nb\u2082\u271d : \u03b2 a\u2081\na\u2082 : \u03b1\nb\u2082 : \u03b2 a\u2082\nh : \u00aca = a\u2081\n\u22a2 { fst := a\u2082, snd := b\u2082 } \u2208 none \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nsimp (config := { contextual := true }) [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\n\u22a2 NodupKeys (kreplace a b l) \u2194 NodupKeys l\n[PROOFSTEP]\nsimp [NodupKeys, keys_kreplace]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\n\u22a2 Pairwise\n    (fun a_1 b_1 =>\n      \u2200 (c : Sigma \u03b2),\n        (c \u2208 if a = a_1.fst then some { fst := a, snd := b } else none) \u2192\n          \u2200 (d : Sigma \u03b2), (d \u2208 if a = b_1.fst then some { fst := a, snd := b } else none) \u2192 a_1 = b_1 \u2227 c = d)\n    l\u2081\n[PROOFSTEP]\nrefine' nd.pairwise_ne.imp _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\n\u22a2 \u2200 {a_1 b_1 : Sigma \u03b2},\n    a_1.fst \u2260 b_1.fst \u2192\n      \u2200 (c : Sigma \u03b2),\n        (c \u2208 if a = a_1.fst then some { fst := a, snd := b } else none) \u2192\n          \u2200 (d : Sigma \u03b2), (d \u2208 if a = b_1.fst then some { fst := a, snd := b } else none) \u2192 a_1 = b_1 \u2227 c = d\n[PROOFSTEP]\nintro x y h z h\u2081 w h\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nz : Sigma \u03b2\nh\u2081 : z \u2208 if a = x.fst then some { fst := a, snd := b } else none\nw : Sigma \u03b2\nh\u2082 : w \u2208 if a = y.fst then some { fst := a, snd := b } else none\n\u22a2 x = y \u2227 z = w\n[PROOFSTEP]\nsplit_ifs at h\u2081 h\u2082  with h_2 h_1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nz w : Sigma \u03b2\nh_2 : a = x.fst\nh\u2081 : z \u2208 some { fst := a, snd := b }\nh_1 : a = y.fst\nh\u2082 : w \u2208 some { fst := a, snd := b }\n\u22a2 x = y \u2227 z = w\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nz w : Sigma \u03b2\nh_2 : a = x.fst\nh\u2081 : z \u2208 some { fst := a, snd := b }\nh_1 : \u00aca = y.fst\nh\u2082 : w \u2208 none\n\u22a2 x = y \u2227 z = w\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nz w : Sigma \u03b2\nh_2 : \u00aca = x.fst\nh\u2081 : z \u2208 none\nh\u271d : a = y.fst\nh\u2082 : w \u2208 some { fst := a, snd := b }\n\u22a2 x = y \u2227 z = w\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nz w : Sigma \u03b2\nh_2 : \u00aca = x.fst\nh\u2081 : z \u2208 none\nh\u271d : \u00aca = y.fst\nh\u2082 : w \u2208 none\n\u22a2 x = y \u2227 z = w\n[PROOFSTEP]\ncases h\u2081\n[GOAL]\ncase pos.refl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nw : Sigma \u03b2\nh_2 : a = x.fst\nh_1 : a = y.fst\nh\u2082 : w \u2208 some { fst := a, snd := b }\n\u22a2 x = y \u2227 { fst := a, snd := b } = w\n[PROOFSTEP]\ncases h\u2082\n[GOAL]\ncase neg.refl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nw : Sigma \u03b2\nh_2 : a = x.fst\nh_1 : \u00aca = y.fst\nh\u2082 : w \u2208 none\n\u22a2 x = y \u2227 { fst := a, snd := b } = w\n[PROOFSTEP]\ncases h\u2082\n[GOAL]\ncase pos.refl.refl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\nh_2 : a = x.fst\nh_1 : a = y.fst\n\u22a2 x = y \u2227 { fst := a, snd := b } = { fst := a, snd := b }\n[PROOFSTEP]\nexact (h (h_2.symm.trans h_1)).elim\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl : List (Sigma \u03b2)\nh : a = s.fst\n\u22a2 kerase a (s :: l) = l\n[PROOFSTEP]\nsimp [kerase, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl : List (Sigma \u03b2)\nh : a \u2260 s.fst\n\u22a2 kerase a (s :: l) = s :: kerase a l\n[PROOFSTEP]\nsimp [kerase, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca \u2208 keys l\n\u22a2 kerase a l = l\n[PROOFSTEP]\ninduction' l with _ _ ih <;> [rfl; (simp [not_or] at h ; simp [h.1, ih h.2])]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca \u2208 keys l\n\u22a2 kerase a l = l\n[PROOFSTEP]\ninduction' l with _ _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh\u271d : \u00aca \u2208 keys l\nh : \u00aca \u2208 keys []\n\u22a2 kerase a [] = []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh\u271d : \u00aca \u2208 keys l\nhead\u271d : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u00aca \u2208 keys tail\u271d \u2192 kerase a tail\u271d = tail\u271d\nh : \u00aca \u2208 keys (head\u271d :: tail\u271d)\n\u22a2 kerase a (head\u271d :: tail\u271d) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp [not_or] at h \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh\u271d : \u00aca \u2208 keys l\nhead\u271d : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u00aca \u2208 keys tail\u271d \u2192 kerase a tail\u271d = tail\u271d\nh : \u00aca = head\u271d.fst \u2227 \u00aca \u2208 keys tail\u271d\n\u22a2 kerase a (head\u271d :: tail\u271d) = head\u271d :: tail\u271d\n[PROOFSTEP]\nsimp [h.1, ih h.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : a \u2208 keys l\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 l = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a l = l\u2081 ++ l\u2082\n[PROOFSTEP]\ninduction l with\n| nil => cases h\n| cons hd tl ih =>\n  by_cases e : a = hd.1\n  \u00b7 subst e\n    exact \u27e8hd.2, [], tl, by simp, by cases hd; rfl, by simp\u27e9\n  \u00b7 simp at h \n    cases' h with h h\n    exact absurd h e\n    rcases ih h with \u27e8b, tl\u2081, tl\u2082, h\u2081, h\u2082, h\u2083\u27e9\n    exact \u27e8b, hd :: tl\u2081, tl\u2082, not_mem_cons_of_ne_of_not_mem e h\u2081, by (rw [h\u2082]; rfl), by simp [e, h\u2083]\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : a \u2208 keys l\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 l = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a l = l\u2081 ++ l\u2082\n[PROOFSTEP]\ninduction l with\n| nil => cases h\n| cons hd tl ih =>\n  by_cases e : a = hd.1\n  \u00b7 subst e\n    exact \u27e8hd.2, [], tl, by simp, by cases hd; rfl, by simp\u27e9\n  \u00b7 simp at h \n    cases' h with h h\n    exact absurd h e\n    rcases ih h with \u27e8b, tl\u2081, tl\u2082, h\u2081, h\u2082, h\u2083\u27e9\n    exact \u27e8b, hd :: tl\u2081, tl\u2082, not_mem_cons_of_ne_of_not_mem e h\u2081, by (rw [h\u2082]; rfl), by simp [e, h\u2083]\u27e9\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nh : a \u2208 keys []\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 [] = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a [] = l\u2081 ++ l\u2082\n[PROOFSTEP]\n\n| nil => cases h\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nh : a \u2208 keys []\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 [] = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a [] = l\u2081 ++ l\u2082\n[PROOFSTEP]\ncases h\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\nh : a \u2208 keys (hd :: tl)\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\n\n| cons hd tl ih =>\n  by_cases e : a = hd.1\n  \u00b7 subst e\n    exact \u27e8hd.2, [], tl, by simp, by cases hd; rfl, by simp\u27e9\n  \u00b7 simp at h \n    cases' h with h h\n    exact absurd h e\n    rcases ih h with \u27e8b, tl\u2081, tl\u2082, h\u2081, h\u2082, h\u2083\u27e9\n    exact \u27e8b, hd :: tl\u2081, tl\u2082, not_mem_cons_of_ne_of_not_mem e h\u2081, by (rw [h\u2082]; rfl), by simp [e, h\u2083]\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\nh : a \u2208 keys (hd :: tl)\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nby_cases e : a = hd.1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\nh : a \u2208 keys (hd :: tl)\ne : a = hd.fst\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nsubst e\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih :\n  hd.fst \u2208 keys tl \u2192\n    \u2203 b l\u2081 l\u2082, \u00achd.fst \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := hd.fst, snd := b } :: l\u2082 \u2227 kerase hd.fst tl = l\u2081 ++ l\u2082\nh : hd.fst \u2208 keys (hd :: tl)\n\u22a2 \u2203 b l\u2081 l\u2082,\n    \u00achd.fst \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := hd.fst, snd := b } :: l\u2082 \u2227 kerase hd.fst (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nexact \u27e8hd.2, [], tl, by simp, by cases hd; rfl, by simp\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih :\n  hd.fst \u2208 keys tl \u2192\n    \u2203 b l\u2081 l\u2082, \u00achd.fst \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := hd.fst, snd := b } :: l\u2082 \u2227 kerase hd.fst tl = l\u2081 ++ l\u2082\nh : hd.fst \u2208 keys (hd :: tl)\n\u22a2 \u00achd.fst \u2208 keys []\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih :\n  hd.fst \u2208 keys tl \u2192\n    \u2203 b l\u2081 l\u2082, \u00achd.fst \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := hd.fst, snd := b } :: l\u2082 \u2227 kerase hd.fst tl = l\u2081 ++ l\u2082\nh : hd.fst \u2208 keys (hd :: tl)\n\u22a2 hd :: tl = [] ++ { fst := hd.fst, snd := hd.snd } :: tl\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\ntl : List (Sigma \u03b2)\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nih :\n  { fst := fst\u271d, snd := snd\u271d }.fst \u2208 keys tl \u2192\n    \u2203 b l\u2081 l\u2082,\n      \u00ac{ fst := fst\u271d, snd := snd\u271d }.fst \u2208 keys l\u2081 \u2227\n        tl = l\u2081 ++ { fst := { fst := fst\u271d, snd := snd\u271d }.fst, snd := b } :: l\u2082 \u2227\n          kerase { fst := fst\u271d, snd := snd\u271d }.fst tl = l\u2081 ++ l\u2082\nh : { fst := fst\u271d, snd := snd\u271d }.fst \u2208 keys ({ fst := fst\u271d, snd := snd\u271d } :: tl)\n\u22a2 { fst := fst\u271d, snd := snd\u271d } :: tl =\n    [] ++ { fst := { fst := fst\u271d, snd := snd\u271d }.fst, snd := { fst := fst\u271d, snd := snd\u271d }.snd } :: tl\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih :\n  hd.fst \u2208 keys tl \u2192\n    \u2203 b l\u2081 l\u2082, \u00achd.fst \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := hd.fst, snd := b } :: l\u2082 \u2227 kerase hd.fst tl = l\u2081 ++ l\u2082\nh : hd.fst \u2208 keys (hd :: tl)\n\u22a2 kerase hd.fst (hd :: tl) = [] ++ tl\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\nh : a \u2208 keys (hd :: tl)\ne : \u00aca = hd.fst\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a = hd.fst \u2228 a \u2208 keys tl\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase neg.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a = hd.fst\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\ncase neg.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a \u2208 keys tl\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nexact absurd h e\n[GOAL]\ncase neg.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a \u2208 keys tl\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nrcases ih h with \u27e8b, tl\u2081, tl\u2082, h\u2081, h\u2082, h\u2083\u27e9\n[GOAL]\ncase neg.inr.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a \u2208 keys tl\nb : \u03b2 a\ntl\u2081 tl\u2082 : List (Sigma \u03b2)\nh\u2081 : \u00aca \u2208 keys tl\u2081\nh\u2082 : tl = tl\u2081 ++ { fst := a, snd := b } :: tl\u2082\nh\u2083 : kerase a tl = tl\u2081 ++ tl\u2082\n\u22a2 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 hd :: tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a (hd :: tl) = l\u2081 ++ l\u2082\n[PROOFSTEP]\nexact \u27e8b, hd :: tl\u2081, tl\u2082, not_mem_cons_of_ne_of_not_mem e h\u2081, by (rw [h\u2082]; rfl), by simp [e, h\u2083]\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a \u2208 keys tl\nb : \u03b2 a\ntl\u2081 tl\u2082 : List (Sigma \u03b2)\nh\u2081 : \u00aca \u2208 keys tl\u2081\nh\u2082 : tl = tl\u2081 ++ { fst := a, snd := b } :: tl\u2082\nh\u2083 : kerase a tl = tl\u2081 ++ tl\u2082\n\u22a2 hd :: tl = hd :: tl\u2081 ++ { fst := a, snd := b } :: tl\u2082\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a \u2208 keys tl\nb : \u03b2 a\ntl\u2081 tl\u2082 : List (Sigma \u03b2)\nh\u2081 : \u00aca \u2208 keys tl\u2081\nh\u2082 : tl = tl\u2081 ++ { fst := a, snd := b } :: tl\u2082\nh\u2083 : kerase a tl = tl\u2081 ++ tl\u2082\n\u22a2 hd :: (tl\u2081 ++ { fst := a, snd := b } :: tl\u2082) = hd :: tl\u2081 ++ { fst := a, snd := b } :: tl\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : a \u2208 keys tl \u2192 \u2203 b l\u2081 l\u2082, \u00aca \u2208 keys l\u2081 \u2227 tl = l\u2081 ++ { fst := a, snd := b } :: l\u2082 \u2227 kerase a tl = l\u2081 ++ l\u2082\ne : \u00aca = hd.fst\nh : a \u2208 keys tl\nb : \u03b2 a\ntl\u2081 tl\u2082 : List (Sigma \u03b2)\nh\u2081 : \u00aca \u2208 keys tl\u2081\nh\u2082 : tl = tl\u2081 ++ { fst := a, snd := b } :: tl\u2082\nh\u2083 : kerase a tl = tl\u2081 ++ tl\u2082\n\u22a2 kerase a (hd :: tl) = hd :: tl\u2081 ++ tl\u2082\n[PROOFSTEP]\nsimp [e, h\u2083]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : a\u2081 \u2260 a\u2082\np\u271d : a\u2081 \u2208 keys l\nw\u271d\u00b2 : \u03b2 a\u2082\nw\u271d\u00b9 w\u271d : List (Sigma \u03b2)\nleft\u271d : \u00aca\u2082 \u2208 keys w\u271d\u00b9\np : a\u2081 \u2208 keys (w\u271d\u00b9 ++ { fst := a\u2082, snd := w\u271d\u00b2 } :: w\u271d)\nq : a\u2082 \u2208 keys (w\u271d\u00b9 ++ { fst := a\u2082, snd := w\u271d\u00b2 } :: w\u271d)\n\u22a2 a\u2081 \u2208 keys (w\u271d\u00b9 ++ w\u271d)\n[PROOFSTEP]\nsimpa [keys, h] using p\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : a\u2081 \u2260 a\u2082\np : a\u2081 \u2208 keys l\nq : \u00aca\u2082 \u2208 keys l\n\u22a2 a\u2081 \u2208 keys (kerase a\u2082 l)\n[PROOFSTEP]\nsimp [q, p]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\n\u22a2 keys (kerase a l) = List.erase (keys l) a\n[PROOFSTEP]\nrw [keys, kerase, erase_eq_eraseP, eraseP_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\n\u22a2 map Sigma.fst (eraseP (fun s => decide (a = s.fst)) l) =\n    map Sigma.fst (eraseP ((fun b => decide (a = b)) \u2218 Sigma.fst) l)\n[PROOFSTEP]\ndsimp [Function.comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nl : List (Sigma \u03b2)\n\u22a2 kerase a (kerase a' l) = kerase a' (kerase a l)\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 kerase a (kerase a' l) = kerase a' (kerase a l)\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\n\u22a2 kerase a (kerase a l) = kerase a (kerase a l)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 kerase a (kerase a' l) = kerase a' (kerase a l)\n[PROOFSTEP]\ninduction' l with x xs\n[GOAL]\ncase neg.nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : \u00aca = a'\n\u22a2 kerase a (kerase a' []) = kerase a' (kerase a [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg.cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : \u00aca = a'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : kerase a (kerase a' xs) = kerase a' (kerase a xs)\n\u22a2 kerase a (kerase a' (x :: xs)) = kerase a' (kerase a (x :: xs))\n[PROOFSTEP]\nby_cases a' = x.1\n[GOAL]\ncase neg.cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : \u00aca = a'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : kerase a (kerase a' xs) = kerase a' (kerase a xs)\n\u22a2 kerase a (kerase a' (x :: xs)) = kerase a' (kerase a (x :: xs))\n[PROOFSTEP]\nby_cases a' = x.1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh\u271d : \u00aca = a'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : kerase a (kerase a' xs) = kerase a' (kerase a xs)\nh : a' = x.fst\n\u22a2 kerase a (kerase a' (x :: xs)) = kerase a' (kerase a (x :: xs))\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\nh : \u00aca = x.fst\ntail_ih\u271d : kerase a (kerase x.fst xs) = kerase x.fst (kerase a xs)\n\u22a2 kerase a (kerase x.fst (x :: xs)) = kerase x.fst (kerase a (x :: xs))\n[PROOFSTEP]\nsimp [kerase_cons_ne h, kerase_cons_eq rfl]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh\u271d : \u00aca = a'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : kerase a (kerase a' xs) = kerase a' (kerase a xs)\nh : \u00aca' = x.fst\n\u22a2 kerase a (kerase a' (x :: xs)) = kerase a' (kerase a (x :: xs))\n[PROOFSTEP]\nby_cases h' : a = x.1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh\u271d : \u00aca = a'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : kerase a (kerase a' xs) = kerase a' (kerase a xs)\nh : \u00aca' = x.fst\nh' : a = x.fst\n\u22a2 kerase a (kerase a' (x :: xs)) = kerase a' (kerase a (x :: xs))\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na' : \u03b1\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\nh\u271d : \u00aca' = x.fst\nh : \u00acx.fst = a'\ntail_ih\u271d : kerase x.fst (kerase a' xs) = kerase a' (kerase x.fst xs)\n\u22a2 kerase x.fst (kerase a' (x :: xs)) = kerase a' (kerase x.fst (x :: xs))\n[PROOFSTEP]\nsimp [kerase_cons_eq rfl, kerase_cons_ne (Ne.symm h)]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh\u271d : \u00aca = a'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : kerase a (kerase a' xs) = kerase a' (kerase a xs)\nh : \u00aca' = x.fst\nh' : \u00aca = x.fst\n\u22a2 kerase a (kerase a' (x :: xs)) = kerase a' (kerase a (x :: xs))\n[PROOFSTEP]\nsimp [kerase_cons_ne, *]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\n\u22a2 \u2200 {a_1 b : Sigma \u03b2}, a_1.fst \u2260 b.fst \u2192 a = a_1.fst \u2192 a = b.fst \u2192 False\n[PROOFSTEP]\nrintro x y h rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nnd : NodupKeys l\u2081\nx y : Sigma \u03b2\nh : x.fst \u2260 y.fst\n\u22a2 x.fst = y.fst \u2192 False\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nnd : NodupKeys l\n\u22a2 \u00aca \u2208 keys (kerase a l)\n[PROOFSTEP]\ninduction l with\n| nil => simp\n| cons hd tl ih =>\n  simp at nd \n  by_cases h : a = hd.1\n  \u00b7 subst h\n    simp [nd.1]\n  \u00b7 simp [h, ih nd.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nnd : NodupKeys l\n\u22a2 \u00aca \u2208 keys (kerase a l)\n[PROOFSTEP]\ninduction l with\n| nil => simp\n| cons hd tl ih =>\n  simp at nd \n  by_cases h : a = hd.1\n  \u00b7 subst h\n    simp [nd.1]\n  \u00b7 simp [h, ih nd.2]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nnd : NodupKeys []\n\u22a2 \u00aca \u2208 keys (kerase a [])\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nnd : NodupKeys []\n\u22a2 \u00aca \u2208 keys (kerase a [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : NodupKeys tl \u2192 \u00aca \u2208 keys (kerase a tl)\nnd : NodupKeys (hd :: tl)\n\u22a2 \u00aca \u2208 keys (kerase a (hd :: tl))\n[PROOFSTEP]\n\n| cons hd tl ih =>\n  simp at nd \n  by_cases h : a = hd.1\n  \u00b7 subst h\n    simp [nd.1]\n  \u00b7 simp [h, ih nd.2]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : NodupKeys tl \u2192 \u00aca \u2208 keys (kerase a tl)\nnd : NodupKeys (hd :: tl)\n\u22a2 \u00aca \u2208 keys (kerase a (hd :: tl))\n[PROOFSTEP]\nsimp at nd \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : NodupKeys tl \u2192 \u00aca \u2208 keys (kerase a tl)\nnd : \u00achd.fst \u2208 keys tl \u2227 NodupKeys tl\n\u22a2 \u00aca \u2208 keys (kerase a (hd :: tl))\n[PROOFSTEP]\nby_cases h : a = hd.1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : NodupKeys tl \u2192 \u00aca \u2208 keys (kerase a tl)\nnd : \u00achd.fst \u2208 keys tl \u2227 NodupKeys tl\nh : a = hd.fst\n\u22a2 \u00aca \u2208 keys (kerase a (hd :: tl))\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nnd : \u00achd.fst \u2208 keys tl \u2227 NodupKeys tl\nih : NodupKeys tl \u2192 \u00achd.fst \u2208 keys (kerase hd.fst tl)\n\u22a2 \u00achd.fst \u2208 keys (kerase hd.fst (hd :: tl))\n[PROOFSTEP]\nsimp [nd.1]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : NodupKeys tl \u2192 \u00aca \u2208 keys (kerase a tl)\nnd : \u00achd.fst \u2208 keys tl \u2227 NodupKeys tl\nh : \u00aca = hd.fst\n\u22a2 \u00aca \u2208 keys (kerase a (hd :: tl))\n[PROOFSTEP]\nsimp [h, ih nd.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nl : List (Sigma \u03b2)\nh : a \u2260 a'\n\u22a2 dlookup a (kerase a' l) = dlookup a l\n[PROOFSTEP]\ninduction l with\n| nil => rfl\n| cons hd tl ih =>\n  cases' hd with ah bh\n  by_cases h\u2081 : a = ah <;> by_cases h\u2082 : a' = ah\n  \u00b7 substs h\u2081 h\u2082\n    cases Ne.irrefl h\n  \u00b7 subst h\u2081\n    simp [h\u2082]\n  \u00b7 subst h\u2082\n    simp [h]\n  \u00b7 simp [h\u2081, h\u2082, ih]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nl : List (Sigma \u03b2)\nh : a \u2260 a'\n\u22a2 dlookup a (kerase a' l) = dlookup a l\n[PROOFSTEP]\ninduction l with\n| nil => rfl\n| cons hd tl ih =>\n  cases' hd with ah bh\n  by_cases h\u2081 : a = ah <;> by_cases h\u2082 : a' = ah\n  \u00b7 substs h\u2081 h\u2082\n    cases Ne.irrefl h\n  \u00b7 subst h\u2081\n    simp [h\u2082]\n  \u00b7 subst h\u2082\n    simp [h]\n  \u00b7 simp [h\u2081, h\u2082, ih]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\n\u22a2 dlookup a (kerase a' []) = dlookup a []\n[PROOFSTEP]\n\n| nil => rfl\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\n\u22a2 dlookup a (kerase a' []) = dlookup a []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\n\u22a2 dlookup a (kerase a' (hd :: tl)) = dlookup a (hd :: tl)\n[PROOFSTEP]\n\n| cons hd tl ih =>\n  cases' hd with ah bh\n  by_cases h\u2081 : a = ah <;> by_cases h\u2082 : a' = ah\n  \u00b7 substs h\u2081 h\u2082\n    cases Ne.irrefl h\n  \u00b7 subst h\u2081\n    simp [h\u2082]\n  \u00b7 subst h\u2082\n    simp [h]\n  \u00b7 simp [h\u2081, h\u2082, ih]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\nhd : Sigma \u03b2\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\n\u22a2 dlookup a (kerase a' (hd :: tl)) = dlookup a (hd :: tl)\n[PROOFSTEP]\ncases' hd with ah bh\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nby_cases h\u2081 : a = ah\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\nh\u2081 : a = ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nby_cases h\u2082 : a' = ah\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\nh\u2081 : \u00aca = ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nby_cases h\u2082 : a' = ah\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\nh\u2081 : a = ah\nh\u2082 : a' = ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nsubsts h\u2081 h\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na' : \u03b1\ntl : List (Sigma \u03b2)\nh : a' \u2260 a'\nih : dlookup a' (kerase a' tl) = dlookup a' tl\nbh : \u03b2 a'\n\u22a2 dlookup a' (kerase a' ({ fst := a', snd := bh } :: tl)) = dlookup a' ({ fst := a', snd := bh } :: tl)\n[PROOFSTEP]\ncases Ne.irrefl h\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\nh\u2081 : a = ah\nh\u2082 : \u00aca' = ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nsubst h\u2081\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nbh : \u03b2 a\nh\u2082 : \u00aca' = a\n\u22a2 dlookup a (kerase a' ({ fst := a, snd := bh } :: tl)) = dlookup a ({ fst := a, snd := bh } :: tl)\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\nh\u2081 : \u00aca = ah\nh\u2082 : a' = ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nsubst h\u2082\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nbh : \u03b2 a'\nh\u2081 : \u00aca = a'\n\u22a2 dlookup a (kerase a' ({ fst := a', snd := bh } :: tl)) = dlookup a ({ fst := a', snd := bh } :: tl)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nh : a \u2260 a'\ntl : List (Sigma \u03b2)\nih : dlookup a (kerase a' tl) = dlookup a tl\nah : \u03b1\nbh : \u03b2 ah\nh\u2081 : \u00aca = ah\nh\u2082 : \u00aca' = ah\n\u22a2 dlookup a (kerase a' ({ fst := ah, snd := bh } :: tl)) = dlookup a ({ fst := ah, snd := bh } :: tl)\n[PROOFSTEP]\nsimp [h\u2081, h\u2082, ih]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx\u271d : List (Sigma \u03b2)\nh : a \u2208 keys []\n\u22a2 kerase a ([] ++ x\u271d) = kerase a [] ++ x\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2081 : a \u2208 keys (s :: l\u2081)\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\nif h\u2082 : a = s.1 then simp [h\u2082]\nelse simp at h\u2081 ; cases' h\u2081 with h\u2081 h\u2081 <;> [exact absurd h\u2081 h\u2082; simp [h\u2082, kerase_append_left h\u2081]]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2081 : a \u2208 keys (s :: l\u2081)\nh\u2082 : a = s.fst\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2081 : a \u2208 keys (s :: l\u2081)\nh\u2082 : \u00aca = s.fst\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\nsimp at h\u2081 \n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2082 : \u00aca = s.fst\nh\u2081 : a = s.fst \u2228 a \u2208 keys l\u2081\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\ncases' h\u2081 with h\u2081 h\u2081 <;> [exact absurd h\u2081 h\u2082; simp [h\u2082, kerase_append_left h\u2081]]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2082 : \u00aca = s.fst\nh\u2081 : a = s.fst \u2228 a \u2208 keys l\u2081\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\ncases' h\u2081 with h\u2081 h\u2081\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2082 : \u00aca = s.fst\nh\u2081 : a = s.fst\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\nexact absurd h\u2081 h\u2082\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh\u2082 : \u00aca = s.fst\nh\u2081 : a \u2208 keys l\u2081\n\u22a2 kerase a (s :: l\u2081 ++ l\u2082) = kerase a (s :: l\u2081) ++ l\u2082\n[PROOFSTEP]\nsimp [h\u2082, kerase_append_left h\u2081]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhead\u271d : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys (head\u271d :: l\u2081)\n\u22a2 kerase a (head\u271d :: l\u2081 ++ l\u2082) = head\u271d :: l\u2081 ++ kerase a l\u2082\n[PROOFSTEP]\nsimp [not_or] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhead\u271d : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca = head\u271d.fst \u2227 \u00aca \u2208 keys l\u2081\n\u22a2 kerase a (head\u271d :: l\u2081 ++ l\u2082) = head\u271d :: l\u2081 ++ kerase a l\u2082\n[PROOFSTEP]\nsimp [h.1, kerase_append_right h.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : a\u2081 = a\u2082\n\u22a2 kerase a\u2082 (kerase a\u2081 l) = kerase a\u2081 (kerase a\u2082 l)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2082 : a\u2082 \u2208 keys l\nb\u2081 : \u03b2 a\u2081\nl\u2081 l\u2082 : List (Sigma \u03b2)\na\u2081_nin_l\u2081 : \u00aca\u2081 \u2208 keys l\u2081\nx\u271d : a\u2082 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nha\u2081 : a\u2081 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nh' : a\u2082 \u2208 keys l\u2081\n\u22a2 kerase a\u2082 (l\u2081 ++ l\u2082) = kerase a\u2081 (kerase a\u2082 (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082))\n[PROOFSTEP]\nsimp [kerase_append_left h', kerase_append_right (mt (mem_keys_kerase_of_ne h).mp a\u2081_nin_l\u2081)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2082 : a\u2082 \u2208 keys l\nb\u2081 : \u03b2 a\u2081\nl\u2081 l\u2082 : List (Sigma \u03b2)\na\u2081_nin_l\u2081 : \u00aca\u2081 \u2208 keys l\u2081\nx\u271d : a\u2082 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nha\u2081 : a\u2081 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nh' : \u00aca\u2082 \u2208 keys l\u2081\n\u22a2 kerase a\u2082 (l\u2081 ++ l\u2082) = kerase a\u2081 (kerase a\u2082 (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082))\n[PROOFSTEP]\nsimp [kerase_append_right h', kerase_append_right a\u2081_nin_l\u2081, @kerase_cons_ne _ _ _ a\u2082 \u27e8a\u2081, b\u2081\u27e9 _ (Ne.symm h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2081 : a\u2081 \u2208 keys l\nha\u2082 : \u00aca\u2082 \u2208 keys l\n\u22a2 kerase a\u2082 (kerase a\u2081 l) = kerase a\u2081 (kerase a\u2082 l)\n[PROOFSTEP]\nsimp [ha\u2082, mt mem_keys_of_mem_keys_kerase ha\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2081 : \u00aca\u2081 \u2208 keys l\n\u22a2 kerase a\u2082 (kerase a\u2081 l) = kerase a\u2081 (kerase a\u2082 l)\n[PROOFSTEP]\nsimp [ha\u2081, mt mem_keys_of_mem_keys_kerase ha\u2081]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\nxs : List (Sigma \u03b2)\n\u22a2 sizeOf (kerase x xs) \u2264 sizeOf xs\n[PROOFSTEP]\nsimp only [SizeOf.sizeOf, _sizeOf_1]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\nxs : List (Sigma \u03b2)\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n[PROOFSTEP]\ninduction' xs with y ys\n[GOAL]\ncase nil\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x []) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\ny : Sigma \u03b2\nys : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x ys) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) ys\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x (y :: ys)) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (y :: ys)\n[PROOFSTEP]\nby_cases x = y.1\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\ny : Sigma \u03b2\nys : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x ys) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) ys\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x (y :: ys)) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (y :: ys)\n[PROOFSTEP]\nby_cases x = y.1\n[GOAL]\ncase pos\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\ny : Sigma \u03b2\nys : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x ys) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) ys\nh : x = y.fst\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x (y :: ys)) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (y :: ys)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : \u03b1\ny : Sigma \u03b2\nys : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x ys) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) ys\nh : \u00acx = y.fst\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x (y :: ys)) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (y :: ys)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kinsert a' b' l) \u2194 a = a' \u2228 a \u2208 keys l\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 a \u2208 keys (kinsert a' b' l) \u2194 a = a' \u2228 a \u2208 keys l\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 a \u2208 keys (kinsert a' b' l) \u2194 a = a' \u2228 a \u2208 keys l\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl : List (Sigma \u03b2)\n\u22a2 dlookup a (kinsert a b l) = some b\n[PROOFSTEP]\nsimp only [kinsert, dlookup_cons_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb' : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a \u2260 a'\n\u22a2 dlookup a (kinsert a' b' l) = dlookup a l\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 kextract a ({ fst := a', snd := b } :: l) =\n    (dlookup a ({ fst := a', snd := b } :: l), kerase a ({ fst := a', snd := b } :: l))\n[PROOFSTEP]\nsimp [kextract]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 (if h : a' = a then (some ((_ : { fst := a', snd := b }.fst = a) \u25b8 b), l)\n    else ((kextract a l).fst, { fst := a', snd := b } :: (kextract a l).snd)) =\n    (dlookup a ({ fst := a', snd := b } :: l), kerase a ({ fst := a', snd := b } :: l))\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 (if h : a' = a then (some ((_ : a' = a) \u25b8 b), l)\n    else ((kextract a l).fst, { fst := a', snd := b } :: (kextract a l).snd)) =\n    (dlookup a ({ fst := a', snd := b } :: l), kerase a ({ fst := a', snd := b } :: l))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a' = a\n\u22a2 (some ((_ : a' = a) \u25b8 b), l) = (dlookup a ({ fst := a', snd := b } :: l), kerase a ({ fst := a', snd := b } :: l))\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb : \u03b2 a\n\u22a2 (some ((_ : a = a) \u25b8 b), l) = (dlookup a ({ fst := a, snd := b } :: l), kerase a ({ fst := a, snd := b } :: l))\n[PROOFSTEP]\nsimp [kerase]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca' = a\n\u22a2 ((kextract a l).fst, { fst := a', snd := b } :: (kextract a l).snd) =\n    (dlookup a ({ fst := a', snd := b } :: l), kerase a ({ fst := a', snd := b } :: l))\n[PROOFSTEP]\nsimp [kextract, Ne.symm h, kextract_eq_dlookup_kerase a l, kerase]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\n\u22a2 NodupKeys (dedupKeys l)\n[PROOFSTEP]\ndsimp [dedupKeys]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) [] l)\n[PROOFSTEP]\ngeneralize hl : nil = l'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\nl' : List (Sigma \u03b2)\nhl : [] = l'\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' l)\n[PROOFSTEP]\nhave : NodupKeys l' := by\n  rw [\u2190 hl]\n  apply nodup_nil\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\nl' : List (Sigma \u03b2)\nhl : [] = l'\n\u22a2 NodupKeys l'\n[PROOFSTEP]\nrw [\u2190 hl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\nl' : List (Sigma \u03b2)\nhl : [] = l'\n\u22a2 NodupKeys []\n[PROOFSTEP]\napply nodup_nil\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\nl' : List (Sigma \u03b2)\nhl : [] = l'\nthis : NodupKeys l'\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' l)\n[PROOFSTEP]\nclear hl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl : List (Sigma \u03b2)\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' l)\n[PROOFSTEP]\ninduction' l with x xs l_ih\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' [])\n[PROOFSTEP]\napply this\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\nl_ih : NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' xs)\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' (x :: xs))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\nxs : List (Sigma \u03b2)\nl_ih : NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' xs)\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' ({ fst := fst\u271d, snd := snd\u271d } :: xs))\n[PROOFSTEP]\nsimp [dedupKeys]\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\nxs : List (Sigma \u03b2)\nl_ih : NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' xs)\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 \u00acfst\u271d \u2208 keys (kerase fst\u271d (foldr (fun x => kinsert x.fst x.snd) l' xs)) \u2227\n    NodupKeys (kerase fst\u271d (foldr (fun x => kinsert x.fst x.snd) l' xs))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.mk.left\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\nxs : List (Sigma \u03b2)\nl_ih : NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' xs)\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 \u00acfst\u271d \u2208 keys (kerase fst\u271d (foldr (fun x => kinsert x.fst x.snd) l' xs))\n[PROOFSTEP]\nsimp [keys_kerase]\n[GOAL]\ncase cons.mk.left\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\nxs : List (Sigma \u03b2)\nl_ih : NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' xs)\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 \u00acfst\u271d \u2208 List.erase (keys (foldr (fun x => kinsert x.fst x.snd) l' xs)) fst\u271d\n[PROOFSTEP]\napply l_ih.not_mem_erase\n[GOAL]\ncase cons.mk.right\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl' : List (Sigma \u03b2)\nthis : NodupKeys l'\nxs : List (Sigma \u03b2)\nl_ih : NodupKeys (foldr (fun x => kinsert x.fst x.snd) l' xs)\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 NodupKeys (kerase fst\u271d (foldr (fun x => kinsert x.fst x.snd) l' xs))\n[PROOFSTEP]\nexact l_ih.kerase _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\n\u22a2 dlookup a (dedupKeys l) = dlookup a l\n[PROOFSTEP]\ninduction' l with l_hd _ l_ih\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 dlookup a (dedupKeys []) = dlookup a []\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl_hd : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\n\u22a2 dlookup a (dedupKeys (l_hd :: tail\u271d)) = dlookup a (l_hd :: tail\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl_hd : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\n\u22a2 dlookup a (dedupKeys (l_hd :: tail\u271d)) = dlookup a (l_hd :: tail\u271d)\n[PROOFSTEP]\ncases' l_hd with a' b\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\na' : \u03b1\nb : \u03b2 a'\n\u22a2 dlookup a (dedupKeys ({ fst := a', snd := b } :: tail\u271d)) = dlookup a ({ fst := a', snd := b } :: tail\u271d)\n[PROOFSTEP]\nby_cases h : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\na' : \u03b1\nb : \u03b2 a'\nh : a = a'\n\u22a2 dlookup a (dedupKeys ({ fst := a', snd := b } :: tail\u271d)) = dlookup a ({ fst := a', snd := b } :: tail\u271d)\n[PROOFSTEP]\nsubst a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\nb : \u03b2 a\n\u22a2 dlookup a (dedupKeys ({ fst := a, snd := b } :: tail\u271d)) = dlookup a ({ fst := a, snd := b } :: tail\u271d)\n[PROOFSTEP]\nrw [dedupKeys_cons, dlookup_kinsert, dlookup_cons_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\na' : \u03b1\nb : \u03b2 a'\nh : \u00aca = a'\n\u22a2 dlookup a (dedupKeys ({ fst := a', snd := b } :: tail\u271d)) = dlookup a ({ fst := a', snd := b } :: tail\u271d)\n[PROOFSTEP]\nrw [dedupKeys_cons, dlookup_kinsert_ne h, l_ih, dlookup_cons_ne]\n[GOAL]\ncase neg.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ntail\u271d : List (Sigma \u03b2)\nl_ih : dlookup a (dedupKeys tail\u271d) = dlookup a tail\u271d\na' : \u03b1\nb : \u03b2 a'\nh : \u00aca = a'\n\u22a2 a \u2260 { fst := a', snd := b }.fst\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nxs : List (Sigma \u03b2)\n\u22a2 sizeOf (dedupKeys xs) \u2264 sizeOf xs\n[PROOFSTEP]\nsimp only [SizeOf.sizeOf, _sizeOf_1]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nxs : List (Sigma \u03b2)\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n[PROOFSTEP]\ninduction' xs with x xs\n[GOAL]\ncase nil\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys []) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []\n[PROOFSTEP]\nsimp [dedupKeys]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys (x :: xs)) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (x :: xs)\n[PROOFSTEP]\nsimp only [dedupKeys_cons, kinsert_def, add_le_add_iff_left, Sigma.eta]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x.fst (dedupKeys xs)) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n[PROOFSTEP]\ntrans\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u22a2 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x.fst (dedupKeys xs)) \u2264 ?m.375896\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u22a2 ?m.375896 \u2264 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u22a2 \u2115\n[PROOFSTEP]\napply sizeOf_kerase\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : \u03b1\u271d \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2\u271d)\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : SizeOf (Sigma \u03b2)\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d :\n  rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (dedupKeys xs) \u2264\n    rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n\u22a2 sizeOf (dedupKeys xs) \u2264 rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nhead\u271d : Sigma \u03b2\nl : List (Sigma \u03b2)\n\u22a2 kunion (head\u271d :: l) [] = head\u271d :: l\n[PROOFSTEP]\nrw [kunion, kerase_nil, kunion_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons s l\u2081 ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons s l\u2081 ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion [] l\u2082) \u2194 a \u2208 keys [] \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion [] l\u2082) \u2194 a \u2208 keys [] \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\nl\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion (s :: l\u2081) l\u2082) \u2194 a \u2208 keys (s :: l\u2081) \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\n\n| cons s l\u2081 ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\nl\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion (s :: l\u2081) l\u2082) \u2194 a \u2208 keys (s :: l\u2081) \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\nby_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\nl\u2082 : List (Sigma \u03b2)\n\u22a2 a \u2208 keys (kunion (s :: l\u2081) l\u2082) \u2194 a \u2208 keys (s :: l\u2081) \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\nby_cases h : a = s.1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\nl\u2082 : List (Sigma \u03b2)\nh : a = s.fst\n\u22a2 a \u2208 keys (kunion (s :: l\u2081) l\u2082) \u2194 a \u2208 keys (s :: l\u2081) \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca = s.fst\n\u22a2 a \u2208 keys (kunion (s :: l\u2081) l\u2082) \u2194 a \u2208 keys (s :: l\u2081) \u2228 a \u2208 keys l\u2082\n[PROOFSTEP]\nsimp [h, ih]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nl : List (Sigma \u03b2)\n\u22a2 kunion (kerase a (s :: tail\u271d)) (kerase a l) = kerase a (kunion (s :: tail\u271d) l)\n[PROOFSTEP]\nby_cases h : a = s.1\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nl : List (Sigma \u03b2)\nh : a = s.fst\n\u22a2 kunion (kerase a (s :: tail\u271d)) (kerase a l) = kerase a (kunion (s :: tail\u271d) l)\n[PROOFSTEP]\nsimp [h, kerase_comm a s.1 l, kunion_kerase]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nl : List (Sigma \u03b2)\nh : \u00aca = s.fst\n\u22a2 kunion (kerase a (s :: tail\u271d)) (kerase a l) = kerase a (kunion (s :: tail\u271d) l)\n[PROOFSTEP]\nsimp [h, kerase_comm a s.1 l, kunion_kerase]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nnd\u2081 : NodupKeys l\u2081\nnd\u2082 : NodupKeys l\u2082\n\u22a2 NodupKeys (List.kunion l\u2081 l\u2082)\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp only [nil_kunion, nd\u2082]\n| cons s l\u2081 ih =>\n  simp at nd\u2081 \n  simp [not_or, nd\u2081.1, nd\u2082, ih nd\u2081.2 (nd\u2082.kerase s.1)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nnd\u2081 : NodupKeys l\u2081\nnd\u2082 : NodupKeys l\u2082\n\u22a2 NodupKeys (List.kunion l\u2081 l\u2082)\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp only [nil_kunion, nd\u2082]\n| cons s l\u2081 ih =>\n  simp at nd\u2081 \n  simp [not_or, nd\u2081.1, nd\u2082, ih nd\u2081.2 (nd\u2082.kerase s.1)]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys []\nnd\u2082 : NodupKeys l\u2082\n\u22a2 NodupKeys (List.kunion [] l\u2082)\n[PROOFSTEP]\n\n| nil => simp only [nil_kunion, nd\u2082]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys []\nnd\u2082 : NodupKeys l\u2082\n\u22a2 NodupKeys (List.kunion [] l\u2082)\n[PROOFSTEP]\nsimp only [nil_kunion, nd\u2082]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, NodupKeys l\u2081 \u2192 NodupKeys l\u2082 \u2192 NodupKeys (List.kunion l\u2081 l\u2082)\nl\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys (s :: l\u2081)\nnd\u2082 : NodupKeys l\u2082\n\u22a2 NodupKeys (List.kunion (s :: l\u2081) l\u2082)\n[PROOFSTEP]\n\n| cons s l\u2081 ih =>\n  simp at nd\u2081 \n  simp [not_or, nd\u2081.1, nd\u2082, ih nd\u2081.2 (nd\u2082.kerase s.1)]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, NodupKeys l\u2081 \u2192 NodupKeys l\u2082 \u2192 NodupKeys (List.kunion l\u2081 l\u2082)\nl\u2082 : List (Sigma \u03b2)\nnd\u2081 : NodupKeys (s :: l\u2081)\nnd\u2082 : NodupKeys l\u2082\n\u22a2 NodupKeys (List.kunion (s :: l\u2081) l\u2082)\n[PROOFSTEP]\nsimp at nd\u2081 \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\ns : Sigma \u03b2\nl\u2081 : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, NodupKeys l\u2081 \u2192 NodupKeys l\u2082 \u2192 NodupKeys (List.kunion l\u2081 l\u2082)\nl\u2082 : List (Sigma \u03b2)\nnd\u2082 : NodupKeys l\u2082\nnd\u2081 : \u00acs.fst \u2208 keys l\u2081 \u2227 NodupKeys l\u2081\n\u22a2 NodupKeys (List.kunion (s :: l\u2081) l\u2082)\n[PROOFSTEP]\nsimp [not_or, nd\u2081.1, nd\u2082, ih nd\u2081.2 (nd\u2082.kerase s.1)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\np : l\u2081 ~ l\u2082\nl : List (Sigma \u03b2)\n\u22a2 kunion l\u2081 l ~ kunion l\u2082 l\n[PROOFSTEP]\ninduction p generalizing l with\n| nil => rfl\n| cons hd _ ih => simp [ih (List.kerase _ _), Perm.cons]\n| swap s\u2081 s\u2082 l => simp [kerase_comm, Perm.swap]\n| trans _ _ ih\u2081\u2082 ih\u2082\u2083 => exact Perm.trans (ih\u2081\u2082 l) (ih\u2082\u2083 l)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\np : l\u2081 ~ l\u2082\nl : List (Sigma \u03b2)\n\u22a2 kunion l\u2081 l ~ kunion l\u2082 l\n[PROOFSTEP]\ninduction p generalizing l with\n| nil => rfl\n| cons hd _ ih => simp [ih (List.kerase _ _), Perm.cons]\n| swap s\u2081 s\u2082 l => simp [kerase_comm, Perm.swap]\n| trans _ _ ih\u2081\u2082 ih\u2082\u2083 => exact Perm.trans (ih\u2081\u2082 l) (ih\u2082\u2083 l)\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 l : List (Sigma \u03b2)\n\u22a2 kunion [] l ~ kunion [] l\n[PROOFSTEP]\n\n| nil => rfl\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 l : List (Sigma \u03b2)\n\u22a2 kunion [] l ~ kunion [] l\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nhd : Sigma \u03b2\nl\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nih : \u2200 (l : List (Sigma \u03b2)), kunion l\u2081\u271d l ~ kunion l\u2082\u271d l\nl : List (Sigma \u03b2)\n\u22a2 kunion (hd :: l\u2081\u271d) l ~ kunion (hd :: l\u2082\u271d) l\n[PROOFSTEP]\n\n| cons hd _ ih => simp [ih (List.kerase _ _), Perm.cons]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nhd : Sigma \u03b2\nl\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\na\u271d : l\u2081\u271d ~ l\u2082\u271d\nih : \u2200 (l : List (Sigma \u03b2)), kunion l\u2081\u271d l ~ kunion l\u2082\u271d l\nl : List (Sigma \u03b2)\n\u22a2 kunion (hd :: l\u2081\u271d) l ~ kunion (hd :: l\u2082\u271d) l\n[PROOFSTEP]\nsimp [ih (List.kerase _ _), Perm.cons]\n[GOAL]\ncase swap\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\ns\u2081 s\u2082 : Sigma \u03b2\nl\u271d : List (Sigma \u03b2)\nl : List (Sigma \u03b2)\n\u22a2 kunion (s\u2082 :: s\u2081 :: l\u271d) l ~ kunion (s\u2081 :: s\u2082 :: l\u271d) l\n[PROOFSTEP]\n\n| swap s\u2081 s\u2082 l => simp [kerase_comm, Perm.swap]\n[GOAL]\ncase swap\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d\u00b9 l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\ns\u2081 s\u2082 : Sigma \u03b2\nl\u271d : List (Sigma \u03b2)\nl : List (Sigma \u03b2)\n\u22a2 kunion (s\u2082 :: s\u2081 :: l\u271d) l ~ kunion (s\u2081 :: s\u2082 :: l\u271d) l\n[PROOFSTEP]\nsimp [kerase_comm, Perm.swap]\n[GOAL]\ncase trans\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nl\u2081\u271d l\u2082\u271d l\u2083\u271d : List (Sigma \u03b2)\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nih\u2081\u2082 : \u2200 (l : List (Sigma \u03b2)), kunion l\u2081\u271d l ~ kunion l\u2082\u271d l\nih\u2082\u2083 : \u2200 (l : List (Sigma \u03b2)), kunion l\u2082\u271d l ~ kunion l\u2083\u271d l\nl : List (Sigma \u03b2)\n\u22a2 kunion l\u2081\u271d l ~ kunion l\u2083\u271d l\n[PROOFSTEP]\n\n| trans _ _ ih\u2081\u2082 ih\u2082\u2083 => exact Perm.trans (ih\u2081\u2082 l) (ih\u2082\u2083 l)\n[GOAL]\ncase trans\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nl\u2081\u271d l\u2082\u271d l\u2083\u271d : List (Sigma \u03b2)\na\u271d\u00b9 : l\u2081\u271d ~ l\u2082\u271d\na\u271d : l\u2082\u271d ~ l\u2083\u271d\nih\u2081\u2082 : \u2200 (l : List (Sigma \u03b2)), kunion l\u2081\u271d l ~ kunion l\u2082\u271d l\nih\u2082\u2083 : \u2200 (l : List (Sigma \u03b2)), kunion l\u2082\u271d l ~ kunion l\u2083\u271d l\nl : List (Sigma \u03b2)\n\u22a2 kunion l\u2081\u271d l ~ kunion l\u2083\u271d l\n[PROOFSTEP]\nexact Perm.trans (ih\u2081\u2082 l) (ih\u2082\u2083 l)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys l\u2081\n\u22a2 dlookup a (kunion l\u2081 l\u2082) = dlookup a l\u2081\n[PROOFSTEP]\ninduction' l\u2081 with s _ ih generalizing l\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys []\n\u22a2 dlookup a (kunion [] l\u2082) = dlookup a []\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys (s :: tail\u271d)\n\u22a2 dlookup a (kunion (s :: tail\u271d) l\u2082) = dlookup a (s :: tail\u271d)\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a = s.fst \u2228 a \u2208 keys tail\u271d\n\u22a2 dlookup a (kunion (s :: tail\u271d) l\u2082) = dlookup a (s :: tail\u271d)\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase cons.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a = s.fst\n\u22a2 dlookup a (kunion (s :: tail\u271d) l\u2082) = dlookup a (s :: tail\u271d)\n[PROOFSTEP]\ncases' s with a'\n[GOAL]\ncase cons.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys tail\u271d\n\u22a2 dlookup a (kunion (s :: tail\u271d) l\u2082) = dlookup a (s :: tail\u271d)\n[PROOFSTEP]\ncases' s with a'\n[GOAL]\ncase cons.inl.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh : a = { fst := a', snd := snd\u271d }.fst\n\u22a2 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) = dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase cons.inl.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nsnd\u271d : \u03b2 a\n\u22a2 dlookup a (kunion ({ fst := a, snd := snd\u271d } :: tail\u271d) l\u2082) = dlookup a ({ fst := a, snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.inr.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys tail\u271d\na' : \u03b1\nsnd\u271d : \u03b2 a'\n\u22a2 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) = dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nrw [kunion_cons]\n[GOAL]\ncase cons.inr.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys tail\u271d\na' : \u03b1\nsnd\u271d : \u03b2 a'\n\u22a2 dlookup a ({ fst := a', snd := snd\u271d } :: kunion tail\u271d (kerase { fst := a', snd := snd\u271d }.fst l\u2082)) =\n    dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nby_cases h' : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys tail\u271d\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh' : a = a'\n\u22a2 dlookup a ({ fst := a', snd := snd\u271d } :: kunion tail\u271d (kerase { fst := a', snd := snd\u271d }.fst l\u2082)) =\n    dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nsubst h'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys tail\u271d\nsnd\u271d : \u03b2 a\n\u22a2 dlookup a ({ fst := a, snd := snd\u271d } :: kunion tail\u271d (kerase { fst := a, snd := snd\u271d }.fst l\u2082)) =\n    dlookup a ({ fst := a, snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d\u00b9 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082\u271d : List (Sigma \u03b2)\nh\u271d : a \u2208 keys l\u2081\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a tail\u271d\nl\u2082 : List (Sigma \u03b2)\nh : a \u2208 keys tail\u271d\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh' : \u00aca = a'\n\u22a2 dlookup a ({ fst := a', snd := snd\u271d } :: kunion tail\u271d (kerase { fst := a', snd := snd\u271d }.fst l\u2082)) =\n    dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d)\n[PROOFSTEP]\nsimp [h', ih h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys l\u2081\n\u22a2 dlookup a (kunion l\u2081 l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons _ _ ih => simp [not_or] at h ; simp [h.1, ih h.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 l\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys l\u2081\n\u22a2 dlookup a (kunion l\u2081 l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons _ _ ih => simp [not_or] at h ; simp [h.1, ih h.2]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys []\n\u22a2 dlookup a (kunion [] l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys []\n\u22a2 dlookup a (kunion [] l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhead\u271d : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, \u00aca \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys (head\u271d :: tail\u271d)\n\u22a2 dlookup a (kunion (head\u271d :: tail\u271d) l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\n\n| cons _ _ ih => simp [not_or] at h ; simp [h.1, ih h.2]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhead\u271d : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, \u00aca \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca \u2208 keys (head\u271d :: tail\u271d)\n\u22a2 dlookup a (kunion (head\u271d :: tail\u271d) l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\nsimp [not_or] at h \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nhead\u271d : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, \u00aca \u2208 keys tail\u271d \u2192 dlookup a (kunion tail\u271d l\u2082) = dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\nh : \u00aca = head\u271d.fst \u2227 \u00aca \u2208 keys tail\u271d\n\u22a2 dlookup a (kunion (head\u271d :: tail\u271d) l\u2082) = dlookup a l\u2082\n[PROOFSTEP]\nsimp [h.1, ih h.2]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\n\u22a2 b \u2208 dlookup a (kunion l\u2081 l\u2082) \u2194 b \u2208 dlookup a l\u2081 \u2228 \u00aca \u2208 keys l\u2081 \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons s _ ih =>\n  cases' s with a'\n  by_cases h\u2081 : a = a'\n  \u00b7 subst h\u2081\n    simp\n  \u00b7 let h\u2082 := @ih (kerase a' l\u2082)\n    simp [h\u2081] at h\u2082 \n    simp [h\u2081, h\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2081 l\u2082 : List (Sigma \u03b2)\n\u22a2 b \u2208 dlookup a (kunion l\u2081 l\u2082) \u2194 b \u2208 dlookup a l\u2081 \u2228 \u00aca \u2208 keys l\u2081 \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\ninduction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons s _ ih =>\n  cases' s with a'\n  by_cases h\u2081 : a = a'\n  \u00b7 subst h\u2081\n    simp\n  \u00b7 let h\u2082 := @ih (kerase a' l\u2082)\n    simp [h\u2081] at h\u2082 \n    simp [h\u2081, h\u2082]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2082 : List (Sigma \u03b2)\n\u22a2 b \u2208 dlookup a (kunion [] l\u2082) \u2194 b \u2208 dlookup a [] \u2228 \u00aca \u2208 keys [] \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\n\n| nil => simp\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\nl\u2082 : List (Sigma \u03b2)\n\u22a2 b \u2208 dlookup a (kunion [] l\u2082) \u2194 b \u2208 dlookup a [] \u2228 \u00aca \u2208 keys [] \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\n\u22a2 b \u2208 dlookup a (kunion (s :: tail\u271d) l\u2082) \u2194 b \u2208 dlookup a (s :: tail\u271d) \u2228 \u00aca \u2208 keys (s :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\n\n| cons s _ ih =>\n  cases' s with a'\n  by_cases h\u2081 : a = a'\n  \u00b7 subst h\u2081\n    simp\n  \u00b7 let h\u2082 := @ih (kerase a' l\u2082)\n    simp [h\u2081] at h\u2082 \n    simp [h\u2081, h\u2082]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : Sigma \u03b2\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\n\u22a2 b \u2208 dlookup a (kunion (s :: tail\u271d) l\u2082) \u2194 b \u2208 dlookup a (s :: tail\u271d) \u2228 \u00aca \u2208 keys (s :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\ncases' s with a'\n[GOAL]\ncase cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\na' : \u03b1\nsnd\u271d : \u03b2 a'\n\u22a2 b \u2208 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) \u2194\n    b \u2208 dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2228\n      \u00aca \u2208 keys ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nby_cases h\u2081 : a = a'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh\u2081 : a = a'\n\u22a2 b \u2208 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) \u2194\n    b \u2208 dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2228\n      \u00aca \u2208 keys ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nsubst h\u2081\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\nsnd\u271d : \u03b2 a\n\u22a2 b \u2208 dlookup a (kunion ({ fst := a, snd := snd\u271d } :: tail\u271d) l\u2082) \u2194\n    b \u2208 dlookup a ({ fst := a, snd := snd\u271d } :: tail\u271d) \u2228\n      \u00aca \u2208 keys ({ fst := a, snd := snd\u271d } :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh\u2081 : \u00aca = a'\n\u22a2 b \u2208 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) \u2194\n    b \u2208 dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2228\n      \u00aca \u2208 keys ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nlet h\u2082 := @ih (kerase a' l\u2082)\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh\u2081 : \u00aca = a'\nh\u2082 : b \u2208 dlookup a (kunion tail\u271d (kerase a' l\u2082)) \u2194\n  b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a (kerase a' l\u2082) :=\n  ih\n\u22a2 b \u2208 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) \u2194\n    b \u2208 dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2228\n      \u00aca \u2208 keys ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nsimp [h\u2081] at h\u2082 \n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ntail\u271d : List (Sigma \u03b2)\nih : \u2200 {l\u2082 : List (Sigma \u03b2)}, b \u2208 dlookup a (kunion tail\u271d l\u2082) \u2194 b \u2208 dlookup a tail\u271d \u2228 \u00aca \u2208 keys tail\u271d \u2227 b \u2208 dlookup a l\u2082\nl\u2082 : List (Sigma \u03b2)\na' : \u03b1\nsnd\u271d : \u03b2 a'\nh\u2081 : \u00aca = a'\nh\u2082 :\n  dlookup a (kunion tail\u271d (kerase a' l\u2082)) = some b \u2194 dlookup a tail\u271d = some b \u2228 \u00aca \u2208 keys tail\u271d \u2227 dlookup a l\u2082 = some b\n\u22a2 b \u2208 dlookup a (kunion ({ fst := a', snd := snd\u271d } :: tail\u271d) l\u2082) \u2194\n    b \u2208 dlookup a ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2228\n      \u00aca \u2208 keys ({ fst := a', snd := snd\u271d } :: tail\u271d) \u2227 b \u2208 dlookup a l\u2082\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Sigma", "llama_tokens": 51723, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.749087201911703, "lm_q2_score": 0.7461389930307512, "lm_q1q2_score": 0.558923170526621}}
{"text": "[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 @UniversallyClosed = universally (topologically @IsClosedMap)\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 UniversallyClosed f \u2194 universally (topologically @IsClosedMap) f\n[PROOFSTEP]\nrw [UniversallyClosed_iff]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 StableUnderComposition @UniversallyClosed\n[PROOFSTEP]\nrw [universallyClosed_eq]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 StableUnderComposition (universally (topologically @IsClosedMap))\n[PROOFSTEP]\nexact StableUnderComposition.universally (fun X Y Z f g hf hg => @IsClosedMap.comp _ _ _ _ _ _ g.1.base f.1.base hg hf)\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 PropertyIsLocalAtTarget @UniversallyClosed\n[PROOFSTEP]\nrw [universallyClosed_eq]\n[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 PropertyIsLocalAtTarget (universally (topologically @IsClosedMap))\n[PROOFSTEP]\napply universallyIsLocalAtTargetOfMorphismRestrict\n[GOAL]\ncase hP\u2081\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 RespectsIso (topologically @IsClosedMap)\n[PROOFSTEP]\nexact\n  StableUnderComposition.respectsIso (fun X Y Z f g hf hg => @IsClosedMap.comp _ _ _ _ _ _ g.1.base f.1.base hg hf)\n    (fun f => (TopCat.homeoOfIso (Scheme.forgetToTop.mapIso f)).isClosedMap)\n[GOAL]\ncase hP\u2082\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) {\u03b9 : Type u_1} (U : \u03b9 \u2192 Opens \u2191\u2191Y.toPresheafedSpace),\n    iSup U = \u22a4 \u2192 (\u2200 (i : \u03b9), topologically (@IsClosedMap) (f \u2223_ U i)) \u2192 topologically (@IsClosedMap) f\n[PROOFSTEP]\nintro X Y f \u03b9 U hU H\n[GOAL]\ncase hP\u2082\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191\u2191Y.toPresheafedSpace\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), topologically (@IsClosedMap) (f \u2223_ U i)\n\u22a2 topologically (@IsClosedMap) f\n[PROOFSTEP]\nsimp_rw [topologically, morphismRestrict_base] at H \n[GOAL]\ncase hP\u2082\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u03b9 : Type u_1\nU : \u03b9 \u2192 Opens \u2191\u2191Y.toPresheafedSpace\nhU : iSup U = \u22a4\nH : \u2200 (i : \u03b9), IsClosedMap (Set.restrictPreimage (U i).carrier \u2191f.val.base)\n\u22a2 topologically (@IsClosedMap) f\n[PROOFSTEP]\nexact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed", "llama_tokens": 991, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789178257654, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.5586676703434901}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nn : \u2115\np q : Partition n\n\u22a2 Decidable (p = q)\n[PROOFSTEP]\nsimp [Partition.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\np q : Partition n\n\u22a2 Decidable (p.parts = q.parts)\n[PROOFSTEP]\nexact decidableEq p.parts q.parts\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nc : Composition n\n\u22a2 sum \u2191c.blocks = n\n[PROOFSTEP]\nrw [Multiset.coe_sum, c.blocks_sum]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 Function.Surjective (ofComposition n)\n[PROOFSTEP]\nrintro \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nn : \u2115\nb : Multiset \u2115\nhb\u2081 : \u2200 {i : \u2115}, i \u2208 b \u2192 0 < i\nhb\u2082 : sum b = n\n\u22a2 \u2203 a, ofComposition n a = { parts := b, parts_pos := hb\u2081, parts_sum := hb\u2082 }\n[PROOFSTEP]\nrcases Quotient.exists_rep b with \u27e8b, rfl\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\nn : \u2115\nb : List \u2115\nhb\u2081 : \u2200 {i : \u2115}, i \u2208 Quotient.mk (List.isSetoid \u2115) b \u2192 0 < i\nhb\u2082 : sum (Quotient.mk (List.isSetoid \u2115) b) = n\n\u22a2 \u2203 a, ofComposition n a = { parts := Quotient.mk (List.isSetoid \u2115) b, parts_pos := hb\u2081, parts_sum := hb\u2082 }\n[PROOFSTEP]\nrefine' \u27e8\u27e8b, fun {i} hi => hb\u2081 hi, _\u27e9, Partition.ext _ _ rfl\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\nn : \u2115\nb : List \u2115\nhb\u2081 : \u2200 {i : \u2115}, i \u2208 Quotient.mk (List.isSetoid \u2115) b \u2192 0 < i\nhb\u2082 : sum (Quotient.mk (List.isSetoid \u2115) b) = n\n\u22a2 List.sum b = n\n[PROOFSTEP]\nsimpa using hb\u2082\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\ni : \u2115\nhi : i \u2208 filter (fun x => x \u2260 0) l\n\u22a2 i \u2260 0\n[PROOFSTEP]\napply of_mem_filter hi\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\n\u22a2 sum (filter (fun x => x \u2260 0) l) = n\n[PROOFSTEP]\nhave lt : l.filter (\u00b7 = 0) + l.filter (\u00b7 \u2260 0) = l := filter_add_not _ l\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\nlt : filter (fun x => x = 0) l + filter (fun x => x \u2260 0) l = l\n\u22a2 sum (filter (fun x => x \u2260 0) l) = n\n[PROOFSTEP]\napply_fun Multiset.sum at lt \n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\nlt : sum (filter (fun x => x = 0) l + filter (fun x => x \u2260 0) l) = sum l\n\u22a2 sum (filter (fun x => x \u2260 0) l) = n\n[PROOFSTEP]\nhave lz : (l.filter (\u00b7 = 0)).sum = 0 := by\n  rw [Multiset.sum_eq_zero_iff]\n  simp\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\nlt : sum (filter (fun x => x = 0) l + filter (fun x => x \u2260 0) l) = sum l\n\u22a2 sum (filter (fun x => x = 0) l) = 0\n[PROOFSTEP]\nrw [Multiset.sum_eq_zero_iff]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\nlt : sum (filter (fun x => x = 0) l + filter (fun x => x \u2260 0) l) = sum l\n\u22a2 \u2200 (x : \u2115), x \u2208 filter (fun x => x = 0) l \u2192 x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : Multiset \u2115\nhl : sum l = n\nlt : sum (filter (fun x => x = 0) l + filter (fun x => x \u2260 0) l) = sum l\nlz : sum (filter (fun x => x = 0) l) = 0\n\u22a2 sum (filter (fun x => x \u2260 0) l) = n\n[PROOFSTEP]\nrwa [sum_add (filter (fun x => x = 0) l) (filter (fun x => \u00acx = 0) l), lz, hl, zero_add] at lt \n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Partition", "llama_tokens": 1395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.787931185683219, "lm_q2_score": 0.7090191214879992, "lm_q1q2_score": 0.5586582770661135}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 Nat.IsSqrt n (sqrt n)\n[PROOFSTEP]\nmatch n with\n| 0 => simp [IsSqrt]\n| 1 => simp [IsSqrt]\n| n + 2 =>\n  have h : \u00ac(n + 2) \u2264 1 := by simp\n  simp only [IsSqrt, sqrt, h, ite_false]\n  refine \u27e8sqrt.iter_sq_le _ _, sqrt.lt_iter_succ_sq _ _ ?_\u27e9\n  simp only [mul_add, add_mul, one_mul, mul_one, \u2190 add_assoc]\n  rw [lt_add_one_iff, add_assoc, \u2190 mul_two]\n  refine le_trans (div_add_mod' (n + 2) 2).ge ?_\n  rw [add_comm, add_le_add_iff_right, add_mod_right]\n  simp only [zero_lt_two, add_div_right, succ_mul_succ_eq]\n  refine le_trans (b := 1) ?_ ?_\n  \u00b7 exact (lt_succ.1 $ mod_lt n zero_lt_two)\n  \u00b7 simp only [le_add_iff_nonneg_left]; exact zero_le _\n[GOAL]\nn : \u2115\n\u22a2 Nat.IsSqrt 0 (sqrt 0)\n[PROOFSTEP]\nsimp [IsSqrt]\n[GOAL]\nn : \u2115\n\u22a2 Nat.IsSqrt 1 (sqrt 1)\n[PROOFSTEP]\nsimp [IsSqrt]\n[GOAL]\nn\u271d n : \u2115\n\u22a2 Nat.IsSqrt (n + 2) (sqrt (n + 2))\n[PROOFSTEP]\nhave h : \u00ac(n + 2) \u2264 1 := by simp\n[GOAL]\nn\u271d n : \u2115\n\u22a2 \u00acn + 2 \u2264 1\n[PROOFSTEP]\nsimp\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 Nat.IsSqrt (n + 2) (sqrt (n + 2))\n[PROOFSTEP]\nsimp only [IsSqrt, sqrt, h, ite_false]\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 sqrt.iter (n + 2) ((n + 2) / 2) * sqrt.iter (n + 2) ((n + 2) / 2) \u2264 n + 2 \u2227\n    n + 2 < (sqrt.iter (n + 2) ((n + 2) / 2) + 1) * (sqrt.iter (n + 2) ((n + 2) / 2) + 1)\n[PROOFSTEP]\nrefine \u27e8sqrt.iter_sq_le _ _, sqrt.lt_iter_succ_sq _ _ ?_\u27e9\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 n + 2 < ((n + 2) / 2 + 1) * ((n + 2) / 2 + 1)\n[PROOFSTEP]\nsimp only [mul_add, add_mul, one_mul, mul_one, \u2190 add_assoc]\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 n + 2 < (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 + (n + 2) / 2 + 1\n[PROOFSTEP]\nrw [lt_add_one_iff, add_assoc, \u2190 mul_two]\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 n + 2 \u2264 (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 * 2\n[PROOFSTEP]\nrefine le_trans (div_add_mod' (n + 2) 2).ge ?_\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 (n + 2) / 2 * 2 + (n + 2) % 2 \u2264 (n + 2) / 2 * ((n + 2) / 2) + (n + 2) / 2 * 2\n[PROOFSTEP]\nrw [add_comm, add_le_add_iff_right, add_mod_right]\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 n % 2 \u2264 (n + 2) / 2 * ((n + 2) / 2)\n[PROOFSTEP]\nsimp only [zero_lt_two, add_div_right, succ_mul_succ_eq]\n[GOAL]\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 n % 2 \u2264 n / 2 * (n / 2) + n / 2 + n / 2 + 1\n[PROOFSTEP]\nrefine le_trans (b := 1) ?_ ?_\n[GOAL]\ncase refine_1\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 n % 2 \u2264 1\n[PROOFSTEP]\nexact (lt_succ.1 $ mod_lt n zero_lt_two)\n[GOAL]\ncase refine_2\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 1 \u2264 n / 2 * (n / 2) + n / 2 + n / 2 + 1\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_left]\n[GOAL]\ncase refine_2\nn\u271d n : \u2115\nh : \u00acn + 2 \u2264 1\n\u22a2 0 \u2264 n / 2 * (n / 2) + n / 2 + n / 2\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\nn : \u2115\n\u22a2 n \u2264 sqrt n * sqrt n + sqrt n + sqrt n\n[PROOFSTEP]\nrw [\u2190 succ_mul]\n[GOAL]\nn : \u2115\n\u22a2 n \u2264 succ (sqrt n) * sqrt n + sqrt n\n[PROOFSTEP]\nexact le_of_lt_succ (lt_succ_sqrt n)\n[GOAL]\nm n : \u2115\n\u22a2 m \u2264 sqrt n \u2194 m ^ 2 \u2264 n\n[PROOFSTEP]\nsimpa only [pow_two] using le_sqrt\n[GOAL]\nn : \u2115\nh : sqrt n = 0\n\u22a2 sqrt n < 1\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn : \u2115\nh : sqrt n = 0\n\u22a2 0 < 1\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\n\u22a2 n = 0 \u2192 sqrt n = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u22a2 sqrt 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn q : \u2115\n\u22a2 q = sqrt n \u2194 q ^ 2 \u2264 n \u2227 n < (q + 1) ^ 2\n[PROOFSTEP]\nsimpa only [pow_two] using eq_sqrt\n[GOAL]\nn : \u2115\nh : sqrt n = 1\n\u22a2 sqrt n < 2\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn : \u2115\nh : sqrt n = 1\n\u22a2 1 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\nh : 1 < n\n\u22a2 n < n * n\n[PROOFSTEP]\nhave := Nat.mul_lt_mul_of_pos_left h (lt_of_succ_lt h)\n[GOAL]\nn : \u2115\nh : 1 < n\nthis : n * 1 < n * n\n\u22a2 n < n * n\n[PROOFSTEP]\nrwa [mul_one] at this \n[GOAL]\nn a : \u2115\nh : a \u2264 n + n\n\u22a2 n * n + a < succ n * succ n\n[PROOFSTEP]\nrw [succ_mul, mul_succ, add_succ, add_assoc]\n[GOAL]\nn a : \u2115\nh : a \u2264 n + n\n\u22a2 n * n + a < succ (n * n + (n + n))\n[PROOFSTEP]\nexact lt_succ_of_le (Nat.add_le_add_left h _)\n[GOAL]\nn : \u2115\n\u22a2 sqrt n * sqrt n + sqrt n \u2264 Nat.mul (succ (succ (sqrt n))) (succ (sqrt n))\n[PROOFSTEP]\nrefine' add_le_add (Nat.mul_le_mul_right _ _) _\n[GOAL]\ncase refine'_1\nn : \u2115\n\u22a2 sqrt n \u2264 succ (succ (sqrt n))\n[PROOFSTEP]\nexact Nat.le_add_right _ 2\n[GOAL]\ncase refine'_2\nn : \u2115\n\u22a2 sqrt n \u2264 succ (succ (sqrt n))\n[PROOFSTEP]\nexact Nat.le_add_right _ 2\n[GOAL]\nx : \u2115\nx\u271d : \u2203 n, n * n = x\nn : \u2115\nhn : n * n = x\n\u22a2 sqrt x * sqrt x = x\n[PROOFSTEP]\nrw [\u2190 hn, sqrt_eq]\n[GOAL]\nx : \u2115\n\u22a2 (\u2203 n, n ^ 2 = x) \u2194 sqrt x ^ 2 = x\n[PROOFSTEP]\nsimpa only [pow_two] using exists_mul_self x\n[GOAL]\nn m : \u2115\nhl : m * m < n\nhr : n < (m + 1) * (m + 1)\n\u22a2 \u00ac\u2203 t, t * t = n\n[PROOFSTEP]\nrintro \u27e8t, rfl\u27e9\n[GOAL]\ncase intro\nm t : \u2115\nhl : m * m < t * t\nhr : t * t < (m + 1) * (m + 1)\n\u22a2 False\n[PROOFSTEP]\nhave h1 : m < t := Nat.mul_self_lt_mul_self_iff.mpr hl\n[GOAL]\ncase intro\nm t : \u2115\nhl : m * m < t * t\nhr : t * t < (m + 1) * (m + 1)\nh1 : m < t\n\u22a2 False\n[PROOFSTEP]\nhave h2 : t < m + 1 := Nat.mul_self_lt_mul_self_iff.mpr hr\n[GOAL]\ncase intro\nm t : \u2115\nhl : m * m < t * t\nhr : t * t < (m + 1) * (m + 1)\nh1 : m < t\nh2 : t < m + 1\n\u22a2 False\n[PROOFSTEP]\nexact (not_lt_of_ge <| le_of_lt_succ h2) h1\n[GOAL]\nn m : \u2115\nhl : m ^ 2 < n\nhr : n < (m + 1) ^ 2\n\u22a2 \u00ac\u2203 t, t ^ 2 = n\n[PROOFSTEP]\nsimpa only [pow_two] using not_exists_sq (by simpa only [pow_two] using hl) (by simpa only [pow_two] using hr)\n[GOAL]\nn m : \u2115\nhl : m ^ 2 < n\nhr : n < (m + 1) ^ 2\n\u22a2 ?m.11228 * ?m.11228 < n\n[PROOFSTEP]\nsimpa only [pow_two] using hl\n[GOAL]\nn m : \u2115\nhl : m ^ 2 < n\nhr : n < (m + 1) ^ 2\n\u22a2 n < (m + 1) * (m + 1)\n[PROOFSTEP]\nsimpa only [pow_two] using hr\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Sqrt", "llama_tokens": 3098, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744939732856, "lm_q2_score": 0.6825737408694988, "lm_q1q2_score": 0.5580548807908331}}
{"text": "[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nk : Fin n\n\u22a2 \u2191(tail (cons y s)) k = \u2191s k\n[PROOFSTEP]\nsimp only [tail_apply, cons_succ]\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\n\u22a2 cons (\u2191t 0) (tail t) = t\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\n\u22a2 \u2191(cons (\u2191t 0) (tail t)) a = \u2191t a\n[PROOFSTEP]\nby_cases c_a : a = 0\n[GOAL]\ncase pos\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\nc_a : a = 0\n\u22a2 \u2191(cons (\u2191t 0) (tail t)) a = \u2191t a\n[PROOFSTEP]\nrw [c_a, cons_zero]\n[GOAL]\ncase neg\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\nc_a : \u00aca = 0\n\u22a2 \u2191(cons (\u2191t 0) (tail t)) a = \u2191t a\n[PROOFSTEP]\nrw [\u2190 Fin.succ_pred a c_a, cons_succ, \u2190 tail_apply]\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\n\u22a2 cons 0 0 = 0\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\n\u22a2 \u2191(cons 0 0) a = \u21910 a\n[PROOFSTEP]\nby_cases c : a = 0\n[GOAL]\ncase pos\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\nc : a = 0\n\u22a2 \u2191(cons 0 0) a = \u21910 a\n[PROOFSTEP]\nsimp [c]\n[GOAL]\ncase neg\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\nc : \u00aca = 0\n\u22a2 \u2191(cons 0 0) a = \u21910 a\n[PROOFSTEP]\nrw [\u2190 Fin.succ_pred a c, cons_succ]\n[GOAL]\ncase neg\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\na : Fin (n + 1)\nc : \u00aca = 0\n\u22a2 \u21910 (Fin.pred a c) = \u21910 (Fin.succ (Fin.pred a c))\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nh : y \u2260 0\n\u22a2 cons y s \u2260 0\n[PROOFSTEP]\ncontrapose! h with c\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nc : cons y s = 0\n\u22a2 y = 0\n[PROOFSTEP]\nrw [\u2190 cons_zero y s, c, Finsupp.coe_zero, Pi.zero_apply]\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nh : s \u2260 0\n\u22a2 cons y s \u2260 0\n[PROOFSTEP]\ncontrapose! h with c\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nc : cons y s = 0\n\u22a2 s = 0\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nc : cons y s = 0\na : Fin n\n\u22a2 \u2191s a = \u21910 a\n[PROOFSTEP]\nsimp [\u2190 cons_succ a y s, c]\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\n\u22a2 cons y s \u2260 0 \u2194 y \u2260 0 \u2228 s \u2260 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.casesOn cons_ne_zero_of_left cons_ne_zero_of_right\u27e9\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nh : cons y s \u2260 0\n\u22a2 y \u2260 0 \u2228 s \u2260 0\n[PROOFSTEP]\nrefine' imp_iff_not_or.1 fun h' c => h _\n[GOAL]\nn : \u2115\ni : Fin n\nM : Type u_1\ninst\u271d : Zero M\ny : M\nt : Fin (n + 1) \u2192\u2080 M\ns : Fin n \u2192\u2080 M\nh : cons y s \u2260 0\nh' : y = 0\nc : s = 0\n\u22a2 cons y s = 0\n[PROOFSTEP]\nrw [h', c, Finsupp.cons_zero_zero]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.Fin", "llama_tokens": 1852, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744939732856, "lm_q2_score": 0.6825737214979745, "lm_q1q2_score": 0.5580548649531689}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nr : R\n\u03b1 : F \u27f6 G\n\u22a2 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), F.map f \u226b (fun X => r \u2022 NatTrans.app \u03b1 X) Y = (fun X => r \u2022 NatTrans.app \u03b1 X) X \u226b G.map f\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nr : R\n\u03b1 : F \u27f6 G\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f\u271d \u226b (fun X => r \u2022 NatTrans.app \u03b1 X) Y\u271d = (fun X => r \u2022 NatTrans.app \u03b1 X) X\u271d \u226b G.map f\u271d\n[PROOFSTEP]\nrw [comp_smul, smul_comp, \u03b1.naturality]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\n\u22a2 \u2200 (b : F \u27f6 G), 1 \u2022 b = b\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nb\u271d : F \u27f6 G\n\u22a2 1 \u2022 b\u271d = b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nb\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (1 \u2022 b\u271d) x\u271d = NatTrans.app b\u271d x\u271d\n[PROOFSTEP]\napply one_smul\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\n\u22a2 \u2200 (x y : R) (b : F \u27f6 G), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nx\u271d y\u271d : R\nb\u271d : F \u27f6 G\n\u22a2 (x\u271d * y\u271d) \u2022 b\u271d = x\u271d \u2022 y\u271d \u2022 b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nx\u271d\u00b9 y\u271d : R\nb\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app ((x\u271d\u00b9 * y\u271d) \u2022 b\u271d) x\u271d = NatTrans.app (x\u271d\u00b9 \u2022 y\u271d \u2022 b\u271d) x\u271d\n[PROOFSTEP]\napply mul_smul\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\n\u22a2 \u2200 (a : R), a \u2022 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\na\u271d : R\n\u22a2 a\u271d \u2022 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\na\u271d : R\nx\u271d : C\n\u22a2 NatTrans.app (a\u271d \u2022 0) x\u271d = NatTrans.app 0 x\u271d\n[PROOFSTEP]\napply smul_zero\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\n\u22a2 \u2200 (a : R) (x y : F \u27f6 G), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\na\u271d : R\nx\u271d y\u271d : F \u27f6 G\n\u22a2 a\u271d \u2022 (x\u271d + y\u271d) = a\u271d \u2022 x\u271d + a\u271d \u2022 y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\na\u271d : R\nx\u271d\u00b9 y\u271d : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (a\u271d \u2022 (x\u271d\u00b9 + y\u271d)) x\u271d = NatTrans.app (a\u271d \u2022 x\u271d\u00b9 + a\u271d \u2022 y\u271d) x\u271d\n[PROOFSTEP]\napply smul_add\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\n\u22a2 \u2200 (r s : R) (x : F \u27f6 G), (r + s) \u2022 x = r \u2022 x + s \u2022 x\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nr\u271d s\u271d : R\nx\u271d : F \u27f6 G\n\u22a2 (r\u271d + s\u271d) \u2022 x\u271d = r\u271d \u2022 x\u271d + s\u271d \u2022 x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nr\u271d s\u271d : R\nx\u271d\u00b9 : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app ((r\u271d + s\u271d) \u2022 x\u271d\u00b9) x\u271d = NatTrans.app (r\u271d \u2022 x\u271d\u00b9 + s\u271d \u2022 x\u271d\u00b9) x\u271d\n[PROOFSTEP]\napply add_smul\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\n\u22a2 \u2200 (x : F \u27f6 G), 0 \u2022 x = 0\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nx\u271d : F \u27f6 G\n\u22a2 0 \u2022 x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nF G : C \u2964 D\nx\u271d\u00b9 : F \u27f6 G\nx\u271d : C\n\u22a2 NatTrans.app (0 \u2022 x\u271d\u00b9) x\u271d = NatTrans.app 0 x\u271d\n[PROOFSTEP]\napply zero_smul\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\n\u22a2 \u2200 (X Y Z : C \u2964 D) (r : R) (f : X \u27f6 Y) (g : Y \u27f6 Z), (r \u2022 f) \u226b g = r \u2022 f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nr\u271d : R\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 (r\u271d \u2022 f\u271d) \u226b g\u271d = r\u271d \u2022 f\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nr\u271d : R\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : C\n\u22a2 NatTrans.app ((r\u271d \u2022 f\u271d) \u226b g\u271d) x\u271d = NatTrans.app (r\u271d \u2022 f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\napply smul_comp\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\n\u22a2 \u2200 (X Y Z : C \u2964 D) (f : X \u27f6 Y) (r : R) (g : Y \u27f6 Z), f \u226b (r \u2022 g) = r \u2022 f \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nf\u271d : X\u271d \u27f6 Y\u271d\nr\u271d : R\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 f\u271d \u226b (r\u271d \u2022 g\u271d) = r\u271d \u2022 f\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u00b3 : Category.{?u.87, u_2} C\ninst\u271d\u00b2 : Category.{?u.91, u_3} D\ninst\u271d\u00b9 : Preadditive D\ninst\u271d : Linear R D\nX\u271d Y\u271d Z\u271d : C \u2964 D\nf\u271d : X\u271d \u27f6 Y\u271d\nr\u271d : R\ng\u271d : Y\u271d \u27f6 Z\u271d\nx\u271d : C\n\u22a2 NatTrans.app (f\u271d \u226b (r\u271d \u2022 g\u271d)) x\u271d = NatTrans.app (r\u271d \u2022 f\u271d \u226b g\u271d) x\u271d\n[PROOFSTEP]\napply comp_smul\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Linear.FunctorCategory", "llama_tokens": 4541, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8807970873650403, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5579059153545638}}
{"text": "[GOAL]\nM : Type u_1\nS T : Set M\ninst\u271d : MulOneClass M\n\u22a2 1 \u2208 centralizer S\n[PROOFSTEP]\nsimp [mem_centralizer_iff]\n[GOAL]\nM : Type u_1\nS T : Set M\ninst\u271d : MulZeroClass M\n\u22a2 0 \u2208 centralizer S\n[PROOFSTEP]\nsimp [mem_centralizer_iff]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : Semigroup M\nha : a \u2208 centralizer S\nhb : b \u2208 centralizer S\ng : M\nhg : g \u2208 S\n\u22a2 g * (a * b) = a * b * g\n[PROOFSTEP]\nrw [mul_assoc, \u2190 hb g hg, \u2190 mul_assoc, ha g hg, mul_assoc]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : Group M\nha : a \u2208 centralizer S\ng : M\nhg : g \u2208 S\n\u22a2 g * a\u207b\u00b9 = a\u207b\u00b9 * g\n[PROOFSTEP]\nrw [mul_inv_eq_iff_eq_mul, mul_assoc, eq_inv_mul_iff_mul_eq, ha g hg]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : Distrib M\nha : a \u2208 centralizer S\nhb : b \u2208 centralizer S\nc : M\nhc : c \u2208 S\n\u22a2 c * (a + b) = (a + b) * c\n[PROOFSTEP]\nrw [add_mul, mul_add, ha c hc, hb c hc]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d\u00b9 : Mul M\ninst\u271d : HasDistribNeg M\nha : a \u2208 centralizer S\nc : M\nhc : c \u2208 S\n\u22a2 c * -a = -a * c\n[PROOFSTEP]\nrw [mul_neg, ha c hc, neg_mul]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : GroupWithZero M\nha : a \u2208 centralizer S\nh : a = 0\n\u22a2 a\u207b\u00b9 \u2208 centralizer S\n[PROOFSTEP]\nrw [h, inv_zero]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : GroupWithZero M\nha : a \u2208 centralizer S\nh : a = 0\n\u22a2 0 \u2208 centralizer S\n[PROOFSTEP]\nexact zero_mem_centralizer S\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : GroupWithZero M\nha : a \u2208 centralizer S\nha0 : a \u2260 0\nc : M\nhc : c \u2208 S\n\u22a2 c * a\u207b\u00b9 = a\u207b\u00b9 * c\n[PROOFSTEP]\nrw [mul_inv_eq_iff_eq_mul\u2080 ha0, mul_assoc, eq_inv_mul_iff_mul_eq\u2080 ha0, ha c hc]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : Group M\nha : a \u2208 centralizer S\nhb : b \u2208 centralizer S\n\u22a2 a / b \u2208 centralizer S\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : Group M\nha : a \u2208 centralizer S\nhb : b \u2208 centralizer S\n\u22a2 a * b\u207b\u00b9 \u2208 centralizer S\n[PROOFSTEP]\nexact mul_mem_centralizer ha (inv_mem_centralizer hb)\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : GroupWithZero M\nha : a \u2208 centralizer S\nhb : b \u2208 centralizer S\n\u22a2 a / b \u2208 centralizer S\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u_1\nS T : Set M\na b : M\ninst\u271d : GroupWithZero M\nha : a \u2208 centralizer S\nhb : b \u2208 centralizer S\n\u22a2 a * b\u207b\u00b9 \u2208 centralizer S\n[PROOFSTEP]\nexact mul_mem_centralizer ha (inv_mem_centralizer\u2080 hb)\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Subsemigroup.Centralizer", "llama_tokens": 1154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.7279754430043072, "lm_q1q2_score": 0.5578429542968859}}
{"text": "[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx : M\n\u22a2 bilin B 0 x = 0\n[PROOFSTEP]\nrw [\u2190 @zero_smul R _ _ _ _ (0 : M), smul_left, zero_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx : M\n\u22a2 bilin B x 0 = 0\n[PROOFSTEP]\nrw [\u2190 @zero_smul R _ _ _ _ (0 : M), smul_right, zero_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx y : M\u2081\n\u22a2 bilin B\u2081 (-x) y = -bilin B\u2081 x y\n[PROOFSTEP]\nrw [\u2190 @neg_one_smul R\u2081 _ _, smul_left, neg_one_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx y : M\u2081\n\u22a2 bilin B\u2081 x (-y) = -bilin B\u2081 x y\n[PROOFSTEP]\nrw [\u2190 @neg_one_smul R\u2081 _ _, smul_right, neg_one_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx y z : M\u2081\n\u22a2 bilin B\u2081 (x - y) z = bilin B\u2081 x z - bilin B\u2081 y z\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg, add_left, neg_left]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx y z : M\u2081\n\u22a2 bilin B\u2081 x (y - z) = bilin B\u2081 x y - bilin B\u2081 x z\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg, add_right, neg_right]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nB D : BilinForm R M\nh : B.bilin = D.bilin\n\u22a2 B = D\n[PROOFSTEP]\ncases B\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nD : BilinForm R M\nbilin\u271d : M \u2192 M \u2192 R\nbilin_add_left\u271d : \u2200 (x y z : M), bilin\u271d (x + y) z = bilin\u271d x z + bilin\u271d y z\nbilin_smul_left\u271d : \u2200 (a : R) (x y : M), bilin\u271d (a \u2022 x) y = a * bilin\u271d x y\nbilin_add_right\u271d : \u2200 (x y z : M), bilin\u271d x (y + z) = bilin\u271d x y + bilin\u271d x z\nbilin_smul_right\u271d : \u2200 (a : R) (x y : M), bilin\u271d x (a \u2022 y) = a * bilin\u271d x y\nh :\n  { bilin := bilin\u271d, bilin_add_left := bilin_add_left\u271d, bilin_smul_left := bilin_smul_left\u271d,\n        bilin_add_right := bilin_add_right\u271d, bilin_smul_right := bilin_smul_right\u271d }.bilin =\n    D.bilin\n\u22a2 { bilin := bilin\u271d, bilin_add_left := bilin_add_left\u271d, bilin_smul_left := bilin_smul_left\u271d,\n      bilin_add_right := bilin_add_right\u271d, bilin_smul_right := bilin_smul_right\u271d } =\n    D\n[PROOFSTEP]\ncases D\n[GOAL]\ncase mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nbilin\u271d\u00b9 : M \u2192 M \u2192 R\nbilin_add_left\u271d\u00b9 : \u2200 (x y z : M), bilin\u271d\u00b9 (x + y) z = bilin\u271d\u00b9 x z + bilin\u271d\u00b9 y z\nbilin_smul_left\u271d\u00b9 : \u2200 (a : R) (x y : M), bilin\u271d\u00b9 (a \u2022 x) y = a * bilin\u271d\u00b9 x y\nbilin_add_right\u271d\u00b9 : \u2200 (x y z : M), bilin\u271d\u00b9 x (y + z) = bilin\u271d\u00b9 x y + bilin\u271d\u00b9 x z\nbilin_smul_right\u271d\u00b9 : \u2200 (a : R) (x y : M), bilin\u271d\u00b9 x (a \u2022 y) = a * bilin\u271d\u00b9 x y\nbilin\u271d : M \u2192 M \u2192 R\nbilin_add_left\u271d : \u2200 (x y z : M), bilin\u271d (x + y) z = bilin\u271d x z + bilin\u271d y z\nbilin_smul_left\u271d : \u2200 (a : R) (x y : M), bilin\u271d (a \u2022 x) y = a * bilin\u271d x y\nbilin_add_right\u271d : \u2200 (x y z : M), bilin\u271d x (y + z) = bilin\u271d x y + bilin\u271d x z\nbilin_smul_right\u271d : \u2200 (a : R) (x y : M), bilin\u271d x (a \u2022 y) = a * bilin\u271d x y\nh :\n  { bilin := bilin\u271d\u00b9, bilin_add_left := bilin_add_left\u271d\u00b9, bilin_smul_left := bilin_smul_left\u271d\u00b9,\n        bilin_add_right := bilin_add_right\u271d\u00b9, bilin_smul_right := bilin_smul_right\u271d\u00b9 }.bilin =\n    { bilin := bilin\u271d, bilin_add_left := bilin_add_left\u271d, bilin_smul_left := bilin_smul_left\u271d,\n        bilin_add_right := bilin_add_right\u271d, bilin_smul_right := bilin_smul_right\u271d }.bilin\n\u22a2 { bilin := bilin\u271d\u00b9, bilin_add_left := bilin_add_left\u271d\u00b9, bilin_smul_left := bilin_smul_left\u271d\u00b9,\n      bilin_add_right := bilin_add_right\u271d\u00b9, bilin_smul_right := bilin_smul_right\u271d\u00b9 } =\n    { bilin := bilin\u271d, bilin_add_left := bilin_add_left\u271d, bilin_smul_left := bilin_smul_left\u271d,\n      bilin_add_right := bilin_add_right\u271d, bilin_smul_right := bilin_smul_right\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nH : \u2200 (x y : M), bilin B x y = bilin D x y\n\u22a2 B.bilin = D.bilin\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nH : \u2200 (x y : M), bilin B x y = bilin D x y\nx\u271d\u00b9 x\u271d : M\n\u22a2 bilin B x\u271d\u00b9 x\u271d = bilin D x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nexact H _ _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nB D : BilinForm R M\nx y z : M\n\u22a2 (fun x y => bilin B x y + bilin D x y) (x + y) z =\n    (fun x y => bilin B x y + bilin D x y) x z + (fun x y => bilin B x y + bilin D x y) y z\n[PROOFSTEP]\nsimp only [add_left, add_left, add_add_add_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nB D : BilinForm R M\na : R\nx y : M\n\u22a2 (fun x y => bilin B x y + bilin D x y) (a \u2022 x) y = a * (fun x y => bilin B x y + bilin D x y) x y\n[PROOFSTEP]\nsimp only [smul_left, smul_left, mul_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nB D : BilinForm R M\nx y z : M\n\u22a2 (fun x y => bilin B x y + bilin D x y) x (y + z) =\n    (fun x y => bilin B x y + bilin D x y) x y + (fun x y => bilin B x y + bilin D x y) x z\n[PROOFSTEP]\nsimp only [add_right, add_right, add_add_add_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\nB D : BilinForm R M\na : R\nx y : M\n\u22a2 (fun x y => bilin B x y + bilin D x y) x (a \u2022 y) = a * (fun x y => bilin B x y + bilin D x y) x y\n[PROOFSTEP]\nsimp only [smul_right, smul_right, mul_add]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\n\u03b1 : Type ?u.105863\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DistribMulAction \u03b1 R\ninst\u271d : SMulCommClass \u03b1 R R\nc : \u03b1\nB : BilinForm R M\nx y z : M\n\u22a2 (fun x y => c \u2022 bilin B x y) (x + y) z = (fun x y => c \u2022 bilin B x y) x z + (fun x y => c \u2022 bilin B x y) y z\n[PROOFSTEP]\nsimp only [add_left, smul_add]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\n\u03b1 : Type ?u.105863\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DistribMulAction \u03b1 R\ninst\u271d : SMulCommClass \u03b1 R R\nc : \u03b1\nB : BilinForm R M\na : R\nx y : M\n\u22a2 (fun x y => c \u2022 bilin B x y) (a \u2022 x) y = a * (fun x y => c \u2022 bilin B x y) x y\n[PROOFSTEP]\nsimp only [smul_left, mul_smul_comm]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\n\u03b1 : Type ?u.105863\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DistribMulAction \u03b1 R\ninst\u271d : SMulCommClass \u03b1 R R\nc : \u03b1\nB : BilinForm R M\nx y z : M\n\u22a2 (fun x y => c \u2022 bilin B x y) x (y + z) = (fun x y => c \u2022 bilin B x y) x y + (fun x y => c \u2022 bilin B x y) x z\n[PROOFSTEP]\nsimp only [add_right, smul_add]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\n\u03b1 : Type ?u.105863\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : DistribMulAction \u03b1 R\ninst\u271d : SMulCommClass \u03b1 R R\nc : \u03b1\nB : BilinForm R M\na : R\nx y : M\n\u22a2 (fun x y => c \u2022 bilin B x y) x (a \u2022 y) = a * (fun x y => c \u2022 bilin B x y) x y\n[PROOFSTEP]\nsimp only [smul_right, mul_smul_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 B : BilinForm R\u2081 M\u2081\nx y z : M\u2081\n\u22a2 (fun x y => -bilin B x y) (x + y) z = (fun x y => -bilin B x y) x z + (fun x y => -bilin B x y) y z\n[PROOFSTEP]\nsimp only [add_left, neg_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 B : BilinForm R\u2081 M\u2081\na : R\u2081\nx y : M\u2081\n\u22a2 (fun x y => -bilin B x y) (a \u2022 x) y = a * (fun x y => -bilin B x y) x y\n[PROOFSTEP]\nsimp only [smul_left, mul_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 B : BilinForm R\u2081 M\u2081\nx y z : M\u2081\n\u22a2 (fun x y => -bilin B x y) x (y + z) = (fun x y => -bilin B x y) x y + (fun x y => -bilin B x y) x z\n[PROOFSTEP]\nsimp only [add_right, neg_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 B : BilinForm R\u2081 M\u2081\na : R\u2081\nx y : M\u2081\n\u22a2 (fun x y => -bilin B x y) x (a \u2022 y) = a * (fun x y => -bilin B x y) x y\n[PROOFSTEP]\nsimp only [smul_right, mul_neg]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 B D : BilinForm R\u2081 M\u2081\nx y z : M\u2081\n\u22a2 (fun x y => bilin B x y - bilin D x y) (x + y) z =\n    (fun x y => bilin B x y - bilin D x y) x z + (fun x y => bilin B x y - bilin D x y) y z\n[PROOFSTEP]\nsimp only [add_left, add_left, add_sub_add_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 B D : BilinForm R\u2081 M\u2081\na : R\u2081\nx y : M\u2081\n\u22a2 (fun x y => bilin B x y - bilin D x y) (a \u2022 x) y = a * (fun x y => bilin B x y - bilin D x y) x y\n[PROOFSTEP]\nsimp only [smul_left, smul_left, mul_sub]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 B D : BilinForm R\u2081 M\u2081\nx y z : M\u2081\n\u22a2 (fun x y => bilin B x y - bilin D x y) x (y + z) =\n    (fun x y => bilin B x y - bilin D x y) x y + (fun x y => bilin B x y - bilin D x y) x z\n[PROOFSTEP]\nsimp only [add_right, add_right, add_sub_add_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD\u271d : BilinForm R M\nD\u2081 B D : BilinForm R\u2081 M\u2081\na : R\u2081\nx y : M\u2081\n\u22a2 (fun x y => bilin B x y - bilin D x y) x (a \u2022 y) = a * (fun x y => bilin B x y - bilin D x y) x y\n[PROOFSTEP]\nsimp only [smul_right, smul_right, mul_sub]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nA\u2081 A\u2082 : BilinForm R M\n\u22a2 (fun A =>\n        { bilin := fun i j => bilin A j i,\n          bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n          bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n          bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n          bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n      (A\u2081 + A\u2082) =\n    (fun A =>\n          { bilin := fun i j => bilin A j i,\n            bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n            bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n            bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n            bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n        A\u2081 +\n      (fun A =>\n          { bilin := fun i j => bilin A j i,\n            bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n            bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n            bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n            bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n        A\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nA\u2081 A\u2082 : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin\n      ((fun A =>\n          { bilin := fun i j => bilin A j i,\n            bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n            bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n            bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n            bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n        (A\u2081 + A\u2082))\n      x\u271d y\u271d =\n    bilin\n      ((fun A =>\n            { bilin := fun i j => bilin A j i,\n              bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n              bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n              bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n              bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n          A\u2081 +\n        (fun A =>\n            { bilin := fun i j => bilin A j i,\n              bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n              bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n              bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n              bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n          A\u2082)\n      x\u271d y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nc : R\u2082\nA : BilinForm R M\n\u22a2 AddHom.toFun\n      {\n        toFun := fun A =>\n          { bilin := fun i j => bilin A j i,\n            bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n            bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n            bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n            bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) },\n        map_add' :=\n          (_ :\n            \u2200 (A\u2081 A\u2082 : BilinForm R M),\n              (fun A =>\n                    { bilin := fun i j => bilin A j i,\n                      bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                      bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                      bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                      bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                  (A\u2081 + A\u2082) =\n                (fun A =>\n                      { bilin := fun i j => bilin A j i,\n                        bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                        bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                        bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                        bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                    A\u2081 +\n                  (fun A =>\n                      { bilin := fun i j => bilin A j i,\n                        bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                        bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                        bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                        bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                    A\u2082) }\n      (c \u2022 A) =\n    \u2191(RingHom.id R\u2082) c \u2022\n      AddHom.toFun\n        {\n          toFun := fun A =>\n            { bilin := fun i j => bilin A j i,\n              bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n              bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n              bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n              bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) },\n          map_add' :=\n            (_ :\n              \u2200 (A\u2081 A\u2082 : BilinForm R M),\n                (fun A =>\n                      { bilin := fun i j => bilin A j i,\n                        bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                        bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                        bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                        bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                    (A\u2081 + A\u2082) =\n                  (fun A =>\n                        { bilin := fun i j => bilin A j i,\n                          bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                          bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                          bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                          bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                      A\u2081 +\n                    (fun A =>\n                        { bilin := fun i j => bilin A j i,\n                          bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                          bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                          bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                          bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                      A\u2082) }\n        A\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nc : R\u2082\nA : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin\n      (AddHom.toFun\n        {\n          toFun := fun A =>\n            { bilin := fun i j => bilin A j i,\n              bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n              bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n              bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n              bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) },\n          map_add' :=\n            (_ :\n              \u2200 (A\u2081 A\u2082 : BilinForm R M),\n                (fun A =>\n                      { bilin := fun i j => bilin A j i,\n                        bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                        bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                        bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                        bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                    (A\u2081 + A\u2082) =\n                  (fun A =>\n                        { bilin := fun i j => bilin A j i,\n                          bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                          bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                          bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                          bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                      A\u2081 +\n                    (fun A =>\n                        { bilin := fun i j => bilin A j i,\n                          bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                          bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                          bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                          bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                      A\u2082) }\n        (c \u2022 A))\n      x\u271d y\u271d =\n    bilin\n      (\u2191(RingHom.id R\u2082) c \u2022\n        AddHom.toFun\n          {\n            toFun := fun A =>\n              { bilin := fun i j => bilin A j i,\n                bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) },\n            map_add' :=\n              (_ :\n                \u2200 (A\u2081 A\u2082 : BilinForm R M),\n                  (fun A =>\n                        { bilin := fun i j => bilin A j i,\n                          bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                          bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                          bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                          bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                      (A\u2081 + A\u2082) =\n                    (fun A =>\n                          { bilin := fun i j => bilin A j i,\n                            bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                            bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                            bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                            bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                        A\u2081 +\n                      (fun A =>\n                          { bilin := fun i j => bilin A j i,\n                            bilin_add_left := (_ : \u2200 (x y z : M), bilin A z (x + y) = bilin A z x + bilin A z y),\n                            bilin_smul_left := (_ : \u2200 (a : R) (x y : M), bilin A y (a \u2022 x) = a * bilin A y x),\n                            bilin_add_right := (_ : \u2200 (x y z : M), bilin A (y + z) x = bilin A y x + bilin A z x),\n                            bilin_smul_right := (_ : \u2200 (a : R) (x y : M), bilin A (a \u2022 y) x = a * bilin A y x) })\n                        A\u2082) }\n          A)\n      x\u271d y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nA : BilinForm R M\n\u22a2 \u2191(flipHomAux R\u2082) (\u2191(flipHomAux R\u2082) A) = A\n[PROOFSTEP]\next A\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nA\u271d : BilinForm R M\nA y\u271d : M\n\u22a2 bilin (\u2191(flipHomAux R\u2082) (\u2191(flipHomAux R\u2082) A\u271d)) A y\u271d = bilin A\u271d A y\u271d\n[PROOFSTEP]\nsimp [flipHomAux]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\n\u22a2 LinearEquiv.trans (flipHom R\u2082) (flipHom R\u2082) = LinearEquiv.refl R\u2082 (BilinForm R M)\n[PROOFSTEP]\next A\n[GOAL]\ncase h.H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\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\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d : Algebra R\u2082 R\nA : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin (\u2191(LinearEquiv.trans (flipHom R\u2082) (flipHom R\u2082)) A) x\u271d y\u271d =\n    bilin (\u2191(LinearEquiv.refl R\u2082 (BilinForm R M)) A) x\u271d y\u271d\n[PROOFSTEP]\nsimp\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA : BilinForm R M\nx\u2081 x\u2082 x : M\n\u22a2 \u2191(toLinHomAux\u2081 A (x\u2081 + x\u2082)) x = \u2191(toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) x\n[PROOFSTEP]\nsimp only [toLinHomAux\u2081, LinearMap.coe_mk, LinearMap.add_apply, add_left, AddHom.coe_mk]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA : BilinForm R M\nc : R\u2082\nx : M\n\u22a2 \u2200 (x_1 : M),\n    \u2191(AddHom.toFun\n            { toFun := toLinHomAux\u2081 A,\n              map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n            (c \u2022 x))\n        x_1 =\n      \u2191(\u2191(RingHom.id R\u2082) c \u2022\n            AddHom.toFun\n              { toFun := toLinHomAux\u2081 A,\n                map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n              x)\n        x_1\n[PROOFSTEP]\ndsimp [toLinHomAux\u2081]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA : BilinForm R M\nc : R\u2082\nx : M\n\u22a2 \u2200 (x_1 : M), bilin A (c \u2022 x) x_1 = c \u2022 bilin A x x_1\n[PROOFSTEP]\nintros\n  -- Porting note: moved out of `simp only`\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA : BilinForm R M\nc : R\u2082\nx x\u271d : M\n\u22a2 bilin A (c \u2022 x) x\u271d = c \u2022 bilin A x x\u271d\n[PROOFSTEP]\nrw [\u2190 algebraMap_smul R c x]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA : BilinForm R M\nc : R\u2082\nx x\u271d : M\n\u22a2 bilin A (\u2191(algebraMap R\u2082 R) c \u2022 x) x\u271d = c \u2022 bilin A x x\u271d\n[PROOFSTEP]\nsimp only [Algebra.smul_def, LinearMap.coe_mk, LinearMap.smul_apply, smul_left]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA\u2081 A\u2082 : BilinForm R M\nx : M\n\u22a2 \u2191(toLinHomAux\u2082 (A\u2081 + A\u2082)) x = \u2191(toLinHomAux\u2082 A\u2081 + toLinHomAux\u2082 A\u2082) x\n[PROOFSTEP]\ndsimp only [toLinHomAux\u2081, toLinHomAux\u2082]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA\u2081 A\u2082 : BilinForm R M\nx : M\n\u22a2 \u2191{\n          toAddHom :=\n            {\n              toFun := fun x =>\n                {\n                  toAddHom :=\n                    { toFun := fun y => bilin (A\u2081 + A\u2082) x y,\n                      map_add' :=\n                        (_ : \u2200 (y z : M), bilin (A\u2081 + A\u2082) x (y + z) = bilin (A\u2081 + A\u2082) x y + bilin (A\u2081 + A\u2082) x z) },\n                  map_smul' := (_ : \u2200 (c : R) (y : M), bilin (A\u2081 + A\u2082) x (c \u2022 y) = c * bilin (A\u2081 + A\u2082) x y) },\n              map_add' :=\n                (_ :\n                  \u2200 (x\u2081 x\u2082 : M),\n                    toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) = toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R\u2082) (x : M),\n                AddHom.toFun\n                    { toFun := toLinHomAux\u2081 (A\u2081 + A\u2082),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x\u2081 x\u2082 : M),\n                            toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) = toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) }\n                    (c \u2022 x) =\n                  \u2191(RingHom.id R\u2082) c \u2022\n                    AddHom.toFun\n                      { toFun := toLinHomAux\u2081 (A\u2081 + A\u2082),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x\u2081 x\u2082 : M),\n                              toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) =\n                                toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) }\n                      x) }\n      x =\n    \u2191({\n            toAddHom :=\n              {\n                toFun := fun x =>\n                  {\n                    toAddHom :=\n                      { toFun := fun y => bilin A\u2081 x y,\n                        map_add' := (_ : \u2200 (y z : M), bilin A\u2081 x (y + z) = bilin A\u2081 x y + bilin A\u2081 x z) },\n                    map_smul' := (_ : \u2200 (c : R) (y : M), bilin A\u2081 x (c \u2022 y) = c * bilin A\u2081 x y) },\n                map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R\u2082) (x : M),\n                  AddHom.toFun\n                      { toFun := toLinHomAux\u2081 A\u2081,\n                        map_add' :=\n                          (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) }\n                      (c \u2022 x) =\n                    \u2191(RingHom.id R\u2082) c \u2022\n                      AddHom.toFun\n                        { toFun := toLinHomAux\u2081 A\u2081,\n                          map_add' :=\n                            (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) }\n                        x) } +\n          {\n            toAddHom :=\n              {\n                toFun := fun x =>\n                  {\n                    toAddHom :=\n                      { toFun := fun y => bilin A\u2082 x y,\n                        map_add' := (_ : \u2200 (y z : M), bilin A\u2082 x (y + z) = bilin A\u2082 x y + bilin A\u2082 x z) },\n                    map_smul' := (_ : \u2200 (c : R) (y : M), bilin A\u2082 x (c \u2022 y) = c * bilin A\u2082 x y) },\n                map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R\u2082) (x : M),\n                  AddHom.toFun\n                      { toFun := toLinHomAux\u2081 A\u2082,\n                        map_add' :=\n                          (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) }\n                      (c \u2022 x) =\n                    \u2191(RingHom.id R\u2082) c \u2022\n                      AddHom.toFun\n                        { toFun := toLinHomAux\u2081 A\u2082,\n                          map_add' :=\n                            (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) }\n                        x) })\n      x\n[PROOFSTEP]\napply LinearMap.ext\n[GOAL]\ncase 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA\u2081 A\u2082 : BilinForm R M\nx : M\n\u22a2 \u2200 (x_1 : M),\n    \u2191(\u2191{\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun y => bilin (A\u2081 + A\u2082) x y,\n                            map_add' :=\n                              (_ :\n                                \u2200 (y z : M), bilin (A\u2081 + A\u2082) x (y + z) = bilin (A\u2081 + A\u2082) x y + bilin (A\u2081 + A\u2082) x z) },\n                        map_smul' := (_ : \u2200 (c : R) (y : M), bilin (A\u2081 + A\u2082) x (c \u2022 y) = c * bilin (A\u2081 + A\u2082) x y) },\n                    map_add' :=\n                      (_ :\n                        \u2200 (x\u2081 x\u2082 : M),\n                          toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) = toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R\u2082) (x : M),\n                      AddHom.toFun\n                          { toFun := toLinHomAux\u2081 (A\u2081 + A\u2082),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x\u2081 x\u2082 : M),\n                                  toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) =\n                                    toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) }\n                          (c \u2022 x) =\n                        \u2191(RingHom.id R\u2082) c \u2022\n                          AddHom.toFun\n                            { toFun := toLinHomAux\u2081 (A\u2081 + A\u2082),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x\u2081 x\u2082 : M),\n                                    toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) =\n                                      toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) }\n                            x) }\n            x)\n        x_1 =\n      \u2191(\u2191({\n                  toAddHom :=\n                    {\n                      toFun := fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun y => bilin A\u2081 x y,\n                              map_add' := (_ : \u2200 (y z : M), bilin A\u2081 x (y + z) = bilin A\u2081 x y + bilin A\u2081 x z) },\n                          map_smul' := (_ : \u2200 (c : R) (y : M), bilin A\u2081 x (c \u2022 y) = c * bilin A\u2081 x y) },\n                      map_add' :=\n                        (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (c : R\u2082) (x : M),\n                        AddHom.toFun\n                            { toFun := toLinHomAux\u2081 A\u2081,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) }\n                            (c \u2022 x) =\n                          \u2191(RingHom.id R\u2082) c \u2022\n                            AddHom.toFun\n                              { toFun := toLinHomAux\u2081 A\u2081,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x\u2081 x\u2082 : M),\n                                      toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) }\n                              x) } +\n                {\n                  toAddHom :=\n                    {\n                      toFun := fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun y => bilin A\u2082 x y,\n                              map_add' := (_ : \u2200 (y z : M), bilin A\u2082 x (y + z) = bilin A\u2082 x y + bilin A\u2082 x z) },\n                          map_smul' := (_ : \u2200 (c : R) (y : M), bilin A\u2082 x (c \u2022 y) = c * bilin A\u2082 x y) },\n                      map_add' :=\n                        (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (c : R\u2082) (x : M),\n                        AddHom.toFun\n                            { toFun := toLinHomAux\u2081 A\u2082,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) }\n                            (c \u2022 x) =\n                          \u2191(RingHom.id R\u2082) c \u2022\n                            AddHom.toFun\n                              { toFun := toLinHomAux\u2081 A\u2082,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x\u2081 x\u2082 : M),\n                                      toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) }\n                              x) })\n            x)\n        x_1\n[PROOFSTEP]\nintro y\n[GOAL]\ncase 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nA\u2081 A\u2082 : BilinForm R M\nx y : M\n\u22a2 \u2191(\u2191{\n              toAddHom :=\n                {\n                  toFun := fun x =>\n                    {\n                      toAddHom :=\n                        { toFun := fun y => bilin (A\u2081 + A\u2082) x y,\n                          map_add' :=\n                            (_ : \u2200 (y z : M), bilin (A\u2081 + A\u2082) x (y + z) = bilin (A\u2081 + A\u2082) x y + bilin (A\u2081 + A\u2082) x z) },\n                      map_smul' := (_ : \u2200 (c : R) (y : M), bilin (A\u2081 + A\u2082) x (c \u2022 y) = c * bilin (A\u2081 + A\u2082) x y) },\n                  map_add' :=\n                    (_ :\n                      \u2200 (x\u2081 x\u2082 : M),\n                        toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) = toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) },\n              map_smul' :=\n                (_ :\n                  \u2200 (c : R\u2082) (x : M),\n                    AddHom.toFun\n                        { toFun := toLinHomAux\u2081 (A\u2081 + A\u2082),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x\u2081 x\u2082 : M),\n                                toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) =\n                                  toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) }\n                        (c \u2022 x) =\n                      \u2191(RingHom.id R\u2082) c \u2022\n                        AddHom.toFun\n                          { toFun := toLinHomAux\u2081 (A\u2081 + A\u2082),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x\u2081 x\u2082 : M),\n                                  toLinHomAux\u2081 (A\u2081 + A\u2082) (x\u2081 + x\u2082) =\n                                    toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2081 + toLinHomAux\u2081 (A\u2081 + A\u2082) x\u2082) }\n                          x) }\n          x)\n      y =\n    \u2191(\u2191({\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun y => bilin A\u2081 x y,\n                            map_add' := (_ : \u2200 (y z : M), bilin A\u2081 x (y + z) = bilin A\u2081 x y + bilin A\u2081 x z) },\n                        map_smul' := (_ : \u2200 (c : R) (y : M), bilin A\u2081 x (c \u2022 y) = c * bilin A\u2081 x y) },\n                    map_add' :=\n                      (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R\u2082) (x : M),\n                      AddHom.toFun\n                          { toFun := toLinHomAux\u2081 A\u2081,\n                            map_add' :=\n                              (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) }\n                          (c \u2022 x) =\n                        \u2191(RingHom.id R\u2082) c \u2022\n                          AddHom.toFun\n                            { toFun := toLinHomAux\u2081 A\u2081,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2081 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2081 x\u2081 + toLinHomAux\u2081 A\u2081 x\u2082) }\n                            x) } +\n              {\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun y => bilin A\u2082 x y,\n                            map_add' := (_ : \u2200 (y z : M), bilin A\u2082 x (y + z) = bilin A\u2082 x y + bilin A\u2082 x z) },\n                        map_smul' := (_ : \u2200 (c : R) (y : M), bilin A\u2082 x (c \u2022 y) = c * bilin A\u2082 x y) },\n                    map_add' :=\n                      (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R\u2082) (x : M),\n                      AddHom.toFun\n                          { toFun := toLinHomAux\u2081 A\u2082,\n                            map_add' :=\n                              (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) }\n                          (c \u2022 x) =\n                        \u2191(RingHom.id R\u2082) c \u2022\n                          AddHom.toFun\n                            { toFun := toLinHomAux\u2081 A\u2082,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A\u2082 (x\u2081 + x\u2082) = toLinHomAux\u2081 A\u2082 x\u2081 + toLinHomAux\u2081 A\u2082 x\u2082) }\n                            x) })\n          x)\n      y\n[PROOFSTEP]\nsimp only [toLinHomAux\u2082, toLinHomAux\u2081, LinearMap.coe_mk, LinearMap.add_apply, add_apply, AddHom.coe_mk]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nc : R\u2082\nA : BilinForm R M\n\u22a2 AddHom.toFun\n      { toFun := toLinHomAux\u2082,\n        map_add' := (_ : \u2200 (A\u2081 A\u2082 : BilinForm R M), toLinHomAux\u2082 (A\u2081 + A\u2082) = toLinHomAux\u2082 A\u2081 + toLinHomAux\u2082 A\u2082) }\n      (c \u2022 A) =\n    \u2191(RingHom.id R\u2082) c \u2022\n      AddHom.toFun\n        { toFun := toLinHomAux\u2082,\n          map_add' := (_ : \u2200 (A\u2081 A\u2082 : BilinForm R M), toLinHomAux\u2082 (A\u2081 + A\u2082) = toLinHomAux\u2082 A\u2081 + toLinHomAux\u2082 A\u2082) }\n        A\n[PROOFSTEP]\ndsimp [toLinHomAux\u2081, toLinHomAux\u2082]\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\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nc : R\u2082\nA : BilinForm R M\n\u22a2 {\n      toAddHom :=\n        {\n          toFun := fun x =>\n            {\n              toAddHom :=\n                { toFun := fun y => c \u2022 bilin A x y,\n                  map_add' := (_ : \u2200 (y z : M), bilin (c \u2022 A) x (y + z) = bilin (c \u2022 A) x y + bilin (c \u2022 A) x z) },\n              map_smul' := (_ : \u2200 (c_1 : R) (y : M), bilin (c \u2022 A) x (c_1 \u2022 y) = c_1 * bilin (c \u2022 A) x y) },\n          map_add' :=\n            (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) },\n      map_smul' :=\n        (_ :\n          \u2200 (c_1 : R\u2082) (x : M),\n            AddHom.toFun\n                { toFun := toLinHomAux\u2081 (c \u2022 A),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x\u2081 x\u2082 : M),\n                        toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                (c_1 \u2022 x) =\n              \u2191(RingHom.id R\u2082) c_1 \u2022\n                AddHom.toFun\n                  { toFun := toLinHomAux\u2081 (c \u2022 A),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x\u2081 x\u2082 : M),\n                          toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                  x) } =\n    c \u2022\n      {\n        toAddHom :=\n          {\n            toFun := fun x =>\n              {\n                toAddHom :=\n                  { toFun := fun y => bilin A x y,\n                    map_add' := (_ : \u2200 (y z : M), bilin A x (y + z) = bilin A x y + bilin A x z) },\n                map_smul' := (_ : \u2200 (c : R) (y : M), bilin A x (c \u2022 y) = c * bilin A x y) },\n            map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) },\n        map_smul' :=\n          (_ :\n            \u2200 (c : R\u2082) (x : M),\n              AddHom.toFun\n                  { toFun := toLinHomAux\u2081 A,\n                    map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                  (c \u2022 x) =\n                \u2191(RingHom.id R\u2082) c \u2022\n                  AddHom.toFun\n                    { toFun := toLinHomAux\u2081 A,\n                      map_add' :=\n                        (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                    x) }\n[PROOFSTEP]\napply LinearMap.ext\n[GOAL]\ncase 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nc : R\u2082\nA : BilinForm R M\n\u22a2 \u2200 (x : M),\n    \u2191{\n            toAddHom :=\n              {\n                toFun := fun x =>\n                  {\n                    toAddHom :=\n                      { toFun := fun y => c \u2022 bilin A x y,\n                        map_add' :=\n                          (_ : \u2200 (y z : M), bilin (c \u2022 A) x (y + z) = bilin (c \u2022 A) x y + bilin (c \u2022 A) x z) },\n                    map_smul' := (_ : \u2200 (c_1 : R) (y : M), bilin (c \u2022 A) x (c_1 \u2022 y) = c_1 * bilin (c \u2022 A) x y) },\n                map_add' :=\n                  (_ :\n                    \u2200 (x\u2081 x\u2082 : M),\n                      toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) },\n            map_smul' :=\n              (_ :\n                \u2200 (c_1 : R\u2082) (x : M),\n                  AddHom.toFun\n                      { toFun := toLinHomAux\u2081 (c \u2022 A),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x\u2081 x\u2082 : M),\n                              toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                      (c_1 \u2022 x) =\n                    \u2191(RingHom.id R\u2082) c_1 \u2022\n                      AddHom.toFun\n                        { toFun := toLinHomAux\u2081 (c \u2022 A),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x\u2081 x\u2082 : M),\n                                toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                        x) }\n        x =\n      \u2191(c \u2022\n            {\n              toAddHom :=\n                {\n                  toFun := fun x =>\n                    {\n                      toAddHom :=\n                        { toFun := fun y => bilin A x y,\n                          map_add' := (_ : \u2200 (y z : M), bilin A x (y + z) = bilin A x y + bilin A x z) },\n                      map_smul' := (_ : \u2200 (c : R) (y : M), bilin A x (c \u2022 y) = c * bilin A x y) },\n                  map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) },\n              map_smul' :=\n                (_ :\n                  \u2200 (c : R\u2082) (x : M),\n                    AddHom.toFun\n                        { toFun := toLinHomAux\u2081 A,\n                          map_add' :=\n                            (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                        (c \u2022 x) =\n                      \u2191(RingHom.id R\u2082) c \u2022\n                        AddHom.toFun\n                          { toFun := toLinHomAux\u2081 A,\n                            map_add' :=\n                              (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                          x) })\n        x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nc : R\u2082\nA : BilinForm R M\nx : M\n\u22a2 \u2191{\n          toAddHom :=\n            {\n              toFun := fun x =>\n                {\n                  toAddHom :=\n                    { toFun := fun y => c \u2022 bilin A x y,\n                      map_add' := (_ : \u2200 (y z : M), bilin (c \u2022 A) x (y + z) = bilin (c \u2022 A) x y + bilin (c \u2022 A) x z) },\n                  map_smul' := (_ : \u2200 (c_1 : R) (y : M), bilin (c \u2022 A) x (c_1 \u2022 y) = c_1 * bilin (c \u2022 A) x y) },\n              map_add' :=\n                (_ :\n                  \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) },\n          map_smul' :=\n            (_ :\n              \u2200 (c_1 : R\u2082) (x : M),\n                AddHom.toFun\n                    { toFun := toLinHomAux\u2081 (c \u2022 A),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x\u2081 x\u2082 : M),\n                            toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                    (c_1 \u2022 x) =\n                  \u2191(RingHom.id R\u2082) c_1 \u2022\n                    AddHom.toFun\n                      { toFun := toLinHomAux\u2081 (c \u2022 A),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x\u2081 x\u2082 : M),\n                              toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                      x) }\n      x =\n    \u2191(c \u2022\n          {\n            toAddHom :=\n              {\n                toFun := fun x =>\n                  {\n                    toAddHom :=\n                      { toFun := fun y => bilin A x y,\n                        map_add' := (_ : \u2200 (y z : M), bilin A x (y + z) = bilin A x y + bilin A x z) },\n                    map_smul' := (_ : \u2200 (c : R) (y : M), bilin A x (c \u2022 y) = c * bilin A x y) },\n                map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R\u2082) (x : M),\n                  AddHom.toFun\n                      { toFun := toLinHomAux\u2081 A,\n                        map_add' :=\n                          (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                      (c \u2022 x) =\n                    \u2191(RingHom.id R\u2082) c \u2022\n                      AddHom.toFun\n                        { toFun := toLinHomAux\u2081 A,\n                          map_add' :=\n                            (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                        x) })\n      x\n[PROOFSTEP]\napply LinearMap.ext\n[GOAL]\ncase h.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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nc : R\u2082\nA : BilinForm R M\nx : M\n\u22a2 \u2200 (x_1 : M),\n    \u2191(\u2191{\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun y => c \u2022 bilin A x y,\n                            map_add' :=\n                              (_ : \u2200 (y z : M), bilin (c \u2022 A) x (y + z) = bilin (c \u2022 A) x y + bilin (c \u2022 A) x z) },\n                        map_smul' := (_ : \u2200 (c_1 : R) (y : M), bilin (c \u2022 A) x (c_1 \u2022 y) = c_1 * bilin (c \u2022 A) x y) },\n                    map_add' :=\n                      (_ :\n                        \u2200 (x\u2081 x\u2082 : M),\n                          toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c_1 : R\u2082) (x : M),\n                      AddHom.toFun\n                          { toFun := toLinHomAux\u2081 (c \u2022 A),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x\u2081 x\u2082 : M),\n                                  toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                          (c_1 \u2022 x) =\n                        \u2191(RingHom.id R\u2082) c_1 \u2022\n                          AddHom.toFun\n                            { toFun := toLinHomAux\u2081 (c \u2022 A),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x\u2081 x\u2082 : M),\n                                    toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) =\n                                      toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                            x) }\n            x)\n        x_1 =\n      \u2191(\u2191(c \u2022\n                {\n                  toAddHom :=\n                    {\n                      toFun := fun x =>\n                        {\n                          toAddHom :=\n                            { toFun := fun y => bilin A x y,\n                              map_add' := (_ : \u2200 (y z : M), bilin A x (y + z) = bilin A x y + bilin A x z) },\n                          map_smul' := (_ : \u2200 (c : R) (y : M), bilin A x (c \u2022 y) = c * bilin A x y) },\n                      map_add' :=\n                        (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (c : R\u2082) (x : M),\n                        AddHom.toFun\n                            { toFun := toLinHomAux\u2081 A,\n                              map_add' :=\n                                (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                            (c \u2022 x) =\n                          \u2191(RingHom.id R\u2082) c \u2022\n                            AddHom.toFun\n                              { toFun := toLinHomAux\u2081 A,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                              x) })\n            x)\n        x_1\n[PROOFSTEP]\nintro y\n[GOAL]\ncase h.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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nD : BilinForm R M\nD\u2081 : BilinForm R\u2081 M\u2081\ninst\u271d\u00b2 : Algebra R\u2082 R\ninst\u271d\u00b9 : Module R\u2082 M\ninst\u271d : IsScalarTower R\u2082 R M\nc : R\u2082\nA : BilinForm R M\nx y : M\n\u22a2 \u2191(\u2191{\n              toAddHom :=\n                {\n                  toFun := fun x =>\n                    {\n                      toAddHom :=\n                        { toFun := fun y => c \u2022 bilin A x y,\n                          map_add' :=\n                            (_ : \u2200 (y z : M), bilin (c \u2022 A) x (y + z) = bilin (c \u2022 A) x y + bilin (c \u2022 A) x z) },\n                      map_smul' := (_ : \u2200 (c_1 : R) (y : M), bilin (c \u2022 A) x (c_1 \u2022 y) = c_1 * bilin (c \u2022 A) x y) },\n                  map_add' :=\n                    (_ :\n                      \u2200 (x\u2081 x\u2082 : M),\n                        toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) },\n              map_smul' :=\n                (_ :\n                  \u2200 (c_1 : R\u2082) (x : M),\n                    AddHom.toFun\n                        { toFun := toLinHomAux\u2081 (c \u2022 A),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x\u2081 x\u2082 : M),\n                                toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                        (c_1 \u2022 x) =\n                      \u2191(RingHom.id R\u2082) c_1 \u2022\n                        AddHom.toFun\n                          { toFun := toLinHomAux\u2081 (c \u2022 A),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x\u2081 x\u2082 : M),\n                                  toLinHomAux\u2081 (c \u2022 A) (x\u2081 + x\u2082) = toLinHomAux\u2081 (c \u2022 A) x\u2081 + toLinHomAux\u2081 (c \u2022 A) x\u2082) }\n                          x) }\n          x)\n      y =\n    \u2191(\u2191(c \u2022\n              {\n                toAddHom :=\n                  {\n                    toFun := fun x =>\n                      {\n                        toAddHom :=\n                          { toFun := fun y => bilin A x y,\n                            map_add' := (_ : \u2200 (y z : M), bilin A x (y + z) = bilin A x y + bilin A x z) },\n                        map_smul' := (_ : \u2200 (c : R) (y : M), bilin A x (c \u2022 y) = c * bilin A x y) },\n                    map_add' := (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R\u2082) (x : M),\n                      AddHom.toFun\n                          { toFun := toLinHomAux\u2081 A,\n                            map_add' :=\n                              (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                          (c \u2022 x) =\n                        \u2191(RingHom.id R\u2082) c \u2022\n                          AddHom.toFun\n                            { toFun := toLinHomAux\u2081 A,\n                              map_add' :=\n                                (_ : \u2200 (x\u2081 x\u2082 : M), toLinHomAux\u2081 A (x\u2081 + x\u2082) = toLinHomAux\u2081 A x\u2081 + toLinHomAux\u2081 A x\u2082) }\n                            x) })\n          x)\n      y\n[PROOFSTEP]\nsimp only [toLinHomAux\u2082, toLinHomAux\u2081, LinearMap.coe_mk, LinearMap.smul_apply, smul_apply, AddHom.coe_mk]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nf : M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082\nx y z : M\u2082\n\u22a2 (fun x y => \u2191(\u2191f x) y) (x + y) z = (fun x y => \u2191(\u2191f x) y) x z + (fun x y => \u2191(\u2191f x) y) y z\n[PROOFSTEP]\nsimp only\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nf : M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082\nx y z : M\u2082\n\u22a2 \u2191(\u2191f (x + y)) z = \u2191(\u2191f x) z + \u2191(\u2191f y) z\n[PROOFSTEP]\nexact (LinearMap.map_add f x y).symm \u25b8 LinearMap.add_apply (f x) (f y) z\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nf : M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082\na : R\u2082\nx y : M\u2082\n\u22a2 (fun x y => \u2191(\u2191f x) y) (a \u2022 x) y = a * (fun x y => \u2191(\u2191f x) y) x y\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, LinearMap.smul_apply, smul_eq_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nsrc\u271d : BilinForm R\u2082 M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082 := toLinHom R\u2082\nB : BilinForm R\u2082 M\u2082\n\u22a2 LinearMap.toBilinAux\n      (AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R\u2082) (x : BilinForm R\u2082 M\u2082),\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]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nsrc\u271d : BilinForm R\u2082 M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082 := toLinHom R\u2082\nB : BilinForm R\u2082 M\u2082\nx\u271d y\u271d : M\u2082\n\u22a2 bilin\n      (LinearMap.toBilinAux\n        (AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R\u2082) (x : BilinForm R\u2082 M\u2082),\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      x\u271d y\u271d =\n    bilin B x\u271d y\u271d\n[PROOFSTEP]\nsimp [LinearMap.toBilinAux]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nsrc\u271d : BilinForm R\u2082 M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082 := toLinHom R\u2082\nB : M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R\u2082) (x : BilinForm R\u2082 M\u2082),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R\u2082) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (LinearMap.toBilinAux B) =\n    B\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nsrc\u271d : BilinForm R\u2082 M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082 := toLinHom R\u2082\nB : M\u2082 \u2192\u2097[R\u2082] M\u2082 \u2192\u2097[R\u2082] R\u2082\nx\u271d\u00b9 x\u271d : M\u2082\n\u22a2 \u2191(\u2191(AddHom.toFun\n              { toAddHom := src\u271d.toAddHom,\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R\u2082) (x : BilinForm R\u2082 M\u2082),\n                        AddHom.toFun src\u271d.toAddHom (r \u2022 x) =\n                          \u2191(RingHom.id R\u2082) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n              (LinearMap.toBilinAux B))\n          x\u271d\u00b9)\n      x\u271d =\n    \u2191(\u2191B x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp [LinearMap.toBilinAux]\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\nR' : Type u_11\ninst\u271d\u00b3 : CommSemiring R'\ninst\u271d\u00b2 : Algebra R' R\ninst\u271d\u00b9 : Module R' M\ninst\u271d : IsScalarTower R' R M\nf : R \u2192\u2097[R'] R'\nB : BilinForm R M\nx y z : M\n\u22a2 (fun x y => \u2191f (BilinForm.bilin B x y)) (x + y) z =\n    (fun x y => \u2191f (BilinForm.bilin B x y)) x z + (fun x y => \u2191f (BilinForm.bilin B x y)) y z\n[PROOFSTEP]\nsimp only [BilinForm.add_left, map_add]\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\nR' : Type u_11\ninst\u271d\u00b3 : CommSemiring R'\ninst\u271d\u00b2 : Algebra R' R\ninst\u271d\u00b9 : Module R' M\ninst\u271d : IsScalarTower R' R M\nf : R \u2192\u2097[R'] R'\nB : BilinForm R M\nr : R'\nx y : M\n\u22a2 (fun x y => \u2191f (BilinForm.bilin B x y)) (r \u2022 x) y = r * (fun x y => \u2191f (BilinForm.bilin B x y)) x y\n[PROOFSTEP]\nsimp only\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\nR' : Type u_11\ninst\u271d\u00b3 : CommSemiring R'\ninst\u271d\u00b2 : Algebra R' R\ninst\u271d\u00b9 : Module R' M\ninst\u271d : IsScalarTower R' R M\nf : R \u2192\u2097[R'] R'\nB : BilinForm R M\nr : R'\nx y : M\n\u22a2 \u2191f (BilinForm.bilin B (r \u2022 x) y) = r * \u2191f (BilinForm.bilin B x y)\n[PROOFSTEP]\nrw [\u2190 smul_one_smul R r (_ : M), BilinForm.smul_left, smul_one_mul r (_ : R), map_smul, smul_eq_mul]\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\nR' : Type u_11\ninst\u271d\u00b3 : CommSemiring R'\ninst\u271d\u00b2 : Algebra R' R\ninst\u271d\u00b9 : Module R' M\ninst\u271d : IsScalarTower R' R M\nf : R \u2192\u2097[R'] R'\nB : BilinForm R M\nx y z : M\n\u22a2 (fun x y => \u2191f (BilinForm.bilin B x y)) x (y + z) =\n    (fun x y => \u2191f (BilinForm.bilin B x y)) x y + (fun x y => \u2191f (BilinForm.bilin B x y)) x z\n[PROOFSTEP]\nsimp only [BilinForm.add_right, map_add]\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\nR' : Type u_11\ninst\u271d\u00b3 : CommSemiring R'\ninst\u271d\u00b2 : Algebra R' R\ninst\u271d\u00b9 : Module R' M\ninst\u271d : IsScalarTower R' R M\nf : R \u2192\u2097[R'] R'\nB : BilinForm R M\nr : R'\nx y : M\n\u22a2 (fun x y => \u2191f (BilinForm.bilin B x y)) x (r \u2022 y) = r * (fun x y => \u2191f (BilinForm.bilin B x y)) x y\n[PROOFSTEP]\nsimp only\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\nR' : Type u_11\ninst\u271d\u00b3 : CommSemiring R'\ninst\u271d\u00b2 : Algebra R' R\ninst\u271d\u00b9 : Module R' M\ninst\u271d : IsScalarTower R' R M\nf : R \u2192\u2097[R'] R'\nB : BilinForm R M\nr : R'\nx y : M\n\u22a2 \u2191f (BilinForm.bilin B x (r \u2022 y)) = r * \u2191f (BilinForm.bilin B x y)\n[PROOFSTEP]\nrw [\u2190 smul_one_smul R r (_ : M), BilinForm.smul_right, smul_one_mul r (_ : R), map_smul, smul_eq_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\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nx y z : M\n\u22a2 (fun x y => bilin B (\u2191l x) (\u2191r y)) (x + y) z =\n    (fun x y => bilin B (\u2191l x) (\u2191r y)) x z + (fun x y => bilin B (\u2191l x) (\u2191r y)) y z\n[PROOFSTEP]\nsimp only [LinearMap.map_add, add_left]\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nx : R\ny z : M\n\u22a2 (fun x y => bilin B (\u2191l x) (\u2191r y)) (x \u2022 y) z = x * (fun x y => bilin B (\u2191l x) (\u2191r y)) y z\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, smul_left]\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nx y z : M\n\u22a2 (fun x y => bilin B (\u2191l x) (\u2191r y)) x (y + z) =\n    (fun x y => bilin B (\u2191l x) (\u2191r y)) x y + (fun x y => bilin B (\u2191l x) (\u2191r y)) x z\n[PROOFSTEP]\nsimp only [LinearMap.map_add, add_right]\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nx : R\ny z : M\n\u22a2 (fun x y => bilin B (\u2191l x) (\u2191r y)) y (x \u2022 z) = x * (fun x y => bilin B (\u2191l x) (\u2191r y)) y z\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, smul_right]\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nr : M \u2192\u2097[R] M\n\u22a2 comp B LinearMap.id r = compRight B r\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nr : M \u2192\u2097[R] M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B LinearMap.id r) x\u271d y\u271d = bilin (compRight B r) x\u271d y\u271d\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\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nl : M \u2192\u2097[R] M\n\u22a2 comp B l LinearMap.id = compLeft B l\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nl : M \u2192\u2097[R] M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B l LinearMap.id) x\u271d y\u271d = bilin (compLeft B l) x\u271d y\u271d\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\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\n\u22a2 compLeft B LinearMap.id = B\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin (compLeft B LinearMap.id) x\u271d y\u271d = bilin B x\u271d y\u271d\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\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\n\u22a2 compRight B LinearMap.id = B\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin (compRight B LinearMap.id) x\u271d y\u271d = bilin B x\u271d y\u271d\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\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\n\u22a2 comp B LinearMap.id LinearMap.id = B\n[PROOFSTEP]\next\n[GOAL]\ncase H\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\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B LinearMap.id LinearMap.id) x\u271d y\u271d = bilin B x\u271d y\u271d\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\n\u22a2 comp B\u2081 l r = comp B\u2082 l r \u2194 B\u2081 = B\u2082\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 : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\n\u22a2 comp B\u2081 l r = comp B\u2082 l r \u2192 B\u2081 = B\u2082\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 : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\n\u22a2 B\u2081 = B\u2082 \u2192 comp B\u2081 l r = comp B\u2082 l r\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 : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : comp B\u2081 l r = comp B\u2082 l r\n\u22a2 B\u2081 = B\u2082\n[PROOFSTEP]\next x y\n[GOAL]\ncase mp.H\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : comp B\u2081 l r = comp B\u2082 l r\nx y : M'\n\u22a2 bilin B\u2081 x y = bilin B\u2082 x y\n[PROOFSTEP]\ncases' h\u2097 x with x' hx\n[GOAL]\ncase mp.H.intro\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : comp B\u2081 l r = comp B\u2082 l r\nx y : M'\nx' : M\nhx : \u2191l x' = x\n\u22a2 bilin B\u2081 x y = bilin B\u2082 x y\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase mp.H.intro\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : comp B\u2081 l r = comp B\u2082 l r\ny : M'\nx' : M\n\u22a2 bilin B\u2081 (\u2191l x') y = bilin B\u2082 (\u2191l x') y\n[PROOFSTEP]\ncases' h\u1d63 y with y' hy\n[GOAL]\ncase mp.H.intro.intro\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : comp B\u2081 l r = comp B\u2082 l r\ny : M'\nx' y' : M\nhy : \u2191r y' = y\n\u22a2 bilin B\u2081 (\u2191l x') y = bilin B\u2082 (\u2191l x') y\n[PROOFSTEP]\nsubst hy\n[GOAL]\ncase mp.H.intro.intro\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : comp B\u2081 l r = comp B\u2082 l r\nx' y' : M\n\u22a2 bilin B\u2081 (\u2191l x') (\u2191r y') = bilin B\u2082 (\u2191l x') (\u2191r y')\n[PROOFSTEP]\nrw [\u2190 comp_apply, \u2190 comp_apply, 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 : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB\u2081 B\u2082 : BilinForm R M'\nl r : M \u2192\u2097[R] M'\nh\u2097 : Function.Surjective \u2191l\nh\u1d63 : Function.Surjective \u2191r\nh : B\u2081 = B\u2082\n\u22a2 comp B\u2081 l r = comp B\u2082 l r\n[PROOFSTEP]\nrw [h]\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\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nB B' : BilinForm R\u2082 M\u2082\nx y : M\u2082'\n\u22a2 bilin ((fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B')) x y =\n    bilin\n      ((fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n        (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B')\n      x y\n[PROOFSTEP]\nsimp only [comp_apply, add_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\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nB : R\u2082\nB' : BilinForm R\u2082 M\u2082\nx y : M\u2082'\n\u22a2 bilin\n      (AddHom.toFun\n        { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n          map_add' :=\n            (_ :\n              \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                  (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                    (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') }\n        (B \u2022 B'))\n      x y =\n    bilin\n      (\u2191(RingHom.id R\u2082) B \u2022\n        AddHom.toFun\n          { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n            map_add' :=\n              (_ :\n                \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                  (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                    (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                      (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') }\n          B')\n      x y\n[PROOFSTEP]\nsimp [comp_apply, smul_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\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nB : BilinForm R\u2082 M\u2082\nx y : M\u2082\n\u22a2 bilin\n      ((fun B => comp B \u2191e \u2191e)\n        (AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n                  map_add' :=\n                    (_ :\n                      \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                        (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                          (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                            (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') },\n              map_smul' :=\n                (_ :\n                  \u2200 (B : R\u2082) (B' : BilinForm R\u2082 M\u2082),\n                    AddHom.toFun\n                        { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n                          map_add' :=\n                            (_ :\n                              \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                                (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                                  (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                                    (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') }\n                        (B \u2022 B') =\n                      \u2191(RingHom.id R\u2082) B \u2022\n                        AddHom.toFun\n                          { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n                            map_add' :=\n                              (_ :\n                                \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                                  (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                                    (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                                      (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') }\n                          B') }.toAddHom\n          B))\n      x y =\n    bilin B x y\n[PROOFSTEP]\nsimp only [comp_apply, LinearEquiv.coe_coe, e.symm_apply_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\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nB : BilinForm R\u2082 M\u2082'\nx y : M\u2082'\n\u22a2 bilin\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n                map_add' :=\n                  (_ :\n                    \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                      (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                        (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                          (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') },\n            map_smul' :=\n              (_ :\n                \u2200 (B : R\u2082) (B' : BilinForm R\u2082 M\u2082),\n                  AddHom.toFun\n                      { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n                        map_add' :=\n                          (_ :\n                            \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                              (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                                (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                                  (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') }\n                      (B \u2022 B') =\n                    \u2191(RingHom.id R\u2082) B \u2022\n                      AddHom.toFun\n                        { toFun := fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e),\n                          map_add' :=\n                            (_ :\n                              \u2200 (B B' : BilinForm R\u2082 M\u2082),\n                                (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) (B + B') =\n                                  (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B +\n                                    (fun B => comp B \u2191(LinearEquiv.symm e) \u2191(LinearEquiv.symm e)) B') }\n                        B') }.toAddHom\n        ((fun B => comp B \u2191e \u2191e) B))\n      x y =\n    bilin B x y\n[PROOFSTEP]\nsimp only [comp_apply, LinearEquiv.coe_coe, e.apply_symm_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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\n\u22a2 LinearEquiv.symm (congr e) = congr (LinearEquiv.symm e)\n[PROOFSTEP]\next\n[GOAL]\ncase 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nx\u271d\u00b9 : BilinForm R\u2082 M\u2082'\nx\u271d y\u271d : M\u2082\n\u22a2 bilin (\u2191(LinearEquiv.symm (congr e)) x\u271d\u00b9) x\u271d y\u271d = bilin (\u2191(congr (LinearEquiv.symm e)) x\u271d\u00b9) x\u271d y\u271d\n[PROOFSTEP]\nsimp only [congr_apply, LinearEquiv.symm_symm]\n[GOAL]\ncase 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nx\u271d\u00b9 : BilinForm R\u2082 M\u2082'\nx\u271d y\u271d : M\u2082\n\u22a2 bilin (\u2191(LinearEquiv.symm (congr e)) x\u271d\u00b9) x\u271d y\u271d = bilin x\u271d\u00b9 (\u2191e x\u271d) (\u2191e y\u271d)\n[PROOFSTEP]\nrfl\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nf g : M\u2082 \u2192\u2097[R\u2082] R\u2082\nx y z : M\u2082\n\u22a2 (fun x y => \u2191f x * \u2191g y) (x + y) z = (fun x y => \u2191f x * \u2191g y) x z + (fun x y => \u2191f x * \u2191g y) y z\n[PROOFSTEP]\nsimp only [LinearMap.map_add, add_mul]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nf g : M\u2082 \u2192\u2097[R\u2082] R\u2082\nx : R\u2082\ny z : M\u2082\n\u22a2 (fun x y => \u2191f x * \u2191g y) (x \u2022 y) z = x * (fun x y => \u2191f x * \u2191g y) y z\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, smul_eq_mul, 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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nf g : M\u2082 \u2192\u2097[R\u2082] R\u2082\nx y z : M\u2082\n\u22a2 (fun x y => \u2191f x * \u2191g y) x (y + z) = (fun x y => \u2191f x * \u2191g y) x y + (fun x y => \u2191f x * \u2191g y) x z\n[PROOFSTEP]\nsimp only [LinearMap.map_add, mul_add]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nf g : M\u2082 \u2192\u2097[R\u2082] R\u2082\nx : R\u2082\ny z : M\u2082\n\u22a2 (fun x y => \u2191f x * \u2191g y) y (x \u2022 z) = x * (fun x y => \u2191f x * \u2191g y) y z\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, smul_eq_mul, mul_left_comm]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nx y : M\u2084\na : R\u2084\nha : a \u2260 0\n\u22a2 IsOrtho G (a \u2022 x) y \u2194 IsOrtho G x y\n[PROOFSTEP]\ndsimp only [IsOrtho]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nx y : M\u2084\na : R\u2084\nha : a \u2260 0\n\u22a2 bilin G (a \u2022 x) y = 0 \u2194 bilin G x y = 0\n[PROOFSTEP]\nrw [smul_left, mul_eq_zero, or_iff_right ha]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nx y : M\u2084\na : R\u2084\nha : a \u2260 0\n\u22a2 IsOrtho G x (a \u2022 y) \u2194 IsOrtho G x y\n[PROOFSTEP]\ndsimp only [IsOrtho]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nx y : M\u2084\na : R\u2084\nha : a \u2260 0\n\u22a2 bilin G x (a \u2022 y) = 0 \u2194 bilin G x y = 0\n[PROOFSTEP]\nrw [smul_right, mul_eq_zero, or_iff_right ha]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\n\u22a2 LinearIndependent K v\n[PROOFSTEP]\nclassical\nrw [linearIndependent_iff']\nintro s w hs i hi\nhave : B (s.sum fun i : n => w i \u2022 v i) (v i) = 0 := by rw [hs, zero_left]\nhave hsum : (s.sum fun j : n => w j * B (v j) (v i)) = w i * B (v i) (v i) :=\n  by\n  apply Finset.sum_eq_single_of_mem i hi\n  intro j _ hij\n  rw [iIsOrtho_def.1 hv\u2081 _ _ hij, mul_zero]\nsimp_rw [sum_left, smul_left, hsum] at this \nexact eq_zero_of_ne_zero_of_mul_right_eq_zero (hv\u2082 i) this\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\n\u22a2 LinearIndependent K v\n[PROOFSTEP]\nrw [linearIndependent_iff']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\n\u22a2 \u2200 (s : Finset n) (g : n \u2192 K), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : n), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nintro s w hs i hi\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\n\u22a2 w i = 0\n[PROOFSTEP]\nhave : B (s.sum fun i : n => w i \u2022 v i) (v i) = 0 := by rw [hs, zero_left]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\n\u22a2 bilin B (\u2211 i in s, w i \u2022 v i) (v i) = 0\n[PROOFSTEP]\nrw [hs, zero_left]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\nthis : bilin B (\u2211 i in s, w i \u2022 v i) (v i) = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nhave hsum : (s.sum fun j : n => w j * B (v j) (v i)) = w i * B (v i) (v i) :=\n  by\n  apply Finset.sum_eq_single_of_mem i hi\n  intro j _ hij\n  rw [iIsOrtho_def.1 hv\u2081 _ _ hij, mul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\nthis : bilin B (\u2211 i in s, w i \u2022 v i) (v i) = 0\n\u22a2 \u2211 j in s, w j * bilin B (v j) (v i) = w i * bilin B (v i) (v i)\n[PROOFSTEP]\napply Finset.sum_eq_single_of_mem i hi\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\nthis : bilin B (\u2211 i in s, w i \u2022 v i) (v i) = 0\n\u22a2 \u2200 (b : n), b \u2208 s \u2192 b \u2260 i \u2192 w b * bilin B (v b) (v i) = 0\n[PROOFSTEP]\nintro j _ hij\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\nthis : bilin B (\u2211 i in s, w i \u2022 v i) (v i) = 0\nj : n\na\u271d : j \u2208 s\nhij : j \u2260 i\n\u22a2 w j * bilin B (v j) (v i) = 0\n[PROOFSTEP]\nrw [iIsOrtho_def.1 hv\u2081 _ _ hij, mul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\nthis : bilin B (\u2211 i in s, w i \u2022 v i) (v i) = 0\nhsum : \u2211 j in s, w j * bilin B (v j) (v i) = w i * bilin B (v i) (v i)\n\u22a2 w i = 0\n[PROOFSTEP]\nsimp_rw [sum_left, smul_left, hsum] at this \n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nR\u2084 : Type u_13\nM\u2084 : Type u_14\ninst\u271d\u00b3 : Ring R\u2084\ninst\u271d\u00b2 : IsDomain R\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : Module R\u2084 M\u2084\nG : BilinForm R\u2084 M\u2084\nn : Type w\nB : BilinForm K V\nv : n \u2192 V\nhv\u2081 : iIsOrtho B v\nhv\u2082 : \u2200 (i : n), \u00acIsOrtho B (v i) (v i)\ns : Finset n\nw : n \u2192 K\nhs : \u2211 i in s, w i \u2022 v i = 0\ni : n\nhi : i \u2208 s\nhsum : \u2211 j in s, w j * bilin B (v j) (v i) = w i * bilin B (v i) (v i)\nthis : w i * bilin B (v i) (v i) = 0\n\u22a2 w i = 0\n[PROOFSTEP]\nexact eq_zero_of_ne_zero_of_mul_right_eq_zero (hv\u2082 i) this\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nF\u2082 : BilinForm R\u2082 M\u2082\n\u03b9 : Type u_13\nb : Basis \u03b9 R\u2082 M\u2082\nx y : M\u2082\n\u22a2 (Finsupp.sum (\u2191b.repr x) fun i xi => Finsupp.sum (\u2191b.repr y) fun j yj => xi \u2022 yj \u2022 bilin B\u2082 (\u2191b i) (\u2191b j)) =\n    bilin B\u2082 x y\n[PROOFSTEP]\nconv_rhs => rw [\u2190 b.total_repr x, \u2190 b.total_repr y]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nF\u2082 : BilinForm R\u2082 M\u2082\n\u03b9 : Type u_13\nb : Basis \u03b9 R\u2082 M\u2082\nx y : M\u2082\n| bilin B\u2082 x y\n[PROOFSTEP]\nrw [\u2190 b.total_repr x, \u2190 b.total_repr y]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nF\u2082 : BilinForm R\u2082 M\u2082\n\u03b9 : Type u_13\nb : Basis \u03b9 R\u2082 M\u2082\nx y : M\u2082\n| bilin B\u2082 x y\n[PROOFSTEP]\nrw [\u2190 b.total_repr x, \u2190 b.total_repr y]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nF\u2082 : BilinForm R\u2082 M\u2082\n\u03b9 : Type u_13\nb : Basis \u03b9 R\u2082 M\u2082\nx y : M\u2082\n| bilin B\u2082 x y\n[PROOFSTEP]\nrw [\u2190 b.total_repr x, \u2190 b.total_repr y]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nF\u2082 : BilinForm R\u2082 M\u2082\n\u03b9 : Type u_13\nb : Basis \u03b9 R\u2082 M\u2082\nx y : M\u2082\n\u22a2 (Finsupp.sum (\u2191b.repr x) fun i xi => Finsupp.sum (\u2191b.repr y) fun j yj => xi \u2022 yj \u2022 bilin B\u2082 (\u2191b i) (\u2191b j)) =\n    bilin B\u2082 (\u2191(Finsupp.total \u03b9 M\u2082 R\u2082 \u2191b) (\u2191b.repr x)) (\u2191(Finsupp.total \u03b9 M\u2082 R\u2082 \u2191b) (\u2191b.repr y))\n[PROOFSTEP]\nsimp_rw [Finsupp.total_apply, Finsupp.sum, sum_left, sum_right, smul_left, smul_right, smul_eq_mul]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2074 : AddCommMonoid M\u2082'\ninst\u271d\u00b3 : AddCommMonoid M\u2082''\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082''\ninst\u271d : Algebra R\u2082 R\nx\u271d\u00b9 x\u271d : M\n\u22a2 bilin B x\u271d\u00b9 x\u271d = bilin B x\u271d x\u271d\u00b9 \u2194 bilin (\u2191(flipHom R\u2082) B) x\u271d\u00b9 x\u271d = bilin B x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nexact eq_comm\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nH : IsAlt B\u2081\nx y : M\u2081\n\u22a2 -bilin B\u2081 x y = bilin B\u2081 y x\n[PROOFSTEP]\nhave H1 : B\u2081 (x + y) (x + y) = 0 := self_eq_zero H (x + y)\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nH : IsAlt B\u2081\nx y : M\u2081\nH1 : bilin B\u2081 (x + y) (x + y) = 0\n\u22a2 -bilin B\u2081 x y = bilin B\u2081 y x\n[PROOFSTEP]\nrw [add_left, add_right, add_right, self_eq_zero H, self_eq_zero H, zero_add, add_zero, add_eq_zero_iff_neg_eq] at H1 \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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nH : IsAlt B\u2081\nx y : M\u2081\nH1 : -bilin B\u2081 x y = bilin B\u2081 y x\n\u22a2 -bilin B\u2081 x y = bilin B\u2081 y x\n[PROOFSTEP]\nexact H1\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nH : IsAlt B\u2081\n\u22a2 IsRefl B\u2081\n[PROOFSTEP]\nintro x y h\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : AddCommMonoid M\u2082''\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : Module R\u2082 M\u2082''\nH : IsAlt B\u2081\nx y : M\u2081\nh : bilin B\u2081 x y = 0\n\u22a2 bilin B\u2081 y x = 0\n[PROOFSTEP]\nrw [\u2190 neg_eq H, h, neg_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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\n\u22a2 IsAdjointPair B F f g \u2194 compLeft F f = compRight B g\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\n\u22a2 IsAdjointPair B F f g \u2192 compLeft F f = compRight B g\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\n\u22a2 compLeft F f = compRight B g \u2192 IsAdjointPair B F f g\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\nh : IsAdjointPair B F f g\n\u22a2 compLeft F f = compRight B g\n[PROOFSTEP]\next x\n[GOAL]\ncase mp.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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\nh : IsAdjointPair B F f g\nx y\u271d : M\n\u22a2 bilin (compLeft F f) x y\u271d = bilin (compRight B g) x y\u271d\n[PROOFSTEP]\nsimp only [compLeft_apply, compRight_apply]\n[GOAL]\ncase mp.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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\nh : IsAdjointPair B F f g\nx y\u271d : M\n\u22a2 bilin F (\u2191f x) y\u271d = bilin B x (\u2191g y\u271d)\n[PROOFSTEP]\napply h\n[GOAL]\ncase mpr\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\nh : compLeft F f = compRight B g\n\u22a2 IsAdjointPair B F f g\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase mpr\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\nh : compLeft F f = compRight B g\nx y : M\n\u22a2 bilin F (\u2191f x) y = bilin B x (\u2191g y)\n[PROOFSTEP]\nrw [\u2190 compLeft_apply, \u2190 compRight_apply]\n[GOAL]\ncase mpr\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng\u271d g' : M' \u2192\u2097[R] M\nf g : Module.End R M\nh : compLeft F f = compRight B g\nx y : M\n\u22a2 bilin (compLeft F f) x y = bilin (compRight B g) x y\n[PROOFSTEP]\nrw [h]\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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nx : M\ny : M'\n\u22a2 bilin B' (\u21910 x) y = bilin B x (\u21910 y)\n[PROOFSTEP]\nsimp only [BilinForm.zero_left, BilinForm.zero_right, LinearMap.zero_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 : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : AddCommMonoid M\u2082''\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB' : BilinForm R M'\nf f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nh : IsAdjointPair B B' f g\nh' : IsAdjointPair B B' f' g'\nx : M\ny : M'\n\u22a2 bilin B' (\u2191(f + f') x) y = bilin B x (\u2191(g + g') y)\n[PROOFSTEP]\nrw [LinearMap.add_apply, LinearMap.add_apply, add_left, add_right, h, h']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b3 : AddCommMonoid M'\ninst\u271d\u00b2 : Module R M'\nB' : BilinForm R M'\nf f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b9 : AddCommGroup M\u2081'\ninst\u271d : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nh : IsAdjointPair B\u2081 B\u2081' f\u2081 g\u2081\nh' : IsAdjointPair B\u2081 B\u2081' f\u2081' g\u2081'\nx : M\u2081\ny : M\u2081'\n\u22a2 bilin B\u2081' (\u2191(f\u2081 - f\u2081') x) y = bilin B\u2081 x (\u2191(g\u2081 - g\u2081') y)\n[PROOFSTEP]\nrw [LinearMap.sub_apply, LinearMap.sub_apply, sub_left, sub_right, h, h']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u00b2 : Semiring R\ninst\u271d\u00b2\u00b9 : AddCommMonoid M\ninst\u271d\u00b2\u2070 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2079 : Ring R\u2081\ninst\u271d\u00b9\u2078 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2077 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2076 : CommSemiring R\u2082\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b3 : CommRing R\u2083\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : AddCommGroup V\ninst\u271d\u2078 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2077 : AddCommMonoid M\u2082'\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : Module R\u2082 M\u2082'\ninst\u271d\u2074 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u00b3 : AddCommMonoid M'\ninst\u271d\u00b2 : Module R M'\nB' : BilinForm R M'\nf f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b9 : AddCommGroup M\u2081'\ninst\u271d : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nc : R\u2082\nh : IsAdjointPair B\u2082 B\u2082' f\u2082 g\u2082\nx : M\u2082\ny : M\u2082'\n\u22a2 bilin B\u2082' (\u2191(c \u2022 f\u2082) x) y = bilin B\u2082 x (\u2191(c \u2022 g\u2082) y)\n[PROOFSTEP]\nrw [LinearMap.smul_apply, LinearMap.smul_apply, smul_left, smul_right, h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf f'\u271d : M \u2192\u2097[R] M'\ng g'\u271d : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nf' : M' \u2192\u2097[R] M''\ng' : M'' \u2192\u2097[R] M'\nh : IsAdjointPair B B' f g\nh' : IsAdjointPair B' B'' f' g'\nx : M\ny : M''\n\u22a2 bilin B'' (\u2191(LinearMap.comp f' f) x) y = bilin B x (\u2191(LinearMap.comp g g') y)\n[PROOFSTEP]\nrw [LinearMap.comp_apply, LinearMap.comp_apply, h', h]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f'\u271d : M \u2192\u2097[R] M'\ng\u271d g'\u271d : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nf g f' g' : Module.End R M\nh : IsAdjointPair B B f g\nh' : IsAdjointPair B B f' g'\nx y : M\n\u22a2 bilin B (\u2191(f * f') x) y = bilin B x (\u2191(g' * g) y)\n[PROOFSTEP]\nrw [LinearMap.mul_apply, LinearMap.mul_apply, h, h']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\nf : Module.End R\u2082 M\u2082\n\u22a2 f \u2208 isPairSelfAdjointSubmodule B\u2082 F\u2082 \u2194 IsPairSelfAdjoint B\u2082 F\u2082 f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\n\u22a2 IsPairSelfAdjoint B\u2082 F\u2082 f \u2194\n    IsPairSelfAdjoint (comp B\u2082 \u2191e \u2191e) (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)\n[PROOFSTEP]\nhave h\u2097 : (F\u2082.comp \u2191e \u2191e).compLeft (e.symm.conj f) = (F\u2082.compLeft f).comp \u2191e \u2191e :=\n  by\n  ext\n  simp [LinearEquiv.symm_conj_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\n\u22a2 compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nx\u271d y\u271d : M\u2082'\n\u22a2 bilin (compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)) x\u271d y\u271d =\n    bilin (comp (compLeft F\u2082 f) \u2191e \u2191e) x\u271d y\u271d\n[PROOFSTEP]\nsimp [LinearEquiv.symm_conj_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nh\u2097 : compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\n\u22a2 IsPairSelfAdjoint B\u2082 F\u2082 f \u2194\n    IsPairSelfAdjoint (comp B\u2082 \u2191e \u2191e) (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)\n[PROOFSTEP]\nhave h\u1d63 : (B\u2082.comp \u2191e \u2191e).compRight (e.symm.conj f) = (B\u2082.compRight f).comp \u2191e \u2191e :=\n  by\n  ext\n  simp [LinearEquiv.conj_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nh\u2097 : compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\n\u22a2 compRight (comp B\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B\u2082 f) \u2191e \u2191e\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nh\u2097 : compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\nx\u271d y\u271d : M\u2082'\n\u22a2 bilin (compRight (comp B\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)) x\u271d y\u271d =\n    bilin (comp (compRight B\u2082 f) \u2191e \u2191e) x\u271d y\u271d\n[PROOFSTEP]\nsimp [LinearEquiv.conj_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nh\u2097 : compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\nh\u1d63 : compRight (comp B\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B\u2082 f) \u2191e \u2191e\n\u22a2 IsPairSelfAdjoint B\u2082 F\u2082 f \u2194\n    IsPairSelfAdjoint (comp B\u2082 \u2191e \u2191e) (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)\n[PROOFSTEP]\nhave he : Function.Surjective (\u21d1(\u2191e : M\u2082' \u2192\u2097[R\u2082] M\u2082) : M\u2082' \u2192 M\u2082) := e.surjective\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nh\u2097 : compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\nh\u1d63 : compRight (comp B\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B\u2082 f) \u2191e \u2191e\nhe : Function.Surjective \u2191\u2191e\n\u22a2 IsPairSelfAdjoint B\u2082 F\u2082 f \u2194\n    IsPairSelfAdjoint (comp B\u2082 \u2191e \u2191e) (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)\n[PROOFSTEP]\nshow BilinForm.IsAdjointPair _ _ _ _ \u2194 BilinForm.IsAdjointPair _ _ _ _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\ne : M\u2082' \u2243\u2097[R\u2082] M\u2082\nf : Module.End R\u2082 M\u2082\nh\u2097 : compLeft (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compLeft F\u2082 f) \u2191e \u2191e\nh\u1d63 : compRight (comp B\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f) = comp (compRight B\u2082 f) \u2191e \u2191e\nhe : Function.Surjective \u2191\u2191e\n\u22a2 IsAdjointPair B\u2082 F\u2082 f f \u2194\n    IsAdjointPair (comp B\u2082 \u2191e \u2191e) (comp F\u2082 \u2191e \u2191e) (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)\n      (\u2191(LinearEquiv.conj (LinearEquiv.symm e)) f)\n[PROOFSTEP]\nrw [isAdjointPair_iff_compLeft_eq_compRight, isAdjointPair_iff_compLeft_eq_compRight, h\u1d63, h\u2097, comp_inj _ _ he he]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\nf : Module.End R\u2081 M\u2081\n\u22a2 (\u2200 (x y : M\u2081), bilin B\u2081 (\u2191f x) y = bilin B\u2081 x (\u2191(-f) y)) \u2194 \u2200 (x y : M\u2081), bilin B\u2081 (\u2191f x) y = bilin (-B\u2081) x (\u2191f y)\n[PROOFSTEP]\nsimp only [LinearMap.neg_apply, BilinForm.neg_apply, BilinForm.neg_right]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\nB\u2083 : BilinForm R\u2083 M\u2083\nf : Module.End R\u2083 M\u2083\n\u22a2 f \u2208 skewAdjointSubmodule B\u2083 \u2194 IsSkewAdjoint B\u2083 f\n[PROOFSTEP]\nrw [isSkewAdjoint_iff_neg_self_adjoint]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2074 : Semiring R\ninst\u271d\u00b2\u00b3 : AddCommMonoid M\ninst\u271d\u00b2\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b2\u00b9 : Ring R\u2081\ninst\u271d\u00b2\u2070 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2078 : CommSemiring R\u2082\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2075 : CommRing R\u2083\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b9\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\nM\u2082'' : Type u_12\ninst\u271d\u2079 : AddCommMonoid M\u2082'\ninst\u271d\u2078 : AddCommMonoid M\u2082''\ninst\u271d\u2077 : Module R\u2082 M\u2082'\ninst\u271d\u2076 : Module R\u2082 M\u2082''\nF : BilinForm R M\nM' : Type u_13\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : Module R M'\nB' : BilinForm R M'\nf\u271d f' : M \u2192\u2097[R] M'\ng g' : M' \u2192\u2097[R] M\nM\u2081' : Type u_14\ninst\u271d\u00b3 : AddCommGroup M\u2081'\ninst\u271d\u00b2 : Module R\u2081 M\u2081'\nB\u2081' : BilinForm R\u2081 M\u2081'\nf\u2081 f\u2081' : M\u2081 \u2192\u2097[R\u2081] M\u2081'\ng\u2081 g\u2081' : M\u2081' \u2192\u2097[R\u2081] M\u2081\nB\u2082' : BilinForm R\u2082 M\u2082'\nf\u2082 f\u2082' : M\u2082 \u2192\u2097[R\u2082] M\u2082'\ng\u2082 g\u2082' : M\u2082' \u2192\u2097[R\u2082] M\u2082\nM'' : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\nB'' : BilinForm R M''\nF\u2082 : BilinForm R\u2082 M\u2082\nB\u2083 : BilinForm R\u2083 M\u2083\nf : Module.End R\u2083 M\u2083\n\u22a2 f \u2208 skewAdjointSubmodule B\u2083 \u2194 IsAdjointPair (-B\u2083) B\u2083 f f\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nB : BilinForm R M\nN : Submodule R M\nx y : M\nhx : x \u2208 {m | \u2200 (n : M), n \u2208 N \u2192 IsOrtho B n m}\nhy : y \u2208 {m | \u2200 (n : M), n \u2208 N \u2192 IsOrtho B n m}\nn : M\nhn : n \u2208 N\n\u22a2 IsOrtho B n (x + y)\n[PROOFSTEP]\nrw [IsOrtho, add_right, show B n x = 0 from hx n hn, show B n y = 0 from hy n hn, zero_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nB : BilinForm R M\nN : Submodule R M\nc : R\nx : M\nhx :\n  x \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {m | \u2200 (n : M), n \u2208 N \u2192 IsOrtho B n m},\n              add_mem' :=\n                (_ :\n                  \u2200 {x y : M},\n                    x \u2208 {m | \u2200 (n : M), n \u2208 N \u2192 IsOrtho B n m} \u2192\n                      y \u2208 {m | \u2200 (n : M), n \u2208 N \u2192 IsOrtho B n m} \u2192 \u2200 (n : M), n \u2208 N \u2192 IsOrtho B n (x + y)) },\n          zero_mem' := (_ : \u2200 (x : M), x \u2208 N \u2192 IsOrtho B x 0) }.toAddSubsemigroup.carrier\nn : M\nhn : n \u2208 N\n\u22a2 IsOrtho B n (c \u2022 x)\n[PROOFSTEP]\nrw [IsOrtho, smul_right, show B n x = 0 from hx n hn, mul_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u22a2 Submodule.span K {x} \u2293 orthogonal B (Submodule.span K {x}) = \u22a5\n[PROOFSTEP]\nrw [\u2190 Finset.coe_singleton]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u22a2 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x}) = \u22a5\n[PROOFSTEP]\nrefine' eq_bot_iff.2 fun y h => _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\ny : V\nh : y \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\n\u22a2 y \u2208 \u22a5\n[PROOFSTEP]\nrcases mem_span_finset.1 h.1 with \u27e8\u03bc, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\n\u22a2 \u2211 i in {x}, \u03bc i \u2022 i \u2208 \u22a5\n[PROOFSTEP]\nhave := h.2 x ?_\n[GOAL]\ncase intro.refine_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\nthis : IsOrtho B x (\u2211 i in {x}, \u03bc i \u2022 i)\n\u22a2 \u2211 i in {x}, \u03bc i \u2022 i \u2208 \u22a5\n[PROOFSTEP]\nrw [Finset.sum_singleton] at this \u22a2\n[GOAL]\ncase intro.refine_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\nthis : IsOrtho B x (\u03bc x \u2022 x)\n\u22a2 \u03bc x \u2022 x \u2208 \u22a5\n[PROOFSTEP]\nsuffices h\u03bczero : \u03bc x = 0\n[GOAL]\ncase intro.refine_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\nthis : IsOrtho B x (\u03bc x \u2022 x)\nh\u03bczero : \u03bc x = 0\n\u22a2 \u03bc x \u2022 x \u2208 \u22a5\n[PROOFSTEP]\nrw [h\u03bczero, zero_smul, Submodule.mem_bot]\n[GOAL]\ncase h\u03bczero\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\nthis : IsOrtho B x (\u03bc x \u2022 x)\n\u22a2 \u03bc x = 0\n[PROOFSTEP]\nchange B x (\u03bc x \u2022 x) = 0 at this \n[GOAL]\ncase h\u03bczero\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\nthis : bilin B x (\u03bc x \u2022 x) = 0\n\u22a2 \u03bc x = 0\n[PROOFSTEP]\nrw [smul_right] at this \n[GOAL]\ncase h\u03bczero\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\nthis : \u03bc x * bilin B x x = 0\n\u22a2 \u03bc x = 0\n[PROOFSTEP]\nexact eq_zero_of_ne_zero_of_mul_right_eq_zero hx this\n[GOAL]\ncase intro.refine_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\n\u22a2 x \u2208 Submodule.span K \u2191{x}\n[PROOFSTEP]\nrw [Submodule.mem_span]\n[GOAL]\ncase intro.refine_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u03bc : V \u2192 K\nh : \u2211 i in {x}, \u03bc i \u2022 i \u2208 Submodule.span K \u2191{x} \u2293 orthogonal B (Submodule.span K \u2191{x})\n\u22a2 \u2200 (p : Submodule K V), \u2191{x} \u2286 \u2191p \u2192 x \u2208 p\n[PROOFSTEP]\nexact fun _ hp => hp <| Finset.mem_singleton_self _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\n\u22a2 orthogonal B (Submodule.span K {x}) = LinearMap.ker (\u2191(\u2191toLin B) x)\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx y : V\n\u22a2 y \u2208 orthogonal B (Submodule.span K {x}) \u2194 y \u2208 LinearMap.ker (\u2191(\u2191toLin B) x)\n[PROOFSTEP]\nsimp_rw [mem_orthogonal_iff, LinearMap.mem_ker, Submodule.mem_span_singleton]\n[GOAL]\ncase h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx y : V\n\u22a2 (\u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 IsOrtho B n y) \u2194 \u2191(\u2191(\u2191toLin B) x) y = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx y : V\n\u22a2 (\u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 IsOrtho B n y) \u2192 \u2191(\u2191(\u2191toLin B) x) y = 0\n[PROOFSTEP]\nexact fun h => h x \u27e81, one_smul _ _\u27e9\n[GOAL]\ncase h.mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx y : V\n\u22a2 \u2191(\u2191(\u2191toLin B) x) y = 0 \u2192 \u2200 (n : V), (\u2203 a, a \u2022 x = n) \u2192 IsOrtho B n y\n[PROOFSTEP]\nrintro h _ \u27e8z, rfl\u27e9\n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx y : V\nh : \u2191(\u2191(\u2191toLin B) x) y = 0\nz : K\n\u22a2 IsOrtho B (z \u2022 x) y\n[PROOFSTEP]\nrw [IsOrtho, smul_left, mul_eq_zero]\n[GOAL]\ncase h.mpr.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx y : V\nh : \u2191(\u2191(\u2191toLin B) x) y = 0\nz : K\n\u22a2 z = 0 \u2228 bilin B x y = 0\n[PROOFSTEP]\nexact Or.intro_right _ h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u22a2 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x}) = \u22a4\n[PROOFSTEP]\nrw [orthogonal_span_singleton_eq_toLin_ker]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nN L : Submodule R M\nB : BilinForm K V\nx : V\nhx : \u00acIsOrtho B x x\n\u22a2 Submodule.span K {x} \u2294 LinearMap.ker (\u2191(\u2191toLin B) x) = \u22a4\n[PROOFSTEP]\nexact LinearMap.span_singleton_sup_ker_eq_top _ hx\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nh : Nondegenerate (\u2191(congr e) B)\n\u22a2 Nondegenerate B\n[PROOFSTEP]\nconvert h.congr e.symm\n[GOAL]\ncase h.e'_6\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\ne : M\u2082 \u2243\u2097[R\u2082] M\u2082'\nh : Nondegenerate (\u2191(congr e) B)\n\u22a2 B = \u2191(congr (LinearEquiv.symm e)) (\u2191(congr e) B)\n[PROOFSTEP]\nrw [congr_congr, e.self_trans_symm, congr_refl, LinearEquiv.refl_apply]\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\n\u22a2 Nondegenerate B \u2194 LinearMap.ker (\u2191toLin B) = \u22a5\n[PROOFSTEP]\nrw [LinearMap.ker_eq_bot']\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\n\u22a2 Nondegenerate B \u2194 \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\n\u22a2 Nondegenerate B \u2192 \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\n\u22a2 (\u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0) \u2192 Nondegenerate 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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : Nondegenerate B\n\u22a2 \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\n[PROOFSTEP]\nrefine' fun m hm => h _ fun x => _\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : Nondegenerate B\nm : M\u2082\nhm : \u2191(\u2191toLin B) m = 0\nx : M\u2082\n\u22a2 bilin B m x = 0\n[PROOFSTEP]\nrw [\u2190 toLin_apply, hm]\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : Nondegenerate B\nm : M\u2082\nhm : \u2191(\u2191toLin B) m = 0\nx : M\u2082\n\u22a2 \u21910 x = 0\n[PROOFSTEP]\nrfl\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\n\u22a2 Nondegenerate B\n[PROOFSTEP]\nintro m hm\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\nm : M\u2082\nhm : \u2200 (n : M\u2082), bilin B m n = 0\n\u22a2 m = 0\n[PROOFSTEP]\napply h\n[GOAL]\ncase mpr.a\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\nm : M\u2082\nhm : \u2200 (n : M\u2082), bilin B m n = 0\n\u22a2 \u2191(\u2191toLin B) m = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase mpr.a.h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2082 M\u2082\nh : \u2200 (m : M\u2082), \u2191(\u2191toLin B) m = 0 \u2192 m = 0\nm : M\u2082\nhm : \u2200 (n : M\u2082), bilin B m n = 0\nx : M\u2082\n\u22a2 \u2191(\u2191(\u2191toLin B) m) x = \u21910 x\n[PROOFSTEP]\nexact hm x\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : IsRefl B\nW : Submodule R\u2081 M\u2081\nhW : Disjoint W (orthogonal B W)\n\u22a2 Nondegenerate (restrict B W)\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9 b\u2081\n[GOAL]\ncase mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : IsRefl B\nW : Submodule R\u2081 M\u2081\nhW : Disjoint W (orthogonal B W)\nx : M\u2081\nhx : x \u2208 W\nb\u2081 : \u2200 (n : { x // x \u2208 W }), bilin (restrict B W) { val := x, property := hx } n = 0\n\u22a2 { val := x, property := hx } = 0\n[PROOFSTEP]\nrw [Submodule.mk_eq_zero, \u2190 Submodule.mem_bot R\u2081]\n[GOAL]\ncase mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : IsRefl B\nW : Submodule R\u2081 M\u2081\nhW : Disjoint W (orthogonal B W)\nx : M\u2081\nhx : x \u2208 W\nb\u2081 : \u2200 (n : { x // x \u2208 W }), bilin (restrict B W) { val := x, property := hx } n = 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nrefine' hW.le_bot \u27e8hx, fun y hy => _\u27e9\n[GOAL]\ncase mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : IsRefl B\nW : Submodule R\u2081 M\u2081\nhW : Disjoint W (orthogonal B W)\nx : M\u2081\nhx : x \u2208 W\nb\u2081 : \u2200 (n : { x // x \u2208 W }), bilin (restrict B W) { val := x, property := hx } n = 0\ny : M\u2081\nhy : y \u2208 W\n\u22a2 IsOrtho B y x\n[PROOFSTEP]\nspecialize b\u2081 \u27e8y, hy\u27e9\n[GOAL]\ncase mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : IsRefl B\nW : Submodule R\u2081 M\u2081\nhW : Disjoint W (orthogonal B W)\nx : M\u2081\nhx : x \u2208 W\ny : M\u2081\nhy : y \u2208 W\nb\u2081 : bilin (restrict B W) { val := x, property := hx } { val := y, property := hy } = 0\n\u22a2 IsOrtho B y x\n[PROOFSTEP]\nrw [restrict_apply, Submodule.coe_mk, Submodule.coe_mk] at b\u2081 \n[GOAL]\ncase mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : IsRefl B\nW : Submodule R\u2081 M\u2081\nhW : Disjoint W (orthogonal B W)\nx : M\u2081\nhx : x \u2208 W\ny : M\u2081\nhy : y \u2208 W\nb\u2081 : bilin B x y = 0\n\u22a2 IsOrtho B y x\n[PROOFSTEP]\nexact isOrtho_def.mpr (b x y b\u2081)\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\n\u22a2 \u00acIsOrtho B (\u2191v i) (\u2191v i)\n[PROOFSTEP]\nintro ho\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\n\u22a2 False\n[PROOFSTEP]\nrefine' v.ne_zero i (hB (v i) fun m => _)\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nm : M\n\u22a2 bilin B (\u2191v i) m = 0\n[PROOFSTEP]\nobtain \u27e8vi, rfl\u27e9 := v.repr.symm.surjective m\n[GOAL]\ncase intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\n\u22a2 bilin B (\u2191v i) (\u2191(LinearEquiv.symm v.repr) vi) = 0\n[PROOFSTEP]\nrw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, sum_right]\n[GOAL]\ncase intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\n\u22a2 \u2211 i_1 in vi.support, bilin B (\u2191v i) (\u2191vi i_1 \u2022 \u2191v i_1) = 0\n[PROOFSTEP]\napply Finset.sum_eq_zero\n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\n\u22a2 \u2200 (x : n), x \u2208 vi.support \u2192 bilin B (\u2191v i) (\u2191vi x \u2022 \u2191v x) = 0\n[PROOFSTEP]\nrintro j -\n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nj : n\n\u22a2 bilin B (\u2191v i) (\u2191vi j \u2022 \u2191v j) = 0\n[PROOFSTEP]\nrw [smul_right]\n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nj : n\n\u22a2 \u2191vi j * bilin B (\u2191v i) (\u2191v j) = 0\n[PROOFSTEP]\nconvert mul_zero (vi j) using 2\n[GOAL]\ncase h.e'_2.h.e'_6\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nj : n\n\u22a2 bilin B (\u2191v i) (\u2191v j) = 0\n[PROOFSTEP]\nobtain rfl | hij := eq_or_ne i j\n[GOAL]\ncase h.e'_2.h.e'_6.inl\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\n\u22a2 bilin B (\u2191v i) (\u2191v i) = 0\n[PROOFSTEP]\nexact ho\n[GOAL]\ncase h.e'_2.h.e'_6.inr\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d : Nontrivial R\nB : BilinForm R M\nv : Basis n R M\nh : iIsOrtho B \u2191v\nhB : Nondegenerate B\ni : n\nho : IsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nj : n\nhij : i \u2260 j\n\u22a2 bilin B (\u2191v i) (\u2191v j) = 0\n[PROOFSTEP]\nexact h hij\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\n\u22a2 Nondegenerate B \u2194 \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\n[PROOFSTEP]\nrefine' \u27e8hO.not_isOrtho_basis_self_of_nondegenerate, fun ho m hB => _\u27e9\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nm : M\nhB : \u2200 (n : M), bilin B m n = 0\n\u22a2 m = 0\n[PROOFSTEP]\nobtain \u27e8vi, rfl\u27e9 := v.repr.symm.surjective m\n[GOAL]\ncase intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nhB : \u2200 (n_1 : M), bilin B (\u2191(LinearEquiv.symm v.repr) vi) n_1 = 0\n\u22a2 \u2191(LinearEquiv.symm v.repr) vi = 0\n[PROOFSTEP]\nrw [LinearEquiv.map_eq_zero_iff]\n[GOAL]\ncase intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nhB : \u2200 (n_1 : M), bilin B (\u2191(LinearEquiv.symm v.repr) vi) n_1 = 0\n\u22a2 vi = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nhB : \u2200 (n_1 : M), bilin B (\u2191(LinearEquiv.symm v.repr) vi) n_1 = 0\ni : n\n\u22a2 \u2191vi i = \u21910 i\n[PROOFSTEP]\nrw [Finsupp.zero_apply]\n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\nhB : \u2200 (n_1 : M), bilin B (\u2191(LinearEquiv.symm v.repr) vi) n_1 = 0\ni : n\n\u22a2 \u2191vi i = 0\n[PROOFSTEP]\nspecialize hB (v i)\n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : bilin B (\u2191(LinearEquiv.symm v.repr) vi) (\u2191v i) = 0\n\u22a2 \u2191vi i = 0\n[PROOFSTEP]\nsimp_rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, sum_left, smul_left] at hB \n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\n\u22a2 \u2191vi i = 0\n[PROOFSTEP]\nrw [Finset.sum_eq_single i] at hB \n[GOAL]\ncase intro.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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2191vi i * bilin B (\u2191v i) (\u2191v i) = 0\n\u22a2 \u2191vi i = 0\n[PROOFSTEP]\nexact eq_zero_of_ne_zero_of_mul_right_eq_zero (ho i) hB\n[GOAL]\ncase intro.h.h\u2080\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\n\u22a2 \u2200 (b : n), b \u2208 vi.support \u2192 b \u2260 i \u2192 \u2191vi b * bilin B (\u2191v b) (\u2191v i) = 0\n[PROOFSTEP]\nintro j _ hij\n[GOAL]\ncase intro.h.h\u2080\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\nj : n\na\u271d : j \u2208 vi.support\nhij : j \u2260 i\n\u22a2 \u2191vi j * bilin B (\u2191v j) (\u2191v i) = 0\n[PROOFSTEP]\nconvert mul_zero (vi j) using 2\n[GOAL]\ncase h.e'_2.h.e'_6\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\nj : n\na\u271d : j \u2208 vi.support\nhij : j \u2260 i\n\u22a2 bilin B (\u2191v j) (\u2191v i) = 0\n[PROOFSTEP]\nexact hO hij\n[GOAL]\ncase intro.h.h\u2081\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\n\u22a2 \u00aci \u2208 vi.support \u2192 \u2191vi i * bilin B (\u2191v i) (\u2191v i) = 0\n[PROOFSTEP]\nintro hi\n[GOAL]\ncase intro.h.h\u2081\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\nhi : \u00aci \u2208 vi.support\n\u22a2 \u2191vi i * bilin B (\u2191v i) (\u2191v i) = 0\n[PROOFSTEP]\nconvert zero_mul (M\u2080 := R) _ using 2\n[GOAL]\ncase h.e'_2.h.e'_5\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\nM\u2082' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid M\u2082'\ninst\u271d\u00b2 : Module R\u2082 M\u2082'\nn : Type w\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nB : BilinForm R M\nv : Basis n R M\nhO : iIsOrtho B \u2191v\nho : \u2200 (i : n), \u00acIsOrtho B (\u2191v i) (\u2191v i)\nvi : n \u2192\u2080 R\ni : n\nhB : \u2211 x in vi.support, \u2191vi x * bilin B (\u2191v x) (\u2191v i) = 0\nhi : \u00aci \u2208 vi.support\n\u22a2 \u2191vi i = 0\n[PROOFSTEP]\nexact Finsupp.not_mem_support_iff.mp hi\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\n\u22a2 Submodule.map (Submodule.subtype W) (LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W)) = W \u2293 orthogonal B \u22a4\n[PROOFSTEP]\next x\n[GOAL]\ncase h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\n\u22a2 x \u2208 Submodule.map (Submodule.subtype W) (LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W)) \u2194 x \u2208 W \u2293 orthogonal B \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\n\u22a2 x \u2208 Submodule.map (Submodule.subtype W) (LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W)) \u2192 x \u2208 W \u2293 orthogonal B \u22a4\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\n\u22a2 x \u2208 W \u2293 orthogonal B \u22a4 \u2192 x \u2208 Submodule.map (Submodule.subtype W) (LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W))\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 Submodule.map (Submodule.subtype W) (LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W))\n\u22a2 x \u2208 W \u2293 orthogonal B \u22a4\n[PROOFSTEP]\nrcases hx with \u27e8\u27e8x, hx\u27e9, hker, rfl\u27e9\n[GOAL]\ncase h.mp.intro.mk.intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : { val := x, property := hx } \u2208 \u2191(LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W))\n\u22a2 \u2191(Submodule.subtype W) { val := x, property := hx } \u2208 W \u2293 orthogonal B \u22a4\n[PROOFSTEP]\nerw [LinearMap.mem_ker] at hker \n[GOAL]\ncase h.mp.intro.mk.intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\n\u22a2 \u2191(Submodule.subtype W) { val := x, property := hx } \u2208 W \u2293 orthogonal B \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp.intro.mk.intro.left\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\n\u22a2 \u2191(Submodule.subtype W) { val := x, property := hx } \u2208 \u2191W\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase h.mp.intro.mk.intro.right\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\n\u22a2 \u2191(Submodule.subtype W) { val := x, property := hx } \u2208 \u2191(orthogonal B \u22a4)\n[PROOFSTEP]\nintro y _\n[GOAL]\ncase h.mp.intro.mk.intro.right\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\ny : V\na\u271d : y \u2208 \u22a4\n\u22a2 IsOrtho B y (\u2191(Submodule.subtype W) { val := x, property := hx })\n[PROOFSTEP]\nrw [IsOrtho, b]\n[GOAL]\ncase h.mp.intro.mk.intro.right.a\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\ny : V\na\u271d : y \u2208 \u22a4\n\u22a2 bilin B (\u2191(Submodule.subtype W) { val := x, property := hx }) y = 0\n[PROOFSTEP]\nchange (B.toLin.domRestrict W) \u27e8x, hx\u27e9 y = 0\n[GOAL]\ncase h.mp.intro.mk.intro.right.a\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\ny : V\na\u271d : y \u2208 \u22a4\n\u22a2 \u2191(\u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx }) y = 0\n[PROOFSTEP]\nrw [hker]\n[GOAL]\ncase h.mp.intro.mk.intro.right.a\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W\nhker : \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := hx } = 0\ny : V\na\u271d : y \u2208 \u22a4\n\u22a2 \u21910 y = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W \u2293 orthogonal B \u22a4\n\u22a2 x \u2208 Submodule.map (Submodule.subtype W) (LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W))\n[PROOFSTEP]\nsimp_rw [Submodule.mem_map, LinearMap.mem_ker]\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W \u2293 orthogonal B \u22a4\n\u22a2 \u2203 y, \u2191(LinearMap.domRestrict (\u2191toLin B) W) y = 0 \u2227 \u2191(Submodule.subtype W) y = x\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, hx.1\u27e9, _, rfl\u27e9\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W \u2293 orthogonal B \u22a4\n\u22a2 \u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := (_ : x \u2208 \u2191W) } = 0\n[PROOFSTEP]\next y\n[GOAL]\ncase h.mpr.h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W \u2293 orthogonal B \u22a4\ny : V\n\u22a2 \u2191(\u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := x, property := (_ : x \u2208 \u2191W) }) y = \u21910 y\n[PROOFSTEP]\nchange B x y = 0\n[GOAL]\ncase h.mpr.h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W \u2293 orthogonal B \u22a4\ny : V\n\u22a2 bilin B x y = 0\n[PROOFSTEP]\nrw [b]\n[GOAL]\ncase h.mpr.h.a\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nb : IsRefl B\nx : V\nhx : x \u2208 W \u2293 orthogonal B \u22a4\ny : V\n\u22a2 bilin B y x = 0\n[PROOFSTEP]\nexact hx.2 _ Submodule.mem_top\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\n\u22a2 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W)) = orthogonal B W\n[PROOFSTEP]\next x\n[GOAL]\ncase h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\n\u22a2 x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W)) \u2194 x \u2208 orthogonal B W\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\n\u22a2 x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W)) \u2192 x \u2208 orthogonal B W\n[PROOFSTEP]\nrw [mem_orthogonal_iff]\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\n\u22a2 x \u2208 orthogonal B W \u2192 x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W))\n[PROOFSTEP]\nrw [mem_orthogonal_iff]\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\n\u22a2 x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W)) \u2192\n    \u2200 (n : V), n \u2208 W \u2192 IsOrtho B n x\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\n\u22a2 (\u2200 (n : V), n \u2208 W \u2192 IsOrtho B n x) \u2192\n    x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W))\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W))\n\u22a2 \u2200 (n : V), n \u2208 W \u2192 IsOrtho B n x\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W))\ny : V\nhy : y \u2208 W\n\u22a2 IsOrtho B y x\n[PROOFSTEP]\nrw [Submodule.mem_dualCoannihilator] at hx \n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : \u2200 (\u03c6 : Module.Dual K V), \u03c6 \u2208 LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W) \u2192 \u2191\u03c6 x = 0\ny : V\nhy : y \u2208 W\n\u22a2 IsOrtho B y x\n[PROOFSTEP]\nrefine' hx (B.toLin.domRestrict W \u27e8y, hy\u27e9) \u27e8\u27e8y, hy\u27e9, rfl\u27e9\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : \u2200 (n : V), n \u2208 W \u2192 IsOrtho B n x\n\u22a2 x \u2208 Submodule.dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W))\n[PROOFSTEP]\nrw [Submodule.mem_dualCoannihilator]\n[GOAL]\ncase h.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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : \u2200 (n : V), n \u2208 W \u2192 IsOrtho B n x\n\u22a2 \u2200 (\u03c6 : Module.Dual K V), \u03c6 \u2208 LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W) \u2192 \u2191\u03c6 x = 0\n[PROOFSTEP]\nrintro _ \u27e8\u27e8w, hw\u27e9, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : \u2200 (n : V), n \u2208 W \u2192 IsOrtho B n x\nw : V\nhw : w \u2208 W\n\u22a2 \u2191(\u2191(LinearMap.domRestrict (\u2191toLin B) W) { val := w, property := hw }) x = 0\n[PROOFSTEP]\nexact hx w hw\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 orthogonal B W } =\n    finrank K V + finrank K { x // x \u2208 W \u2293 orthogonal B \u22a4 }\n[PROOFSTEP]\nrw [\u2190 toLin_restrict_ker_eq_inf_orthogonal _ _ b\u2081, \u2190 toLin_restrict_range_dualCoannihilator_eq_orthogonal _ _,\n  finrank_map_subtype_eq]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\n\u22a2 finrank K { x // x \u2208 W } +\n      finrank K { x // x \u2208 dualCoannihilator (LinearMap.range (LinearMap.domRestrict (\u2191toLin B) W)) } =\n    finrank K V + finrank K { x // x \u2208 LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W) }\n[PROOFSTEP]\nconv_rhs =>\n  rw [\u2190 @Subspace.finrank_add_finrank_dualCoannihilator_eq K V _ _ _ _ (LinearMap.range (B.toLin.domRestrict W)),\n    add_comm, \u2190 add_assoc, add_comm (finrank K (LinearMap.ker (B.toLin.domRestrict W))),\n    LinearMap.finrank_range_add_finrank_ker]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\n| finrank K V + finrank K { x // x \u2208 LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W) }\n[PROOFSTEP]\nrw [\u2190 @Subspace.finrank_add_finrank_dualCoannihilator_eq K V _ _ _ _ (LinearMap.range (B.toLin.domRestrict W)),\n    add_comm, \u2190 add_assoc, add_comm (finrank K (LinearMap.ker (B.toLin.domRestrict W))),\n    LinearMap.finrank_range_add_finrank_ker]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\n| finrank K V + finrank K { x // x \u2208 LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W) }\n[PROOFSTEP]\nrw [\u2190 @Subspace.finrank_add_finrank_dualCoannihilator_eq K V _ _ _ _ (LinearMap.range (B.toLin.domRestrict W)),\n    add_comm, \u2190 add_assoc, add_comm (finrank K (LinearMap.ker (B.toLin.domRestrict W))),\n    LinearMap.finrank_range_add_finrank_ker]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\n| finrank K V + finrank K { x // x \u2208 LinearMap.ker (LinearMap.domRestrict (\u2191toLin B) W) }\n[PROOFSTEP]\nrw [\u2190 @Subspace.finrank_add_finrank_dualCoannihilator_eq K V _ _ _ _ (LinearMap.range (B.toLin.domRestrict W)),\n  add_comm, \u2190 add_assoc, add_comm (finrank K (LinearMap.ker (B.toLin.domRestrict W))),\n  LinearMap.finrank_range_add_finrank_ker]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\n\u22a2 IsCompl W (orthogonal B W)\n[PROOFSTEP]\nhave : W \u2293 B.orthogonal W = \u22a5 := by\n  rw [eq_bot_iff]\n  intro x hx\n  obtain \u27e8hx\u2081, hx\u2082\u27e9 := mem_inf.1 hx\n  refine' Subtype.mk_eq_mk.1 (b\u2082 \u27e8x, hx\u2081\u27e9 _)\n  rintro \u27e8n, hn\u27e9\n  rw [restrict_apply, coe_mk, coe_mk, b\u2081]\n  exact hx\u2082 n hn\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\n\u22a2 W \u2293 orthogonal B W = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\n\u22a2 W \u2293 orthogonal B W \u2264 \u22a5\n[PROOFSTEP]\nintro x hx\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nx : V\nhx : x \u2208 W \u2293 orthogonal B W\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nobtain \u27e8hx\u2081, hx\u2082\u27e9 := mem_inf.1 hx\n[GOAL]\ncase intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nx : V\nhx : x \u2208 W \u2293 orthogonal B W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 orthogonal B W\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nrefine' Subtype.mk_eq_mk.1 (b\u2082 \u27e8x, hx\u2081\u27e9 _)\n[GOAL]\ncase intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nx : V\nhx : x \u2208 W \u2293 orthogonal B W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 orthogonal B W\n\u22a2 \u2200 (n : { x // x \u2208 W }), bilin (restrict B W) { val := x, property := hx\u2081 } n = 0\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase intro.mk\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nx : V\nhx : x \u2208 W \u2293 orthogonal B W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 orthogonal B W\nn : V\nhn : n \u2208 W\n\u22a2 bilin (restrict B W) { val := x, property := hx\u2081 } { val := n, property := hn } = 0\n[PROOFSTEP]\nrw [restrict_apply, coe_mk, coe_mk, b\u2081]\n[GOAL]\ncase intro.mk.a\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nx : V\nhx : x \u2208 W \u2293 orthogonal B W\nhx\u2081 : x \u2208 W\nhx\u2082 : x \u2208 orthogonal B W\nn : V\nhn : n \u2208 W\n\u22a2 bilin B n x = 0\n[PROOFSTEP]\nexact hx\u2082 n hn\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n\u22a2 IsCompl W (orthogonal B W)\n[PROOFSTEP]\nrefine' IsCompl.of_eq this (eq_top_of_finrank_eq <| (finrank_le _).antisymm _)\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n\u22a2 finrank K V \u2264 finrank K { x // x \u2208 W \u2294 orthogonal B W }\n[PROOFSTEP]\nconv_rhs => rw [\u2190 add_zero (finrank K _)]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n| finrank K { x // x \u2208 W \u2294 orthogonal B W }\n[PROOFSTEP]\nrw [\u2190 add_zero (finrank K _)]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n| finrank K { x // x \u2208 W \u2294 orthogonal B W }\n[PROOFSTEP]\nrw [\u2190 add_zero (finrank K _)]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n| finrank K { x // x \u2208 W \u2294 orthogonal B W }\n[PROOFSTEP]\nrw [\u2190 add_zero (finrank K _)]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n\u22a2 finrank K V \u2264 finrank K { x // x \u2208 W \u2294 orthogonal B W } + 0\n[PROOFSTEP]\nrw [\u2190 finrank_bot K V, \u2190 this, finrank_sup_add_finrank_inf_eq, finrank_add_finrank_orthogonal b\u2081]\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\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB : BilinForm K V\nW : Subspace K V\nb\u2081 : IsRefl B\nb\u2082 : Nondegenerate (restrict B W)\nthis : W \u2293 orthogonal B W = \u22a5\n\u22a2 finrank K V \u2264 finrank K V + finrank K { x // x \u2208 W \u2293 orthogonal B \u22a4 }\n[PROOFSTEP]\nexact le_self_add\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\nM\u2082' : Type u_11\ninst\u271d\u2074 : AddCommMonoid M\u2082'\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : FiniteDimensional K V\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : BilinForm K V\nhB : Nondegenerate B\nb : Basis \u03b9 K V\nx : V\ni : \u03b9\n\u22a2 \u2191(\u2191(dualBasis B hB b).repr x) i = bilin B x (\u2191b i)\n[PROOFSTEP]\nrw [dualBasis, Basis.map_repr, LinearEquiv.symm_symm, LinearEquiv.trans_apply, Basis.dualBasis_repr, toDual_def]\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\nM\u2082' : Type u_11\ninst\u271d\u2074 : AddCommMonoid M\u2082'\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : FiniteDimensional K V\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : BilinForm K V\nhB : Nondegenerate B\nb : Basis \u03b9 K V\ni j : \u03b9\n\u22a2 bilin B (\u2191(dualBasis B hB b) i) (\u2191b j) = if j = i then 1 else 0\n[PROOFSTEP]\nrw [dualBasis, Basis.map_apply, Basis.coe_dualBasis, \u2190 toDual_def hB, LinearEquiv.apply_symm_apply, Basis.coord_apply,\n  Basis.repr_self, Finsupp.single_apply]\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\nM\u2082' : Type u_11\ninst\u271d\u2074 : AddCommMonoid M\u2082'\ninst\u271d\u00b3 : Module R\u2082 M\u2082'\ninst\u271d\u00b2 : FiniteDimensional K V\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : BilinForm K V\nhB : Nondegenerate B\nsym : IsSymm B\nb : Basis \u03b9 K V\ni j : \u03b9\n\u22a2 bilin B (\u2191b i) (\u2191(dualBasis B hB b) j) = if i = j then 1 else 0\n[PROOFSTEP]\nrw [sym, apply_dualBasis_left]\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\n\u22a2 Nondegenerate (restrict B (orthogonal B (Submodule.span K {x})))\n[PROOFSTEP]\nrefine' fun m hm => Submodule.coe_eq_zero.1 (b\u2081 m.1 fun n => _)\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\nn : V\n\u22a2 bilin B (\u2191m) n = 0\n[PROOFSTEP]\nhave : n \u2208 (K \u2219 x) \u2294 B.orthogonal (K \u2219 x) := (span_singleton_sup_orthogonal_eq_top hx).symm \u25b8 Submodule.mem_top\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\nn : V\nthis : n \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\n\u22a2 bilin B (\u2191m) n = 0\n[PROOFSTEP]\nrcases Submodule.mem_sup.1 this with \u27e8y, hy, z, hz, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\n\u22a2 bilin B (\u2191m) (y + z) = 0\n[PROOFSTEP]\nspecialize hm \u27e8z, hz\u27e9\n[GOAL]\ncase intro.intro.intro.intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm : bilin (restrict B (orthogonal B (Submodule.span K {x}))) m { val := z, property := hz } = 0\n\u22a2 bilin B (\u2191m) (y + z) = 0\n[PROOFSTEP]\nrw [restrict] at hm \n[GOAL]\ncase intro.intro.intro.intro\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B (\u2191m) (y + z) = 0\n[PROOFSTEP]\nerw [add_right, show B m.1 y = 0 by rw [b\u2082]; exact m.2 y hy, hm, add_zero]\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B (\u2191m) y = 0\n[PROOFSTEP]\nrw [b\u2082]\n[GOAL]\ncase a\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B y \u2191m = 0\n[PROOFSTEP]\nexact m.2 y hy\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : Nondegenerate B\n\u03c6 \u03c8 : M\u2081 \u2192\u2097[R\u2081] M\u2081\nh : compLeft B \u03c6 = compLeft B \u03c8\n\u22a2 \u03c6 = \u03c8\n[PROOFSTEP]\next w\n[GOAL]\ncase h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : Nondegenerate B\n\u03c6 \u03c8 : M\u2081 \u2192\u2097[R\u2081] M\u2081\nh : compLeft B \u03c6 = compLeft B \u03c8\nw : M\u2081\n\u22a2 \u2191\u03c6 w = \u2191\u03c8 w\n[PROOFSTEP]\nrefine' eq_of_sub_eq_zero (b _ _)\n[GOAL]\ncase h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : Nondegenerate B\n\u03c6 \u03c8 : M\u2081 \u2192\u2097[R\u2081] M\u2081\nh : compLeft B \u03c6 = compLeft B \u03c8\nw : M\u2081\n\u22a2 \u2200 (n : M\u2081), bilin B (\u2191\u03c6 w - \u2191\u03c8 w) n = 0\n[PROOFSTEP]\nintro v\n[GOAL]\ncase h\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : Nondegenerate B\n\u03c6 \u03c8 : M\u2081 \u2192\u2097[R\u2081] M\u2081\nh : compLeft B \u03c6 = compLeft B \u03c8\nw v : M\u2081\n\u22a2 bilin B (\u2191\u03c6 w - \u2191\u03c8 w) v = 0\n[PROOFSTEP]\nrw [sub_left, \u2190 compLeft_apply, \u2190 compLeft_apply, \u2190 h, sub_self]\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\nM\u2082' : Type u_11\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm R\u2081 M\u2081\nb : Nondegenerate B\n\u03c6 \u03c8\u2081 \u03c8\u2082 : M\u2081 \u2192\u2097[R\u2081] M\u2081\nh\u03c8\u2081 : IsAdjointPair B B \u03c8\u2081 \u03c6\nh\u03c8\u2082 : IsAdjointPair B B \u03c8\u2082 \u03c6\nv w : M\u2081\n\u22a2 bilin (compLeft B \u03c8\u2081) v w = bilin (compLeft B \u03c8\u2082) v w\n[PROOFSTEP]\nrw [compLeft_apply, compLeft_apply, h\u03c8\u2081, h\u03c8\u2082]\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : Nondegenerate B\u2082\nv : V\n\u22a2 \u2191(\u2191toLin B\u2082) (\u2191(symmCompOfNondegenerate B\u2081 B\u2082 b\u2082) v) = \u2191(\u2191toLin B\u2081) v\n[PROOFSTEP]\nerw [symmCompOfNondegenerate, LinearEquiv.apply_symm_apply (B\u2082.toDual b\u2082) _]\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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : Nondegenerate B\u2082\nv w : V\n\u22a2 bilin B\u2082 (\u2191(symmCompOfNondegenerate B\u2081 B\u2082 b\u2082) w) v = bilin B\u2081 w v\n[PROOFSTEP]\nconv_lhs => rw [\u2190 BilinForm.toLin_apply, comp_symmCompOfNondegenerate_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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : Nondegenerate B\u2082\nv w : V\n| bilin B\u2082 (\u2191(symmCompOfNondegenerate B\u2081 B\u2082 b\u2082) w) v\n[PROOFSTEP]\nrw [\u2190 BilinForm.toLin_apply, comp_symmCompOfNondegenerate_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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : Nondegenerate B\u2082\nv w : V\n| bilin B\u2082 (\u2191(symmCompOfNondegenerate B\u2081 B\u2082 b\u2082) w) v\n[PROOFSTEP]\nrw [\u2190 BilinForm.toLin_apply, comp_symmCompOfNondegenerate_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\u271d : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nM\u2082' : Type u_11\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : Nondegenerate B\u2082\nv w : V\n| bilin B\u2082 (\u2191(symmCompOfNondegenerate B\u2081 B\u2082 b\u2082) w) v\n[PROOFSTEP]\nrw [\u2190 BilinForm.toLin_apply, comp_symmCompOfNondegenerate_apply]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.BilinearForm", "llama_tokens": 153991, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289387914176258, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.5573218261494362}}
{"text": "[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix l m \u03b1\nD : Matrix l n \u03b1\ninst\u271d : Invertible A\n\u22a2 fromBlocks A B C D = fromBlocks 1 0 (C * \u215fA) 1 * fromBlocks A 0 0 (D - C * \u215fA * B) * fromBlocks 1 (\u215fA * B) 0 1\n[PROOFSTEP]\nsimp only [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add, Matrix.one_mul, Matrix.mul_one,\n  invOf_mul_self, Matrix.mul_invOf_self_assoc, Matrix.mul_invOf_mul_self_cancel, Matrix.mul_assoc,\n  add_sub_cancel'_right]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix l n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\n\u22a2 \u2191(reindex (Equiv.sumComm l n) (Equiv.sumComm m n)) (fromBlocks A B C D) =\n    \u2191(reindex (Equiv.sumComm l n) (Equiv.sumComm m n))\n      (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n[PROOFSTEP]\nsimpa [reindex_apply, Equiv.sumComm_symm, \u2190 submatrix_mul_equiv _ _ _ (Equiv.sumComm n m), \u2190\n  submatrix_mul_equiv _ _ _ (Equiv.sumComm n l), Equiv.sumComm_apply, fromBlocks_submatrix_sum_swap_sum_swap] using\n  fromBlocks_eq_of_invertible\u2081\u2081 D C B A\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible D\n\u22a2 fromBlocks (\u215fA) (-(\u215fA * B * \u215fD)) 0 \u215fD * fromBlocks A B 0 D = 1\n[PROOFSTEP]\nsimp_rw [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, zero_add, add_zero, Matrix.neg_mul, invOf_mul_self,\n  Matrix.mul_invOf_mul_self_cancel, add_right_neg, fromBlocks_one]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible D\n\u22a2 fromBlocks (\u215fA) 0 (-(\u215fD * C * \u215fA)) \u215fD * fromBlocks A 0 C D = 1\n[PROOFSTEP]\nsimp_rw [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, zero_add, add_zero, Matrix.neg_mul, invOf_mul_self,\n  Matrix.mul_invOf_mul_self_cancel, add_left_neg, fromBlocks_one]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible A\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B 0 D)\n\u22a2 \u215f(fromBlocks A B 0 D) = fromBlocks (\u215fA) (-(\u215fA * B * \u215fD)) 0 \u215fD\n[PROOFSTEP]\nletI := fromBlocksZero\u2082\u2081Invertible A B D\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible A\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis : Invertible (fromBlocks A B 0 D) := fromBlocksZero\u2082\u2081Invertible A B D\n\u22a2 \u215f(fromBlocks A B 0 D) = fromBlocks (\u215fA) (-(\u215fA * B * \u215fD)) 0 \u215fD\n[PROOFSTEP]\nconvert (rfl : \u215f(fromBlocks A B 0 D) = _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible A\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A 0 C D)\n\u22a2 \u215f(fromBlocks A 0 C D) = fromBlocks (\u215fA) 0 (-(\u215fD * C * \u215fA)) \u215fD\n[PROOFSTEP]\nletI := fromBlocksZero\u2081\u2082Invertible A C D\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible A\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis : Invertible (fromBlocks A 0 C D) := fromBlocksZero\u2081\u2082Invertible A C D\n\u22a2 \u215f(fromBlocks A 0 C D) = fromBlocks (\u215fA) 0 (-(\u215fD * C * \u215fA)) \u215fD\n[PROOFSTEP]\nconvert (rfl : \u215f(fromBlocks A 0 C D) = _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\n\u22a2 toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * A = 1\n[PROOFSTEP]\nhave := invOf_mul_self (fromBlocks A B 0 D)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis : \u215f(fromBlocks A B 0 D) * fromBlocks A B 0 D = 1\n\u22a2 toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * A = 1\n[PROOFSTEP]\nrw [\u2190 fromBlocks_toBlocks (\u215f(fromBlocks A B 0 D)), fromBlocks_multiply] at this \n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis :\n  fromBlocks (toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * A + toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) * 0)\n      (toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * B + toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) * D)\n      (toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D) * A + toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) * 0)\n      (toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D) * B + toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) * D) =\n    1\n\u22a2 toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * A = 1\n[PROOFSTEP]\nreplace := congr_arg Matrix.toBlocks\u2081\u2081 this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis :\n  toBlocks\u2081\u2081\n      (fromBlocks (toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * A + toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) * 0)\n        (toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * B + toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) * D)\n        (toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D) * A + toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) * 0)\n        (toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D) * B + toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) * D)) =\n    toBlocks\u2081\u2081 1\n\u22a2 toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) * A = 1\n[PROOFSTEP]\nsimpa only [Matrix.toBlocks_fromBlocks\u2081\u2081, Matrix.mul_zero, add_zero, \u2190 fromBlocks_one] using this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\n\u22a2 D * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) = 1\n[PROOFSTEP]\nhave := mul_invOf_self (fromBlocks A B 0 D)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis : fromBlocks A B 0 D * \u215f(fromBlocks A B 0 D) = 1\n\u22a2 D * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) = 1\n[PROOFSTEP]\nrw [\u2190 fromBlocks_toBlocks (\u215f(fromBlocks A B 0 D)), fromBlocks_multiply] at this \n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis :\n  fromBlocks (A * toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) + B * toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D))\n      (A * toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) + B * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D))\n      (0 * toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) + D * toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D))\n      (0 * toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) + D * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D)) =\n    1\n\u22a2 D * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) = 1\n[PROOFSTEP]\nreplace := congr_arg Matrix.toBlocks\u2082\u2082 this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A B 0 D)\nthis :\n  toBlocks\u2082\u2082\n      (fromBlocks (A * toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) + B * toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D))\n        (A * toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) + B * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D))\n        (0 * toBlocks\u2081\u2081 \u215f(fromBlocks A B 0 D) + D * toBlocks\u2082\u2081 \u215f(fromBlocks A B 0 D))\n        (0 * toBlocks\u2081\u2082 \u215f(fromBlocks A B 0 D) + D * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D))) =\n    toBlocks\u2082\u2082 1\n\u22a2 D * toBlocks\u2082\u2082 \u215f(fromBlocks A B 0 D) = 1\n[PROOFSTEP]\nsimpa only [Matrix.toBlocks_fromBlocks\u2082\u2082, Matrix.zero_mul, zero_add, \u2190 fromBlocks_one] using this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\n\u22a2 A * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) = 1\n[PROOFSTEP]\nhave := mul_invOf_self (fromBlocks A 0 C D)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis : fromBlocks A 0 C D * \u215f(fromBlocks A 0 C D) = 1\n\u22a2 A * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) = 1\n[PROOFSTEP]\nrw [\u2190 fromBlocks_toBlocks (\u215f(fromBlocks A 0 C D)), fromBlocks_multiply] at this \n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis :\n  fromBlocks (A * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) + 0 * toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D))\n      (A * toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) + 0 * toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D))\n      (C * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) + D * toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D))\n      (C * toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) + D * toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D)) =\n    1\n\u22a2 A * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) = 1\n[PROOFSTEP]\nreplace := congr_arg Matrix.toBlocks\u2081\u2081 this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis :\n  toBlocks\u2081\u2081\n      (fromBlocks (A * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) + 0 * toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D))\n        (A * toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) + 0 * toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D))\n        (C * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) + D * toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D))\n        (C * toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) + D * toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D))) =\n    toBlocks\u2081\u2081 1\n\u22a2 A * toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) = 1\n[PROOFSTEP]\nsimpa only [Matrix.toBlocks_fromBlocks\u2081\u2081, Matrix.zero_mul, add_zero, \u2190 fromBlocks_one] using this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\n\u22a2 toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * D = 1\n[PROOFSTEP]\nhave := invOf_mul_self (fromBlocks A 0 C D)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis : \u215f(fromBlocks A 0 C D) * fromBlocks A 0 C D = 1\n\u22a2 toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * D = 1\n[PROOFSTEP]\nrw [\u2190 fromBlocks_toBlocks (\u215f(fromBlocks A 0 C D)), fromBlocks_multiply] at this \n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis :\n  fromBlocks (toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) * A + toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) * C)\n      (toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) * 0 + toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) * D)\n      (toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D) * A + toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * C)\n      (toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D) * 0 + toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * D) =\n    1\n\u22a2 toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * D = 1\n[PROOFSTEP]\nreplace := congr_arg Matrix.toBlocks\u2082\u2082 this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible (fromBlocks A 0 C D)\nthis :\n  toBlocks\u2082\u2082\n      (fromBlocks (toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) * A + toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) * C)\n        (toBlocks\u2081\u2081 \u215f(fromBlocks A 0 C D) * 0 + toBlocks\u2081\u2082 \u215f(fromBlocks A 0 C D) * D)\n        (toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D) * A + toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * C)\n        (toBlocks\u2082\u2081 \u215f(fromBlocks A 0 C D) * 0 + toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * D)) =\n    toBlocks\u2082\u2082 1\n\u22a2 toBlocks\u2082\u2082 \u215f(fromBlocks A 0 C D) * D = 1\n[PROOFSTEP]\nsimpa only [Matrix.toBlocks_fromBlocks\u2082\u2082, Matrix.mul_zero, zero_add, \u2190 fromBlocks_one] using this\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ni : Invertible A \u00d7 Invertible D\n\u22a2 Invertible (fromBlocks A B 0 D)\n[PROOFSTEP]\nletI := i.1\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ni : Invertible A \u00d7 Invertible D\nthis : Invertible A := i.fst\n\u22a2 Invertible (fromBlocks A B 0 D)\n[PROOFSTEP]\nletI := i.2\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\ni : Invertible A \u00d7 Invertible D\nthis\u271d : Invertible A := i.fst\nthis : Invertible D := i.snd\n\u22a2 Invertible (fromBlocks A B 0 D)\n[PROOFSTEP]\nexact fromBlocksZero\u2082\u2081Invertible A B D\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ni : Invertible A \u00d7 Invertible D\n\u22a2 Invertible (fromBlocks A 0 C D)\n[PROOFSTEP]\nletI := i.1\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ni : Invertible A \u00d7 Invertible D\nthis : Invertible A := i.fst\n\u22a2 Invertible (fromBlocks A 0 C D)\n[PROOFSTEP]\nletI := i.2\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ni : Invertible A \u00d7 Invertible D\nthis\u271d : Invertible A := i.fst\nthis : Invertible D := i.snd\n\u22a2 Invertible (fromBlocks A 0 C D)\n[PROOFSTEP]\nexact fromBlocksZero\u2081\u2082Invertible A C D\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\n\u22a2 IsUnit (fromBlocks A B 0 D) \u2194 IsUnit A \u2227 IsUnit D\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, \u2190 nonempty_prod, (fromBlocksZero\u2082\u2081InvertibleEquiv _ _ _).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\n\u22a2 IsUnit (fromBlocks A 0 C D) \u2194 IsUnit A \u2227 IsUnit D\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, \u2190 nonempty_prod, (fromBlocksZero\u2081\u2082InvertibleEquiv _ _ _).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nby_cases hA : IsUnit A\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nhave hD := hAD.mp hA\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\ncases hA.nonempty_invertible\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\nval\u271d : Invertible A\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\ncases hD.nonempty_invertible\n[GOAL]\ncase pos.intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\nval\u271d\u00b9 : Invertible A\nval\u271d : Invertible D\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nletI := fromBlocksZero\u2082\u2081Invertible A B D\n[GOAL]\ncase pos.intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\nval\u271d\u00b9 : Invertible A\nval\u271d : Invertible D\nthis : Invertible (fromBlocks A B 0 D) := fromBlocksZero\u2082\u2081Invertible A B D\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [\u2190 invOf_eq_nonsing_inv, invOf_fromBlocks_zero\u2082\u2081_eq]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : \u00acIsUnit A\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nhave hD := hAD.not.mp hA\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : \u00acIsUnit A\nhD : \u00acIsUnit D\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nhave : \u00acIsUnit (fromBlocks A B 0 D) := isUnit_fromBlocks_zero\u2082\u2081.not.mpr (not_and'.mpr fun _ => hA)\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : \u00acIsUnit A\nhD : \u00acIsUnit D\nthis : \u00acIsUnit (fromBlocks A B 0 D)\n\u22a2 (fromBlocks A B 0 D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 (-(A\u207b\u00b9 * B * D\u207b\u00b9)) 0 D\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ hA, Ring.inverse_non_unit _ hD,\n  Ring.inverse_non_unit _ this, Matrix.zero_mul, neg_zero, fromBlocks_zero]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nby_cases hA : IsUnit A\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nhave hD := hAD.mp hA\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\ncases hA.nonempty_invertible\n[GOAL]\ncase pos.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\nval\u271d : Invertible A\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\ncases hD.nonempty_invertible\n[GOAL]\ncase pos.intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\nval\u271d\u00b9 : Invertible A\nval\u271d : Invertible D\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nletI := fromBlocksZero\u2081\u2082Invertible A C D\n[GOAL]\ncase pos.intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : IsUnit A\nhD : IsUnit D\nval\u271d\u00b9 : Invertible A\nval\u271d : Invertible D\nthis : Invertible (fromBlocks A 0 C D) := fromBlocksZero\u2081\u2082Invertible A C D\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [\u2190 invOf_eq_nonsing_inv, invOf_fromBlocks_zero\u2081\u2082_eq]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : \u00acIsUnit A\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nhave hD := hAD.not.mp hA\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : \u00acIsUnit A\nhD : \u00acIsUnit D\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nhave : \u00acIsUnit (fromBlocks A 0 C D) := isUnit_fromBlocks_zero\u2081\u2082.not.mpr (not_and'.mpr fun _ => hA)\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nhAD : IsUnit A \u2194 IsUnit D\nhA : \u00acIsUnit A\nhD : \u00acIsUnit D\nthis : \u00acIsUnit (fromBlocks A 0 C D)\n\u22a2 (fromBlocks A 0 C D)\u207b\u00b9 = fromBlocks A\u207b\u00b9 0 (-(D\u207b\u00b9 * C * A\u207b\u00b9)) D\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ hA, Ring.inverse_non_unit _ hD,\n  Ring.inverse_non_unit _ this, Matrix.zero_mul, neg_zero, fromBlocks_zero]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\n\u22a2 Invertible (fromBlocks A B C D)\n[PROOFSTEP]\nrefine'\n  Invertible.copy' _ _\n    (fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * B * \u215fD)) (-(\u215fD * C * \u215f(A - B * \u215fD * C)))\n      (\u215fD + \u215fD * C * \u215f(A - B * \u215fD * C) * B * \u215fD))\n    (fromBlocks_eq_of_invertible\u2082\u2082 _ _ _ _) _\n[GOAL]\ncase refine'_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\n\u22a2 Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n[PROOFSTEP]\nletI : Invertible (1 : Matrix n n \u03b1) := invertibleOne\n[GOAL]\ncase refine'_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\nthis : Invertible 1 := invertibleOne\n\u22a2 Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n[PROOFSTEP]\nletI : Invertible (1 : Matrix m m \u03b1) := invertibleOne\n[GOAL]\ncase refine'_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\n\u22a2 Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n[PROOFSTEP]\nrefine' Invertible.mul _ (fromBlocksZero\u2081\u2082Invertible _ _ _)\n[GOAL]\ncase refine'_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\n\u22a2 Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nexact Invertible.mul (fromBlocksZero\u2082\u2081Invertible _ _ _) (fromBlocksZero\u2082\u2081Invertible _ _ _)\n[GOAL]\ncase refine'_2\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\n\u22a2 fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * B * \u215fD)) (-(\u215fD * C * \u215f(A - B * \u215fD * C)))\n      (\u215fD + \u215fD * C * \u215f(A - B * \u215fD * C) * B * \u215fD) =\n    \u215f(fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n[PROOFSTEP]\nshow\n  _ =\n    fromBlocks 1 0 (-(1 * (\u215fD * C) * 1)) 1 *\n      (fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * 0 * \u215fD)) 0 (\u215fD) * fromBlocks 1 (-(1 * (B * \u215fD) * 1)) 0 1)\n        -- combine into a single block matrix\n[GOAL]\ncase refine'_2\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (A - B * \u215fD * C)\n\u22a2 fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * B * \u215fD)) (-(\u215fD * C * \u215f(A - B * \u215fD * C)))\n      (\u215fD + \u215fD * C * \u215f(A - B * \u215fD * C) * B * \u215fD) =\n    fromBlocks 1 0 (-(1 * (\u215fD * C) * 1)) 1 *\n      (fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * 0 * \u215fD)) 0 \u215fD * fromBlocks 1 (-(1 * (B * \u215fD) * 1)) 0 1)\n[PROOFSTEP]\nsimp only [fromBlocks_multiply, invOf_one, Matrix.one_mul, Matrix.mul_one, Matrix.zero_mul, Matrix.mul_zero, add_zero,\n  zero_add, neg_zero, Matrix.mul_neg, Matrix.neg_mul, neg_neg, \u2190 Matrix.mul_assoc, add_comm (\u215fD)]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (D - C * \u215fA * B)\n\u22a2 Invertible (fromBlocks A B C D)\n[PROOFSTEP]\nletI := fromBlocks\u2082\u2082Invertible D C B A\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (D - C * \u215fA * B)\nthis : Invertible (fromBlocks D C B A) := fromBlocks\u2082\u2082Invertible D C B A\n\u22a2 Invertible (fromBlocks A B C D)\n[PROOFSTEP]\nletI iDCBA := submatrixEquivInvertible (fromBlocks D C B A) (Equiv.sumComm _ _) (Equiv.sumComm _ _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (D - C * \u215fA * B)\nthis : Invertible (fromBlocks D C B A) := fromBlocks\u2082\u2082Invertible D C B A\niDCBA : Invertible (submatrix (fromBlocks D C B A) \u2191(Equiv.sumComm m n) \u2191(Equiv.sumComm m n)) :=\n  submatrixEquivInvertible (fromBlocks D C B A) (Equiv.sumComm m n) (Equiv.sumComm m n)\n\u22a2 Invertible (fromBlocks A B C D)\n[PROOFSTEP]\nexact\n  iDCBA.copy' _\n    (fromBlocks (\u215fA + \u215fA * B * \u215f(D - C * \u215fA * B) * C * \u215fA) (-(\u215fA * B * \u215f(D - C * \u215fA * B)))\n      (-(\u215f(D - C * \u215fA * B) * C * \u215fA)) (\u215f(D - C * \u215fA * B)))\n    (fromBlocks_submatrix_sum_swap_sum_swap _ _ _ _).symm (fromBlocks_submatrix_sum_swap_sum_swap _ _ _ _).symm\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible D\ninst\u271d\u00b9 : Invertible (A - B * \u215fD * C)\ninst\u271d : Invertible (fromBlocks A B C D)\n\u22a2 \u215f(fromBlocks A B C D) =\n    fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * B * \u215fD)) (-(\u215fD * C * \u215f(A - B * \u215fD * C)))\n      (\u215fD + \u215fD * C * \u215f(A - B * \u215fD * C) * B * \u215fD)\n[PROOFSTEP]\nletI := fromBlocks\u2082\u2082Invertible A B C D\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible D\ninst\u271d\u00b9 : Invertible (A - B * \u215fD * C)\ninst\u271d : Invertible (fromBlocks A B C D)\nthis : Invertible (fromBlocks A B C D) := fromBlocks\u2082\u2082Invertible A B C D\n\u22a2 \u215f(fromBlocks A B C D) =\n    fromBlocks (\u215f(A - B * \u215fD * C)) (-(\u215f(A - B * \u215fD * C) * B * \u215fD)) (-(\u215fD * C * \u215f(A - B * \u215fD * C)))\n      (\u215fD + \u215fD * C * \u215f(A - B * \u215fD * C) * B * \u215fD)\n[PROOFSTEP]\nconvert (rfl : \u215f(fromBlocks A B C D) = _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible A\ninst\u271d\u00b9 : Invertible (D - C * \u215fA * B)\ninst\u271d : Invertible (fromBlocks A B C D)\n\u22a2 \u215f(fromBlocks A B C D) =\n    fromBlocks (\u215fA + \u215fA * B * \u215f(D - C * \u215fA * B) * C * \u215fA) (-(\u215fA * B * \u215f(D - C * \u215fA * B)))\n      (-(\u215f(D - C * \u215fA * B) * C * \u215fA)) \u215f(D - C * \u215fA * B)\n[PROOFSTEP]\nletI := fromBlocks\u2081\u2081Invertible A B C D\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2079 : Fintype l\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq l\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b2 : Invertible A\ninst\u271d\u00b9 : Invertible (D - C * \u215fA * B)\ninst\u271d : Invertible (fromBlocks A B C D)\nthis : Invertible (fromBlocks A B C D) := fromBlocks\u2081\u2081Invertible A B C D\n\u22a2 \u215f(fromBlocks A B C D) =\n    fromBlocks (\u215fA + \u215fA * B * \u215f(D - C * \u215fA * B) * C * \u215fA) (-(\u215fA * B * \u215f(D - C * \u215fA * B)))\n      (-(\u215f(D - C * \u215fA * B) * C * \u215fA)) \u215f(D - C * \u215fA * B)\n[PROOFSTEP]\nconvert (rfl : \u215f(fromBlocks A B C D) = _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\n\u22a2 Invertible (A - B * \u215fD * C)\n[PROOFSTEP]\nsuffices Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D) by\n  exact (invertibleOfFromBlocksZero\u2081\u2082Invertible (A - B * \u215fD * C) 0 D).1\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis : Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n\u22a2 Invertible (A - B * \u215fD * C)\n[PROOFSTEP]\nexact (invertibleOfFromBlocksZero\u2081\u2082Invertible (A - B * \u215fD * C) 0 D).1\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\n\u22a2 Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nletI : Invertible (1 : Matrix n n \u03b1) := invertibleOne\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis : Invertible 1 := invertibleOne\n\u22a2 Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nletI : Invertible (1 : Matrix m m \u03b1) := invertibleOne\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\n\u22a2 Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nletI iDC : Invertible (fromBlocks 1 0 (\u215fD * C) 1 : Matrix (Sum m n) (Sum m n) \u03b1) := fromBlocksZero\u2081\u2082Invertible _ _ _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\niDC : Invertible (fromBlocks 1 0 (\u215fD * C) 1) := fromBlocksZero\u2081\u2082Invertible 1 (\u215fD * C) 1\n\u22a2 Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nletI iBD : Invertible (fromBlocks 1 (B * \u215fD) 0 1 : Matrix (Sum m n) (Sum m n) \u03b1) := fromBlocksZero\u2082\u2081Invertible _ _ _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\niDC : Invertible (fromBlocks 1 0 (\u215fD * C) 1) := fromBlocksZero\u2081\u2082Invertible 1 (\u215fD * C) 1\niBD : Invertible (fromBlocks 1 (B * \u215fD) 0 1) := fromBlocksZero\u2082\u2081Invertible 1 (B * \u215fD) 1\n\u22a2 Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nletI iBDC := Invertible.copy \u2039_\u203a _ (fromBlocks_eq_of_invertible\u2082\u2082 A B C D).symm\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\niDC : Invertible (fromBlocks 1 0 (\u215fD * C) 1) := fromBlocksZero\u2081\u2082Invertible 1 (\u215fD * C) 1\niBD : Invertible (fromBlocks 1 (B * \u215fD) 0 1) := fromBlocksZero\u2082\u2081Invertible 1 (B * \u215fD) 1\niBDC : Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1) :=\n  Invertible.copy inst\u271d (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n    (_ : fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1 = fromBlocks A B C D)\n\u22a2 Invertible (fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nrefine' (iBD.mulLeft _).symm _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible D\ninst\u271d : Invertible (fromBlocks A B C D)\nthis\u271d : Invertible 1 := invertibleOne\nthis : Invertible 1 := invertibleOne\niDC : Invertible (fromBlocks 1 0 (\u215fD * C) 1) := fromBlocksZero\u2081\u2082Invertible 1 (\u215fD * C) 1\niBD : Invertible (fromBlocks 1 (B * \u215fD) 0 1) := fromBlocksZero\u2082\u2081Invertible 1 (B * \u215fD) 1\niBDC : Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1) :=\n  Invertible.copy inst\u271d (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1)\n    (_ : fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D * fromBlocks 1 0 (\u215fD * C) 1 = fromBlocks A B C D)\n\u22a2 Invertible (fromBlocks 1 (B * \u215fD) 0 1 * fromBlocks (A - B * \u215fD * C) 0 0 D)\n[PROOFSTEP]\nrefine' (iDC.mulRight _).symm iBDC\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (fromBlocks A B C D)\n\u22a2 Invertible (D - C * \u215fA * B)\n[PROOFSTEP]\nletI iABCD' := submatrixEquivInvertible (fromBlocks A B C D) (Equiv.sumComm _ _) (Equiv.sumComm _ _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (fromBlocks A B C D)\niABCD' : Invertible (submatrix (fromBlocks A B C D) \u2191(Equiv.sumComm n m) \u2191(Equiv.sumComm n m)) :=\n  submatrixEquivInvertible (fromBlocks A B C D) (Equiv.sumComm n m) (Equiv.sumComm n m)\n\u22a2 Invertible (D - C * \u215fA * B)\n[PROOFSTEP]\nletI iDCBA := iABCD'.copy _ (fromBlocks_submatrix_sum_swap_sum_swap _ _ _ _).symm\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2078 : Fintype l\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d\u00b9 : Invertible A\ninst\u271d : Invertible (fromBlocks A B C D)\niABCD' : Invertible (submatrix (fromBlocks A B C D) \u2191(Equiv.sumComm n m) \u2191(Equiv.sumComm n m)) :=\n  submatrixEquivInvertible (fromBlocks A B C D) (Equiv.sumComm n m) (Equiv.sumComm n m)\niDCBA : Invertible (fromBlocks D C B A) :=\n  Invertible.copy iABCD' (fromBlocks D C B A)\n    (_ : fromBlocks D C B A = submatrix (fromBlocks A B C D) Sum.swap Sum.swap)\n\u22a2 Invertible (D - C * \u215fA * B)\n[PROOFSTEP]\nrefine' invertibleOfFromBlocks\u2082\u2082Invertible D C B A\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\n\u22a2 IsUnit (fromBlocks A B C D) \u2194 IsUnit (A - B * \u215fD * C)\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, (invertibleEquivFromBlocks\u2082\u2082Invertible A B C D).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 IsUnit (fromBlocks A B C D) \u2194 IsUnit (D - C * \u215fA * B)\n[PROOFSTEP]\nsimp only [\u2190 nonempty_invertible_iff_isUnit, (invertibleEquivFromBlocks\u2081\u2081Invertible A B C D).nonempty_congr]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 det (fromBlocks A B C D) = det A * det (D - C * \u215fA * B)\n[PROOFSTEP]\nrw [fromBlocks_eq_of_invertible\u2081\u2081 (A := A), det_mul, det_mul, det_fromBlocks_zero\u2082\u2081, det_fromBlocks_zero\u2082\u2081,\n  det_fromBlocks_zero\u2081\u2082, det_one, det_one, one_mul, one_mul, mul_one]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\n\u22a2 det (fromBlocks 1 B C D) = det (D - C * B)\n[PROOFSTEP]\nhaveI : Invertible (1 : Matrix m m \u03b1) := invertibleOne\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\nthis : Invertible 1\n\u22a2 det (fromBlocks 1 B C D) = det (D - C * B)\n[PROOFSTEP]\nrw [det_fromBlocks\u2081\u2081, invOf_one, Matrix.mul_one, det_one, one_mul]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\n\u22a2 det (fromBlocks A B C D) = det D * det (A - B * \u215fD * C)\n[PROOFSTEP]\nhave : fromBlocks A B C D = (fromBlocks D C B A).submatrix (Equiv.sumComm _ _) (Equiv.sumComm _ _) :=\n  by\n  ext (i j)\n  cases i <;> cases j <;> rfl\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\n\u22a2 fromBlocks A B C D = submatrix (fromBlocks D C B A) \u2191(Equiv.sumComm m n) \u2191(Equiv.sumComm m n)\n[PROOFSTEP]\next (i j)\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\ni j : m \u2295 n\n\u22a2 fromBlocks A B C D i j = submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) i j\n[PROOFSTEP]\ncases i\n[GOAL]\ncase a.h.inl\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nj : m \u2295 n\nval\u271d : m\n\u22a2 fromBlocks A B C D (Sum.inl val\u271d) j =\n    submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) (Sum.inl val\u271d) j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase a.h.inr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nj : m \u2295 n\nval\u271d : n\n\u22a2 fromBlocks A B C D (Sum.inr val\u271d) j =\n    submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) (Sum.inr val\u271d) j\n[PROOFSTEP]\ncases j\n[GOAL]\ncase a.h.inl.inl\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nval\u271d\u00b9 val\u271d : m\n\u22a2 fromBlocks A B C D (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d) =\n    submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) (Sum.inl val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h.inl.inr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nval\u271d\u00b9 : m\nval\u271d : n\n\u22a2 fromBlocks A B C D (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d) =\n    submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) (Sum.inl val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h.inr.inl\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nval\u271d\u00b9 : n\nval\u271d : m\n\u22a2 fromBlocks A B C D (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d) =\n    submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h.inr.inr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nval\u271d\u00b9 val\u271d : n\n\u22a2 fromBlocks A B C D (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d) =\n    submatrix (fromBlocks D C B A) (\u2191(Equiv.sumComm m n)) (\u2191(Equiv.sumComm m n)) (Sum.inr val\u271d\u00b9) (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible D\nthis : fromBlocks A B C D = submatrix (fromBlocks D C B A) \u2191(Equiv.sumComm m n) \u2191(Equiv.sumComm m n)\n\u22a2 det (fromBlocks A B C D) = det D * det (A - B * \u215fD * C)\n[PROOFSTEP]\nrw [this, det_submatrix_equiv_self, det_fromBlocks\u2081\u2081]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\n\u22a2 det (fromBlocks A B C 1) = det (A - B * C)\n[PROOFSTEP]\nhaveI : Invertible (1 : Matrix n n \u03b1) := invertibleOne\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nthis : Invertible 1\n\u22a2 det (fromBlocks A B C 1) = det (A - B * C)\n[PROOFSTEP]\nrw [det_fromBlocks\u2082\u2082, invOf_one, Matrix.mul_one, det_one, one_mul]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m n \u03b1\nB : Matrix n m \u03b1\n\u22a2 det (1 + A * B) = det (fromBlocks 1 (-A) B 1)\n[PROOFSTEP]\nrw [det_fromBlocks_one\u2082\u2082, Matrix.neg_mul, sub_neg_eq_add]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m n \u03b1\nB : Matrix n m \u03b1\n\u22a2 det (fromBlocks 1 (-A) B 1) = det (1 + B * A)\n[PROOFSTEP]\nrw [det_fromBlocks_one\u2081\u2081, Matrix.mul_neg, sub_neg_eq_add]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m n \u03b1\nB : Matrix n m \u03b1\n\u22a2 det (A * B + 1) = det (B * A + 1)\n[PROOFSTEP]\nrw [add_comm, det_one_add_mul_comm, add_comm]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA : Matrix m n \u03b1\nB : Matrix n m \u03b1\n\u22a2 det (1 - A * B) = det (1 - B * A)\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 Matrix.neg_mul, det_one_add_mul_comm, Matrix.mul_neg, \u2190 sub_eq_add_neg]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2076 : Fintype l\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nu v : m \u2192 \u03b1\n\u22a2 det (1 + col u * row v) = 1 + v \u2b1d\u1d65 u\n[PROOFSTEP]\nrw [det_one_add_mul_comm, det_unique, Pi.add_apply, Pi.add_apply, Matrix.one_apply_eq, Matrix.row_mul_col_apply]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nx : m \u2192 \ud835\udd5c\ny : n \u2192 \ud835\udd5c\ninst\u271d : Invertible A\nhA : IsHermitian A\n\u22a2 vecMul (star (x \u2295\u1d65 y)) (fromBlocks A B B\u1d34 D) \u2b1d\u1d65 (x \u2295\u1d65 y) =\n    vecMul (star (x + mulVec (A\u207b\u00b9 * B) y)) A \u2b1d\u1d65 (x + mulVec (A\u207b\u00b9 * B) y) + vecMul (star y) (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 y\n[PROOFSTEP]\nsimp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul, dotProduct_mulVec, vecMul_sub,\n  Matrix.mul_assoc, vecMul_mulVec, hA.eq, conjTranspose_nonsing_inv, star_mulVec]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nx : m \u2192 \ud835\udd5c\ny : n \u2192 \ud835\udd5c\ninst\u271d : Invertible A\nhA : IsHermitian A\n\u22a2 vecMul (star x) A \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x + (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) D \u2b1d\u1d65 y) =\n    vecMul (star x) A \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x +\n        (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) (B\u1d34 * (A\u207b\u00b9 * B)) \u2b1d\u1d65 y) +\n      (vecMul (star y) D \u2b1d\u1d65 y - vecMul (star y) (B\u1d34 * (A\u207b\u00b9 * B)) \u2b1d\u1d65 y)\n[PROOFSTEP]\nabel\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nx : m \u2192 \ud835\udd5c\ny : n \u2192 \ud835\udd5c\ninst\u271d : Invertible A\nhA : IsHermitian A\n\u22a2 vecMul (star x) A \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x + (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) D \u2b1d\u1d65 y) =\n    vecMul (star x) A \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x +\n        (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) (B\u1d34 * (A\u207b\u00b9 * B)) \u2b1d\u1d65 y) +\n      (vecMul (star y) D \u2b1d\u1d65 y - vecMul (star y) (B\u1d34 * (A\u207b\u00b9 * B)) \u2b1d\u1d65 y)\n[PROOFSTEP]\nabel\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nx : m \u2192 \ud835\udd5c\ny : n \u2192 \ud835\udd5c\ninst\u271d : Invertible D\nhD : IsHermitian D\n\u22a2 vecMul (star (x \u2295\u1d65 y)) (fromBlocks A B B\u1d34 D) \u2b1d\u1d65 (x \u2295\u1d65 y) =\n    vecMul (star (mulVec (D\u207b\u00b9 * B\u1d34) x + y)) D \u2b1d\u1d65 (mulVec (D\u207b\u00b9 * B\u1d34) x + y) + vecMul (star x) (A - B * D\u207b\u00b9 * B\u1d34) \u2b1d\u1d65 x\n[PROOFSTEP]\nsimp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul, dotProduct_mulVec, vecMul_sub,\n  Matrix.mul_assoc, vecMul_mulVec, hD.eq, conjTranspose_nonsing_inv, star_mulVec]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nx : m \u2192 \ud835\udd5c\ny : n \u2192 \ud835\udd5c\ninst\u271d : Invertible D\nhD : IsHermitian D\n\u22a2 vecMul (star x) A \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x + (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) D \u2b1d\u1d65 y) =\n    vecMul (star x) (B * (D\u207b\u00b9 * B\u1d34)) \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x +\n        (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) D \u2b1d\u1d65 y) +\n      (vecMul (star x) A \u2b1d\u1d65 x - vecMul (star x) (B * (D\u207b\u00b9 * B\u1d34)) \u2b1d\u1d65 x)\n[PROOFSTEP]\nabel\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nx : m \u2192 \ud835\udd5c\ny : n \u2192 \ud835\udd5c\ninst\u271d : Invertible D\nhD : IsHermitian D\n\u22a2 vecMul (star x) A \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x + (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) D \u2b1d\u1d65 y) =\n    vecMul (star x) (B * (D\u207b\u00b9 * B\u1d34)) \u2b1d\u1d65 x + vecMul (star y) B\u1d34 \u2b1d\u1d65 x +\n        (vecMul (star x) B \u2b1d\u1d65 y + vecMul (star y) D \u2b1d\u1d65 y) +\n      (vecMul (star x) A \u2b1d\u1d65 x - vecMul (star x) (B * (D\u207b\u00b9 * B\u1d34)) \u2b1d\u1d65 x)\n[PROOFSTEP]\nabel\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\n\u22a2 IsHermitian (Matrix.fromBlocks A B B\u1d34 D) \u2194 IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nhave hBAB : (B\u1d34 * A\u207b\u00b9 * B).IsHermitian :=\n  by\n  apply isHermitian_conjTranspose_mul_mul\n  apply hA.inv\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\n\u22a2 IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\napply isHermitian_conjTranspose_mul_mul\n[GOAL]\ncase hA\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\n\u22a2 IsHermitian A\u207b\u00b9\n[PROOFSTEP]\napply hA.inv\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian (Matrix.fromBlocks A B B\u1d34 D) \u2194 IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nrw [isHermitian_fromBlocks_iff]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian A \u2227 B\u1d34 = B\u1d34 \u2227 B\u1d34\u1d34 = B \u2227 IsHermitian D \u2194 IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian A \u2227 B\u1d34 = B\u1d34 \u2227 B\u1d34\u1d34 = B \u2227 IsHermitian D \u2192 IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\nh : IsHermitian A \u2227 B\u1d34 = B\u1d34 \u2227 B\u1d34\u1d34 = B \u2227 IsHermitian D\n\u22a2 IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\napply IsHermitian.sub h.2.2.2 hBAB\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2192 IsHermitian A \u2227 B\u1d34 = B\u1d34 \u2227 B\u1d34\u1d34 = B \u2227 IsHermitian D\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian A \u2227 B\u1d34 = B\u1d34 \u2227 B\u1d34\u1d34 = B \u2227 IsHermitian D\n[PROOFSTEP]\nrefine' \u27e8hA, rfl, conjTranspose_conjTranspose B, _\u27e9\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian D\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel D]\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 IsHermitian (D - ?mpr + ?mpr)\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : IsHermitian A\nhBAB : IsHermitian (B\u1d34 * A\u207b\u00b9 * B)\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B)\n\u22a2 Matrix n n \ud835\udd5c\n[PROOFSTEP]\napply IsHermitian.add h hBAB\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : IsHermitian D\n\u22a2 IsHermitian (Matrix.fromBlocks A B B\u1d34 D) \u2194 IsHermitian (A - B * D\u207b\u00b9 * B\u1d34)\n[PROOFSTEP]\nrw [\u2190 isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply, fromBlocks_submatrix_sum_swap_sum_swap]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : IsHermitian D\n\u22a2 IsHermitian (Matrix.fromBlocks D B\u1d34 B A) \u2194 IsHermitian (A - B * D\u207b\u00b9 * B\u1d34)\n[PROOFSTEP]\nconvert IsHermitian.fromBlocks\u2081\u2081 _ _ hD\n[GOAL]\ncase h.e'_1.h.e'_4.h.e'_8\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : IsHermitian D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h.e'_4.h.e'_6.h.e'_5.h.e'_5\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2074 : CommRing \ud835\udd5c\ninst\u271d\u00b3 : PartialOrder \ud835\udd5c\ninst\u271d\u00b2 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : IsHermitian D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\n\u22a2 PosSemidef (fromBlocks A B B\u1d34 D) \u2194 PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nrw [PosSemidef, IsHermitian.fromBlocks\u2081\u2081 _ _ hA.1]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\n\u22a2 (IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x) \u2194\n    PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\n\u22a2 (IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x) \u2192\n    PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\n[PROOFSTEP]\nrefine' fun h => \u27e8h.1, fun x => _\u27e9\n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x\nx : n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star x \u2b1d\u1d65 mulVec (D - B\u1d34 * A\u207b\u00b9 * B) x\n[PROOFSTEP]\nhave := h.2 (-(A\u207b\u00b9 * B).mulVec x \u2295\u1d65 x)\n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 star (-mulVec (A\u207b\u00b9 * B) x \u2295\u1d65 x) \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) (-mulVec (A\u207b\u00b9 * B) x \u2295\u1d65 x)\n\u22a2 0 \u2264 star x \u2b1d\u1d65 mulVec (D - B\u1d34 * A\u207b\u00b9 * B) x\n[PROOFSTEP]\nrw [dotProduct_mulVec, schur_complement_eq\u2081\u2081 B D _ _ hA.1, neg_add_self, dotProduct_zero, zero_add] at this \n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 vecMul (star x) (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n\u22a2 0 \u2264 star x \u2b1d\u1d65 mulVec (D - B\u1d34 * A\u207b\u00b9 * B) x\n[PROOFSTEP]\nrw [dotProduct_mulVec]\n[GOAL]\ncase mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x\nx : n \u2192 \ud835\udd5c\nthis : 0 \u2264 vecMul (star x) (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n\u22a2 0 \u2264 vecMul (star x) (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x\n[PROOFSTEP]\nexact this\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\n\u22a2 PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B) \u2192\n    IsHermitian (D - B\u1d34 * A\u207b\u00b9 * B) \u2227 \u2200 (x : m \u2295 n \u2192 \ud835\udd5c), 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x\n[PROOFSTEP]\nrefine' fun h => \u27e8h.1, fun x => _\u27e9\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star x \u2b1d\u1d65 mulVec (fromBlocks A B B\u1d34 D) x\n[PROOFSTEP]\nrw [dotProduct_mulVec, \u2190 Sum.elim_comp_inl_inr x, schur_complement_eq\u2081\u2081 B D _ _ hA.1]\n[GOAL]\ncase mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264\n    vecMul (star (x \u2218 Sum.inl + mulVec (A\u207b\u00b9 * B) (x \u2218 Sum.inr))) A \u2b1d\u1d65 (x \u2218 Sum.inl + mulVec (A\u207b\u00b9 * B) (x \u2218 Sum.inr)) +\n      vecMul (star (x \u2218 Sum.inr)) (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x \u2218 Sum.inr\n[PROOFSTEP]\napply le_add_of_nonneg_of_le\n[GOAL]\ncase mpr.ha\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 vecMul (star (x \u2218 Sum.inl + mulVec (A\u207b\u00b9 * B) (x \u2218 Sum.inr))) A \u2b1d\u1d65 (x \u2218 Sum.inl + mulVec (A\u207b\u00b9 * B) (x \u2218 Sum.inr))\n[PROOFSTEP]\nrw [\u2190 dotProduct_mulVec]\n[GOAL]\ncase mpr.ha\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inl + mulVec (A\u207b\u00b9 * B) (x \u2218 Sum.inr)) \u2b1d\u1d65 mulVec A (x \u2218 Sum.inl + mulVec (A\u207b\u00b9 * B) (x \u2218 Sum.inr))\n[PROOFSTEP]\napply hA.posSemidef.2\n[GOAL]\ncase mpr.hbc\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 vecMul (star (x \u2218 Sum.inr)) (D - B\u1d34 * A\u207b\u00b9 * B) \u2b1d\u1d65 x \u2218 Sum.inr\n[PROOFSTEP]\nrw [\u2190 dotProduct_mulVec (star (x \u2218 Sum.inr))]\n[GOAL]\ncase mpr.hbc\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhA : PosDef A\ninst\u271d : Invertible A\nh : PosSemidef (D - B\u1d34 * A\u207b\u00b9 * B)\nx : m \u2295 n \u2192 \ud835\udd5c\n\u22a2 0 \u2264 star (x \u2218 Sum.inr) \u2b1d\u1d65 mulVec (D - B\u1d34 * A\u207b\u00b9 * B) (x \u2218 Sum.inr)\n[PROOFSTEP]\napply h.2\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 PosSemidef (fromBlocks A B B\u1d34 D) \u2194 PosSemidef (A - B * D\u207b\u00b9 * B\u1d34)\n[PROOFSTEP]\nrw [\u2190 posSemidef_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply, fromBlocks_submatrix_sum_swap_sum_swap]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 PosSemidef (fromBlocks D B\u1d34 B A) \u2194 PosSemidef (A - B * D\u207b\u00b9 * B\u1d34)\n[PROOFSTEP]\nconvert PosSemidef.fromBlocks\u2081\u2081 B\u1d34 A hD\n[GOAL]\ncase h.e'_1.h.e'_7.h.e'_8\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\nfirst\n| infer_instance\n| simp\n[GOAL]\ncase h.e'_1.h.e'_7.h.e'_8\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h.e'_1.h.e'_7.h.e'_8\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h.e'_7.h.e'_6.h.e'_5.h.e'_5\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\nfirst\n| infer_instance\n| simp\n[GOAL]\ncase h.e'_2.h.e'_7.h.e'_6.h.e'_5.h.e'_5\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase h.e'_2.h.e'_7.h.e'_6.h.e'_5.h.e'_5\nl : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\n\ud835\udd5c : Type u_5\ninst\u271d\u2076 : CommRing \ud835\udd5c\ninst\u271d\u2075 : PartialOrder \ud835\udd5c\ninst\u271d\u2074 : StarOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nA : Matrix m m \ud835\udd5c\nB : Matrix m n \ud835\udd5c\nD : Matrix n n \ud835\udd5c\nhD : PosDef D\ninst\u271d : Invertible D\n\u22a2 B = B\u1d34\u1d34\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.SchurComplement", "llama_tokens": 38573, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718435083355188, "lm_q2_score": 0.7217432122827967, "lm_q1q2_score": 0.5570728130857009}}
{"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\na\u271d n\u271d : \u2115\nx\u271d : 0 \u2264 a\u271d\n\u22a2 n\u271d \u2264 id a\u271d \u2194 \u2191n\u271d \u2264 a\u271d\n[PROOFSTEP]\nrw [Nat.cast_id]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\na\u271d n\u271d : \u2115\nx\u271d : 0 \u2264 a\u271d\n\u22a2 n\u271d \u2264 id a\u271d \u2194 n\u271d \u2264 a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn a : \u2115\n\u22a2 id n \u2264 a \u2194 n \u2264 \u2191a\n[PROOFSTEP]\nrw [Nat.cast_id]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nn a : \u2115\n\u22a2 id n \u2264 a \u2194 n \u2264 a\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 a < \u21911 \u2194 a < 1\n[PROOFSTEP]\nrw [Nat.cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nh : \u230aa\u230b\u208a < 1\n\u22a2 a < 1\n[PROOFSTEP]\nexact_mod_cast lt_of_floor_lt h\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a < \u2191\u230aa\u230b\u208a + 1\n[PROOFSTEP]\nsimpa using lt_succ_floor a\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d n a : \u2115\n\u22a2 a \u2264 \u230a\u2191n\u230b\u208a \u2194 a \u2264 n\n[PROOFSTEP]\nrw [le_floor_iff, Nat.cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d n a : \u2115\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nexact n.cast_nonneg\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230a0\u230b\u208a = 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, floor_coe]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230a1\u230b\u208a = 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, floor_coe]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : a \u2264 0\n\u22a2 a = 0 \u2192 \u230aa\u230b\u208a = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\nn : \u2115\nha : 0 \u2264 0\n\u22a2 \u230a0\u230b\u208a = 0\n[PROOFSTEP]\nexact floor_zero\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a \u2264 b\n\u22a2 \u230aa\u230b\u208a \u2264 \u230ab\u230b\u208a\n[PROOFSTEP]\nobtain ha | ha := le_total a 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a \u2264 b\nha : a \u2264 0\n\u22a2 \u230aa\u230b\u208a \u2264 \u230ab\u230b\u208a\n[PROOFSTEP]\nrw [floor_of_nonpos ha]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a \u2264 b\nha : a \u2264 0\n\u22a2 0 \u2264 \u230ab\u230b\u208a\n[PROOFSTEP]\nexact Nat.zero_le _\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a \u2264 b\nha : 0 \u2264 a\n\u22a2 \u230aa\u230b\u208a \u2264 \u230ab\u230b\u208a\n[PROOFSTEP]\nexact le_floor ((floor_le ha).trans h)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\n\u22a2 n \u2264 \u230aa\u230b\u208a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nobtain ha | ha := le_total a 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\nha : a \u2264 0\n\u22a2 n \u2264 \u230aa\u230b\u208a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nrw [floor_of_nonpos ha]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\nha : a \u2264 0\n\u22a2 n \u2264 0 \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nexact iff_of_false (Nat.pos_of_ne_zero hn).not_le (not_le_of_lt <| ha.trans_lt <| cast_pos.2 <| Nat.pos_of_ne_zero hn)\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\nha : 0 \u2264 a\n\u22a2 n \u2264 \u230aa\u230b\u208a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nexact le_floor_iff ha\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 1 \u2264 \u230ax\u230b\u208a \u2194 1 \u2264 x\n[PROOFSTEP]\nexact_mod_cast @le_floor_iff' \u03b1 _ _ x 1 one_ne_zero\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 0 < \u230aa\u230b\u208a \u2194 1 \u2264 a\n[PROOFSTEP]\nrw [Nat.lt_iff_add_one_le, zero_add, le_floor_iff' Nat.one_ne_zero, cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nh : 0 < \u230aa\u230b\u208a\nha : a \u2264 0\n\u22a2 0 < 0\n[PROOFSTEP]\nrwa [floor_of_nonpos ha] at h \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa\u230b\u208a = 0 \u2194 a < 1\n[PROOFSTEP]\nrw [\u2190 lt_one_iff, \u2190 @cast_one \u03b1]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa\u230b\u208a < 1 \u2194 a < \u21911\n[PROOFSTEP]\nexact floor_lt' Nat.one_ne_zero\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 \u230aa\u230b\u208a = n \u2194 \u2191n \u2264 a \u2227 a < \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 le_floor_iff ha, \u2190 Nat.cast_one, \u2190 Nat.cast_add, \u2190 floor_lt ha, Nat.lt_add_one_iff, le_antisymm_iff, and_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u230aa\u230b\u208a = n \u2194 \u2191n \u2264 a \u2227 a < \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 le_floor_iff' hn, \u2190 Nat.cast_one, \u2190 Nat.cast_add, \u2190 floor_lt' (Nat.add_one_ne_zero n), Nat.lt_add_one_iff,\n  le_antisymm_iff, and_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d n : \u2115\nx : \u03b1\nhx : x \u2208 Ico (\u2191n) (\u2191n + 1)\n\u22a2 \u2191\u230ax\u230b\u208a = \u2191n\n[PROOFSTEP]\nexact_mod_cast floor_eq_on_Ico n x hx\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 n + 1 \u2264 \u2308a\u2309\u208a \u2194 \u2191n < a\n[PROOFSTEP]\nrw [\u2190 Nat.lt_ceil, Nat.add_one_le_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 1 \u2264 \u2308a\u2309\u208a \u2194 0 < a\n[PROOFSTEP]\nrw [\u2190 zero_add 1, Nat.add_one_le_ceil_iff, Nat.cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 \u2308a\u2309\u208a \u2264 \u230aa\u230b\u208a + 1\n[PROOFSTEP]\nrw [ceil_le, Nat.cast_add, Nat.cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a \u2264 \u2191\u230aa\u230b\u208a + 1\n[PROOFSTEP]\nexact (lt_floor_add_one a).le\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b1\u271d\ninst\u271d\u00b2 : FloorSemiring \u03b1\u271d\na\u271d : \u03b1\u271d\nn : \u2115\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorSemiring \u03b1\nz : \u2124\na : \u2115\n\u22a2 \u2308\u2191z\u2309\u208a \u2264 a \u2194 Int.toNat z \u2264 a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b1\u271d\ninst\u271d\u00b2 : FloorSemiring \u03b1\u271d\na\u271d : \u03b1\u271d\nn : \u2115\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorSemiring \u03b1\nz : \u2124\na : \u2115\n\u22a2 \u2191z \u2264 \u2191a \u2194 z \u2264 \u2191a\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d n a : \u2115\n\u22a2 \u2308\u2191n\u2309\u208a \u2264 a \u2194 n \u2264 a\n[PROOFSTEP]\nrw [ceil_le, cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u23080\u2309\u208a = 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, ceil_natCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u23081\u2309\u208a = 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, ceil_natCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u2308a\u2309\u208a = 0 \u2194 a \u2264 0\n[PROOFSTEP]\nrw [\u2190 le_zero_iff, ceil_le, Nat.cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 0 < \u2308a\u2309\u208a \u2194 0 < a\n[PROOFSTEP]\nrw [lt_ceil, cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 \u230aa\u230b\u208a \u2264 \u2308a\u2309\u208a\n[PROOFSTEP]\nobtain ha | ha := le_total a 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : a \u2264 0\n\u22a2 \u230aa\u230b\u208a \u2264 \u2308a\u2309\u208a\n[PROOFSTEP]\nrw [floor_of_nonpos ha]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : a \u2264 0\n\u22a2 0 \u2264 \u2308a\u2309\u208a\n[PROOFSTEP]\nexact Nat.zero_le _\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u230aa\u230b\u208a \u2264 \u2308a\u2309\u208a\n[PROOFSTEP]\nexact cast_le.1 ((floor_le ha).trans <| le_ceil _)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a < b\nh' : 0 < b\n\u22a2 \u230aa\u230b\u208a < \u2308b\u2309\u208a\n[PROOFSTEP]\nrcases le_or_lt 0 a with (ha | ha)\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a < b\nh' : 0 < b\nha : 0 \u2264 a\n\u22a2 \u230aa\u230b\u208a < \u2308b\u2309\u208a\n[PROOFSTEP]\nrw [floor_lt ha]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a < b\nh' : 0 < b\nha : 0 \u2264 a\n\u22a2 a < \u2191\u2308b\u2309\u208a\n[PROOFSTEP]\nexact h.trans_le (le_ceil _)\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nh : a < b\nh' : 0 < b\nha : a < 0\n\u22a2 \u230aa\u230b\u208a < \u2308b\u2309\u208a\n[PROOFSTEP]\nrwa [floor_of_nonpos ha.le, lt_ceil, Nat.cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2308a\u2309\u208a = n \u2194 \u2191(n - 1) < a \u2227 a \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 ceil_le, \u2190 not_le, \u2190 ceil_le, not_le, tsub_lt_iff_right (Nat.add_one_le_iff.2 (pos_iff_ne_zero.2 hn)),\n  Nat.lt_add_one_iff, le_antisymm_iff, and_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nha : 0 \u2264 a\n\u22a2 Nat.cast \u207b\u00b9' Ioo a b = Ioo \u230aa\u230b\u208a \u2308b\u2309\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nha : 0 \u2264 a\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ioo a b \u2194 x\u271d \u2208 Ioo \u230aa\u230b\u208a \u2308b\u2309\u208a\n[PROOFSTEP]\nsimp [floor_lt, lt_ceil, ha]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\n\u22a2 Nat.cast \u207b\u00b9' Ico a b = Ico \u2308a\u2309\u208a \u2308b\u2309\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ico a b \u2194 x\u271d \u2208 Ico \u2308a\u2309\u208a \u2308b\u2309\u208a\n[PROOFSTEP]\nsimp [ceil_le, lt_ceil]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 Nat.cast \u207b\u00b9' Ioc a b = Ioc \u230aa\u230b\u208a \u230ab\u230b\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ioc a b \u2194 x\u271d \u2208 Ioc \u230aa\u230b\u208a \u230ab\u230b\u208a\n[PROOFSTEP]\nsimp [floor_lt, le_floor_iff, hb, ha]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nhb : 0 \u2264 b\n\u22a2 Nat.cast \u207b\u00b9' Icc a b = Icc \u2308a\u2309\u208a \u230ab\u230b\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nhb : 0 \u2264 b\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Icc a b \u2194 x\u271d \u2208 Icc \u2308a\u2309\u208a \u230ab\u230b\u208a\n[PROOFSTEP]\nsimp [ceil_le, hb, le_floor_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : 0 \u2264 a\n\u22a2 Nat.cast \u207b\u00b9' Ioi a = Ioi \u230aa\u230b\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : 0 \u2264 a\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ioi a \u2194 x\u271d \u2208 Ioi \u230aa\u230b\u208a\n[PROOFSTEP]\nsimp [floor_lt, ha]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 Nat.cast \u207b\u00b9' Ici a = Ici \u2308a\u2309\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ici a \u2194 x\u271d \u2208 Ici \u2308a\u2309\u208a\n[PROOFSTEP]\nsimp [ceil_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 Nat.cast \u207b\u00b9' Iio a = Iio \u2308a\u2309\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Iio a \u2194 x\u271d \u2208 Iio \u2308a\u2309\u208a\n[PROOFSTEP]\nsimp [lt_ceil]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : 0 \u2264 a\n\u22a2 Nat.cast \u207b\u00b9' Iic a = Iic \u230aa\u230b\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\nha : 0 \u2264 a\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Iic a \u2194 x\u271d \u2208 Iic \u230aa\u230b\u208a\n[PROOFSTEP]\nsimp [le_floor_iff, ha]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 b \u2264 \u230aa + \u2191n\u230b\u208a \u2194 b \u2264 \u230aa\u230b\u208a + n\n[PROOFSTEP]\nrw [le_floor_iff (add_nonneg ha n.cast_nonneg)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 \u2191b \u2264 a + \u2191n \u2194 b \u2264 \u230aa\u230b\u208a + n\n[PROOFSTEP]\nobtain hb | hb := le_total n b\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : n \u2264 b\n\u22a2 \u2191b \u2264 a + \u2191n \u2194 b \u2264 \u230aa\u230b\u208a + n\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := exists_add_of_le hb\n[GOAL]\ncase inl.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn d : \u2115\nhb : n \u2264 n + d\n\u22a2 \u2191(n + d) \u2264 a + \u2191n \u2194 n + d \u2264 \u230aa\u230b\u208a + n\n[PROOFSTEP]\nrw [Nat.cast_add, add_comm n, add_comm (n : \u03b1), add_le_add_iff_right, add_le_add_iff_right, le_floor_iff ha]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : b \u2264 n\n\u22a2 \u2191b \u2264 a + \u2191n \u2194 b \u2264 \u230aa\u230b\u208a + n\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := exists_add_of_le hb\n[GOAL]\ncase inr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nb d : \u2115\nhb : b \u2264 b + d\n\u22a2 \u2191b \u2264 a + \u2191(b + d) \u2194 b \u2264 \u230aa\u230b\u208a + (b + d)\n[PROOFSTEP]\nrw [Nat.cast_add, add_left_comm _ b, add_left_comm _ (b : \u03b1)]\n[GOAL]\ncase inr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nb d : \u2115\nhb : b \u2264 b + d\n\u22a2 \u2191b \u2264 \u2191b + (a + \u2191d) \u2194 b \u2264 b + (\u230aa\u230b\u208a + d)\n[PROOFSTEP]\nrefine' iff_of_true _ le_self_add\n[GOAL]\ncase inr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nb d : \u2115\nhb : b \u2264 b + d\n\u22a2 \u2191b \u2264 \u2191b + (a + \u2191d)\n[PROOFSTEP]\nexact le_add_of_nonneg_right <| ha.trans <| le_add_of_nonneg_right d.cast_nonneg\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 \u230aa + 1\u230b\u208a = \u230aa\u230b\u208a + 1\n[PROOFSTEP]\nrw [\u2190 cast_one, floor_add_nat ha 1]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa - \u2191n\u230b\u208a = \u230aa\u230b\u208a - n\n[PROOFSTEP]\nobtain ha | ha := le_total a 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\nha : a \u2264 0\n\u22a2 \u230aa - \u2191n\u230b\u208a = \u230aa\u230b\u208a - n\n[PROOFSTEP]\nrw [floor_of_nonpos ha, floor_of_nonpos (tsub_nonpos_of_le (ha.trans n.cast_nonneg)), zero_tsub]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 \u230aa - \u2191n\u230b\u208a = \u230aa\u230b\u208a - n\n[PROOFSTEP]\ncases' le_total a n with h h\n[GOAL]\ncase inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nh : a \u2264 \u2191n\n\u22a2 \u230aa - \u2191n\u230b\u208a = \u230aa\u230b\u208a - n\n[PROOFSTEP]\nrw [floor_of_nonpos (tsub_nonpos_of_le h), eq_comm, tsub_eq_zero_iff_le]\n[GOAL]\ncase inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nh : a \u2264 \u2191n\n\u22a2 \u230aa\u230b\u208a \u2264 n\n[PROOFSTEP]\nexact Nat.cast_le.1 ((Nat.floor_le ha).trans h)\n[GOAL]\ncase inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nh : \u2191n \u2264 a\n\u22a2 \u230aa - \u2191n\u230b\u208a = \u230aa\u230b\u208a - n\n[PROOFSTEP]\nrw [eq_tsub_iff_add_eq_of_le (le_floor h), \u2190 floor_add_nat _, tsub_add_cancel_of_le h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn\u271d : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nh : \u2191n \u2264 a\n\u22a2 0 \u2264 a - \u2191n\n[PROOFSTEP]\nexact le_tsub_of_add_le_left ((add_zero _).trans_le h)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ExistsAddOfLE \u03b1\na : \u03b1\n\u22a2 \u230aa - 1\u230b\u208a = \u230aa\u230b\u208a - 1\n[PROOFSTEP]\nexact_mod_cast floor_sub_nat a 1\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 \u2308a + \u2191n\u2309\u208a \u2264 b \u2194 \u2308a\u2309\u208a + n \u2264 b\n[PROOFSTEP]\nrw [\u2190 not_lt, \u2190 not_lt, not_iff_not, lt_ceil]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n\n[PROOFSTEP]\nobtain hb | hb := le_or_lt n b\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : n \u2264 b\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := exists_add_of_le hb\n[GOAL]\ncase inl.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn d : \u2115\nhb : n \u2264 n + d\n\u22a2 \u2191(n + d) < a + \u2191n \u2194 n + d < \u2308a\u2309\u208a + n\n[PROOFSTEP]\nrw [Nat.cast_add, add_comm n, add_comm (n : \u03b1), add_lt_add_iff_right, add_lt_add_iff_right, lt_ceil]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : b < n\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n\n[PROOFSTEP]\nexact iff_of_true (lt_add_of_nonneg_of_lt ha <| cast_lt.2 hb) (lt_add_left _ _ _ hb)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 \u2308a + 1\u2309\u208a = \u2308a\u2309\u208a + 1\n[PROOFSTEP]\nrw [cast_one.symm, ceil_add_nat ha 1]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\n\u22a2 \u2308a + b\u2309\u208a \u2264 \u2308a\u2309\u208a + \u2308b\u2309\u208a\n[PROOFSTEP]\nrw [ceil_le, Nat.cast_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\n\u22a2 a + b \u2264 \u2191\u2308a\u2309\u208a + \u2191\u2308b\u2309\u208a\n[PROOFSTEP]\nexact _root_.add_le_add (le_ceil _) (le_ceil _)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa / \u2191n\u230b\u208a = \u230aa\u230b\u208a / n\n[PROOFSTEP]\ncases' le_total a 0 with ha ha\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : a \u2264 0\n\u22a2 \u230aa / \u2191n\u230b\u208a = \u230aa\u230b\u208a / n\n[PROOFSTEP]\nrw [floor_of_nonpos, floor_of_nonpos ha]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : a \u2264 0\n\u22a2 0 = 0 / n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : a \u2264 0\n\u22a2 a / \u2191n \u2264 0\n[PROOFSTEP]\napply div_nonpos_of_nonpos_of_nonneg ha n.cast_nonneg\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 \u230aa / \u2191n\u230b\u208a = \u230aa\u230b\u208a / n\n[PROOFSTEP]\nobtain rfl | hn := n.eq_zero_or_pos\n[GOAL]\ncase inr.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u230aa / \u21910\u230b\u208a = \u230aa\u230b\u208a / 0\n[PROOFSTEP]\nrw [cast_zero, div_zero, Nat.div_zero, floor_zero]\n[GOAL]\ncase inr.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 \u230aa / \u2191n\u230b\u208a = \u230aa\u230b\u208a / n\n[PROOFSTEP]\nrefine' (floor_eq_iff _).2 _\n[GOAL]\ncase inr.inr.refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 0 \u2264 a / \u2191n\n[PROOFSTEP]\nexact div_nonneg ha n.cast_nonneg\n[GOAL]\ncase inr.inr.refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 \u2191(\u230aa\u230b\u208a / n) \u2264 a / \u2191n \u2227 a / \u2191n < \u2191(\u230aa\u230b\u208a / n) + 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.inr.refine'_2.left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 \u2191(\u230aa\u230b\u208a / n) \u2264 a / \u2191n\n[PROOFSTEP]\nexact cast_div_le.trans (div_le_div_of_le_of_nonneg (floor_le ha) n.cast_nonneg)\n[GOAL]\ncase inr.inr.refine'_2.right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 a / \u2191n < \u2191(\u230aa\u230b\u208a / n) + 1\n[PROOFSTEP]\nrw [div_lt_iff, add_mul, one_mul, \u2190 cast_mul, \u2190 cast_add, \u2190 floor_lt ha]\n[GOAL]\ncase inr.inr.refine'_2.right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 \u230aa\u230b\u208a < \u230aa\u230b\u208a / n * n + n\n[PROOFSTEP]\nexact lt_div_mul_add hn\n[GOAL]\ncase inr.inr.refine'_2.right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\nhn : n > 0\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nexact cast_pos.2 hn\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\nm n : \u2115\n\u22a2 \u230a\u2191m / \u2191n\u230b\u208a = m / n\n[PROOFSTEP]\nconvert floor_div_nat (m : \u03b1) n\n[GOAL]\ncase h.e'_3.h.e'_5\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemifield \u03b1\ninst\u271d : FloorSemiring \u03b1\nm n : \u2115\n\u22a2 m = \u230a\u2191m\u230b\u208a\n[PROOFSTEP]\nrw [m.floor_coe]\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\n\u22a2 Subsingleton (FloorSemiring \u03b1)\n[PROOFSTEP]\nrefine' \u27e8fun H\u2081 H\u2082 => _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\n\u22a2 H\u2081 = H\u2082\n[PROOFSTEP]\nhave : H\u2081.ceil = H\u2082.ceil := funext fun a => (H\u2081.gc_ceil.l_unique H\u2082.gc_ceil) fun n => rfl\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\n\u22a2 H\u2081 = H\u2082\n[PROOFSTEP]\nhave : H\u2081.floor = H\u2082.floor := by\n  ext a\n  cases' lt_or_le a 0 with h h\n  \u00b7 rw [H\u2081.floor_of_neg, H\u2082.floor_of_neg] <;> exact h\n  \u00b7 refine' eq_of_forall_le_iff fun n => _\n    rw [H\u2081.gc_floor, H\u2082.gc_floor] <;> exact h\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\n\u22a2 FloorSemiring.floor = FloorSemiring.floor\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\n\u22a2 FloorSemiring.floor a = FloorSemiring.floor a\n[PROOFSTEP]\ncases' lt_or_le a 0 with h h\n[GOAL]\ncase h.inl\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : a < 0\n\u22a2 FloorSemiring.floor a = FloorSemiring.floor a\n[PROOFSTEP]\nrw [H\u2081.floor_of_neg, H\u2082.floor_of_neg]\n[GOAL]\ncase h.inl\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : a < 0\n\u22a2 a < 0\n[PROOFSTEP]\nexact h\n[GOAL]\ncase h.inl\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : a < 0\n\u22a2 a < 0\n[PROOFSTEP]\nexact h\n[GOAL]\ncase h.inr\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : 0 \u2264 a\n\u22a2 FloorSemiring.floor a = FloorSemiring.floor a\n[PROOFSTEP]\nrefine' eq_of_forall_le_iff fun n => _\n[GOAL]\ncase h.inr\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : 0 \u2264 a\nn : \u2115\n\u22a2 n \u2264 FloorSemiring.floor a \u2194 n \u2264 FloorSemiring.floor a\n[PROOFSTEP]\nrw [H\u2081.gc_floor, H\u2082.gc_floor]\n[GOAL]\ncase h.inr\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a\n[PROOFSTEP]\nexact h\n[GOAL]\ncase h.inr\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis : FloorSemiring.ceil = FloorSemiring.ceil\na : \u03b1\nh : 0 \u2264 a\nn : \u2115\n\u22a2 0 \u2264 a\n[PROOFSTEP]\nexact h\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2081 H\u2082 : FloorSemiring \u03b1\nthis\u271d : FloorSemiring.ceil = FloorSemiring.ceil\nthis : FloorSemiring.floor = FloorSemiring.floor\n\u22a2 H\u2081 = H\u2082\n[PROOFSTEP]\ncases H\u2081\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nH\u2082 : FloorSemiring \u03b1\nfloor\u271d ceil\u271d : \u03b1 \u2192 \u2115\nfloor_of_neg\u271d : \u2200 {a : \u03b1}, a < 0 \u2192 floor\u271d a = 0\ngc_floor\u271d : \u2200 {a : \u03b1} {n : \u2115}, 0 \u2264 a \u2192 (n \u2264 floor\u271d a \u2194 \u2191n \u2264 a)\ngc_ceil\u271d : GaloisConnection ceil\u271d Nat.cast\nthis\u271d : FloorSemiring.ceil = FloorSemiring.ceil\nthis : FloorSemiring.floor = FloorSemiring.floor\n\u22a2 { floor := floor\u271d, ceil := ceil\u271d, floor_of_neg := floor_of_neg\u271d, gc_floor := gc_floor\u271d, gc_ceil := gc_ceil\u271d } = H\u2082\n[PROOFSTEP]\ncases H\u2082\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedSemiring \u03b1\nfloor\u271d\u00b9 ceil\u271d\u00b9 : \u03b1 \u2192 \u2115\nfloor_of_neg\u271d\u00b9 : \u2200 {a : \u03b1}, a < 0 \u2192 floor\u271d\u00b9 a = 0\ngc_floor\u271d\u00b9 : \u2200 {a : \u03b1} {n : \u2115}, 0 \u2264 a \u2192 (n \u2264 floor\u271d\u00b9 a \u2194 \u2191n \u2264 a)\ngc_ceil\u271d\u00b9 : GaloisConnection ceil\u271d\u00b9 Nat.cast\nfloor\u271d ceil\u271d : \u03b1 \u2192 \u2115\nfloor_of_neg\u271d : \u2200 {a : \u03b1}, a < 0 \u2192 floor\u271d a = 0\ngc_floor\u271d : \u2200 {a : \u03b1} {n : \u2115}, 0 \u2264 a \u2192 (n \u2264 floor\u271d a \u2194 \u2191n \u2264 a)\ngc_ceil\u271d : GaloisConnection ceil\u271d Nat.cast\nthis\u271d : FloorSemiring.ceil = FloorSemiring.ceil\nthis : FloorSemiring.floor = FloorSemiring.floor\n\u22a2 { floor := floor\u271d\u00b9, ceil := ceil\u271d\u00b9, floor_of_neg := floor_of_neg\u271d\u00b9, gc_floor := gc_floor\u271d\u00b9, gc_ceil := gc_ceil\u271d\u00b9 } =\n    { floor := floor\u271d, ceil := ceil\u271d, floor_of_neg := floor_of_neg\u271d, gc_floor := gc_floor\u271d, gc_ceil := gc_ceil\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\na b : \u2124\n\u22a2 \u2191a \u2264 b \u2194 a \u2264 id b\n[PROOFSTEP]\nrw [Int.cast_id]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\na b : \u2124\n\u22a2 a \u2264 b \u2194 a \u2264 id b\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\na b : \u2124\n\u22a2 id a \u2264 b \u2194 a \u2264 \u2191b\n[PROOFSTEP]\nrw [Int.cast_id]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\na b : \u2124\n\u22a2 id a \u2264 b \u2194 a \u2264 b\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type ?u.109134\ninst\u271d : LinearOrderedRing \u03b1\nfloor : \u03b1 \u2192 \u2124\ngc_coe_floor : GaloisConnection Int.cast floor\na : \u03b1\nz : \u2124\n\u22a2 (fun a => -floor (-a)) a \u2264 z \u2194 a \u2264 \u2191z\n[PROOFSTEP]\nrw [neg_le, \u2190 gc_coe_floor, Int.cast_neg, neg_le_neg_iff]\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type ?u.110495\ninst\u271d : LinearOrderedRing \u03b1\nceil : \u03b1 \u2192 \u2124\ngc_ceil_coe : GaloisConnection ceil Int.cast\na : \u2124\nz : \u03b1\n\u22a2 \u2191a \u2264 z \u2194 a \u2264 (fun a => -ceil (-a)) z\n[PROOFSTEP]\nrw [le_neg, gc_ceil_coe, Int.cast_neg, neg_le_neg_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nx : \u2124\n\u22a2 fract x = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 0 \u2264 \u230aa\u230b \u2194 0 \u2264 a\n[PROOFSTEP]\nrw [le_floor, Int.cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230aa\u230b \u2264 z - 1 \u2194 a < \u2191z\n[PROOFSTEP]\nrw [\u2190 floor_lt, le_sub_one_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230aa\u230b \u2264 -1 \u2194 a < 0\n[PROOFSTEP]\nrw [\u2190 zero_sub (1 : \u2124), floor_le_sub_one_iff, cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : a \u2264 0\n\u22a2 \u230aa\u230b \u2264 0\n[PROOFSTEP]\nrw [\u2190 @cast_le \u03b1, Int.cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : a \u2264 0\n\u22a2 \u2191\u230aa\u230b \u2264 0\n[PROOFSTEP]\nexact (floor_le a).trans ha\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 a < \u2191\u230aa\u230b + 1\n[PROOFSTEP]\nsimpa only [Int.succ, Int.cast_add, Int.cast_one] using lt_succ_floor a\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d : \u03b1\nz a : \u2124\n\u22a2 a \u2264 \u230a\u2191z\u230b \u2194 a \u2264 z\n[PROOFSTEP]\nrw [le_floor, Int.cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nn : \u2115\na : \u2124\n\u22a2 a \u2264 \u230a\u2191n\u230b \u2194 a \u2264 \u2191n\n[PROOFSTEP]\nrw [le_floor, \u2190 cast_ofNat, cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230a0\u230b = 0\n[PROOFSTEP]\nrw [\u2190 cast_zero, floor_intCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230a1\u230b = 1\n[PROOFSTEP]\nrw [\u2190 cast_one, floor_intCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 0 < \u230aa\u230b \u2194 1 \u2264 a\n[PROOFSTEP]\nrw [Int.lt_iff_add_one_le, zero_add, le_floor, cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d\u00b9 a\u271d : \u03b1\nz a : \u2124\n\u22a2 a \u2264 \u230aa\u271d + \u2191z\u230b \u2194 a \u2264 \u230aa\u271d\u230b + z\n[PROOFSTEP]\nrw [le_floor, \u2190 sub_le_iff_le_add, \u2190 sub_le_iff_le_add, le_floor, Int.cast_sub]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u230aa + 1\u230b = \u230aa\u230b + 1\n[PROOFSTEP]\nrw [\u2190 cast_one, floor_add_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u230aa\u230b + \u230ab\u230b \u2264 \u230aa + b\u230b\n[PROOFSTEP]\nrw [le_floor, Int.cast_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa\u230b + \u2191\u230ab\u230b \u2264 a + b\n[PROOFSTEP]\nexact add_le_add (floor_le _) (floor_le _)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u230aa + b\u230b - 1 \u2264 \u230aa\u230b + \u230ab\u230b\n[PROOFSTEP]\nrw [\u2190 sub_le_iff_le_add, le_floor, Int.cast_sub, sub_le_comm, Int.cast_sub, Int.cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa + b\u230b - 1 - a \u2264 \u2191\u230ab\u230b\n[PROOFSTEP]\nrefine' le_trans _ (sub_one_lt_floor _).le\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa + b\u230b - 1 - a \u2264 b - 1\n[PROOFSTEP]\nrw [sub_le_iff_le_add', \u2190 add_sub_assoc, sub_le_sub_iff_right]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa + b\u230b \u2264 a + b\n[PROOFSTEP]\nexact floor_le _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d : \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230a\u2191z + a\u230b = z + \u230aa\u230b\n[PROOFSTEP]\nsimpa only [add_comm] using floor_add_int a z\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u230aa + \u2191n\u230b = \u230aa\u230b + \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, floor_add_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 \u230a\u2191n + a\u230b = \u2191n + \u230aa\u230b\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, floor_int_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nz : \u2124\n\u22a2 \u230aa - \u2191z\u230b = \u230aa + \u2191(-z)\u230b\n[PROOFSTEP]\nrw [Int.cast_neg, sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u230aa - \u2191n\u230b = \u230aa\u230b - \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, floor_sub_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u230aa - 1\u230b = \u230aa\u230b - 1\n[PROOFSTEP]\nexact_mod_cast floor_sub_nat a 1\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\n\u22a2 |a - b| < 1\n[PROOFSTEP]\nhave : a < \u230aa\u230b + 1 := lt_floor_add_one a\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis : a < \u2191\u230aa\u230b + 1\n\u22a2 |a - b| < 1\n[PROOFSTEP]\nhave : b < \u230ab\u230b + 1 := lt_floor_add_one b\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d : a < \u2191\u230aa\u230b + 1\nthis : b < \u2191\u230ab\u230b + 1\n\u22a2 |a - b| < 1\n[PROOFSTEP]\nhave : (\u230aa\u230b : \u03b1) = \u230ab\u230b := Int.cast_inj.2 h\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b9 : a < \u2191\u230aa\u230b + 1\nthis\u271d : b < \u2191\u230ab\u230b + 1\nthis : \u2191\u230aa\u230b = \u2191\u230ab\u230b\n\u22a2 |a - b| < 1\n[PROOFSTEP]\nhave : (\u230aa\u230b : \u03b1) \u2264 a := floor_le a\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b2 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b9 : b < \u2191\u230ab\u230b + 1\nthis\u271d : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis : \u2191\u230aa\u230b \u2264 a\n\u22a2 |a - b| < 1\n[PROOFSTEP]\nhave : (\u230ab\u230b : \u03b1) \u2264 b := floor_le b\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 |a - b| < 1\n[PROOFSTEP]\nexact abs_sub_lt_iff.2 \u27e8by linarith, by linarith\u27e9\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 a - b < 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_4\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 b - a < 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230aa\u230b = z \u2194 \u2191z \u2264 a \u2227 a < \u2191z + 1\n[PROOFSTEP]\nrw [le_antisymm_iff, le_floor, \u2190 Int.lt_add_one_iff, floor_lt, Int.cast_add, Int.cast_one, and_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u230aa\u230b = 0 \u2194 a \u2208 Ico 0 1\n[PROOFSTEP]\nsimp [floor_eq_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 fract (a + \u2191m) = fract a\n[PROOFSTEP]\nrw [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 a + \u2191m - \u2191\u230aa + \u2191m\u230b = fract a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2115\n\u22a2 fract (a + \u2191m) = fract a\n[PROOFSTEP]\nrw [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2115\n\u22a2 a + \u2191m - \u2191\u230aa + \u2191m\u230b = fract a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 fract (a + 1) = fract a\n[PROOFSTEP]\nexact_mod_cast fract_add_nat a 1\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nm : \u2124\na : \u03b1\n\u22a2 fract (\u2191m + a) = fract a\n[PROOFSTEP]\nrw [add_comm, fract_add_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 fract (\u2191n + a) = fract a\n[PROOFSTEP]\nrw [add_comm, fract_add_nat]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 fract (1 + a) = fract a\n[PROOFSTEP]\nexact_mod_cast fract_nat_add 1 a\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 fract (a - \u2191m) = fract a\n[PROOFSTEP]\nrw [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nm : \u2124\n\u22a2 a - \u2191m - \u2191\u230aa - \u2191m\u230b = fract a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 fract (a - \u2191n) = fract a\n[PROOFSTEP]\nrw [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 a - \u2191n - \u2191\u230aa - \u2191n\u230b = fract a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 fract (a - 1) = fract a\n[PROOFSTEP]\nexact_mod_cast fract_sub_nat a 1\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 fract (a + b) \u2264 fract a + fract b\n[PROOFSTEP]\nrw [fract, fract, fract, sub_add_sub_comm, sub_le_sub_iff_left, \u2190 Int.cast_add, Int.cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u230aa\u230b + \u230ab\u230b \u2264 \u230aa + b\u230b\n[PROOFSTEP]\nexact le_floor_add _ _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 fract a + fract b \u2264 fract (a + b) + 1\n[PROOFSTEP]\nrw [fract, fract, fract, sub_add_sub_comm, sub_add, sub_le_sub_iff_left]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa + b\u230b - 1 \u2264 \u2191\u230aa\u230b + \u2191\u230ab\u230b\n[PROOFSTEP]\nexact_mod_cast le_floor_add_floor a b\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 fract 0 = 0\n[PROOFSTEP]\nrw [fract, floor_zero, cast_zero, sub_self]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 fract 1 = 0\n[PROOFSTEP]\nsimp [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na : \u03b1\nz : \u2124\n\u22a2 fract \u2191z = 0\n[PROOFSTEP]\nunfold fract\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na : \u03b1\nz : \u2124\n\u22a2 \u2191z - \u2191\u230a\u2191z\u230b = 0\n[PROOFSTEP]\nrw [floor_intCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na : \u03b1\nz : \u2124\n\u22a2 \u2191z - \u2191z = 0\n[PROOFSTEP]\nexact sub_self _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nn : \u2115\n\u22a2 fract \u2191n = 0\n[PROOFSTEP]\nsimp [fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u230afract a\u230b = 0\n[PROOFSTEP]\nrw [floor_eq_iff, Int.cast_zero, zero_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 0 \u2264 fract a \u2227 fract a < 1\n[PROOFSTEP]\nexact \u27e8fract_nonneg _, fract_lt_one _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nh : fract a = b\n\u22a2 0 \u2264 b \u2227 b < 1 \u2227 \u2203 z, a - b = \u2191z\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nh : fract a = b\n\u22a2 0 \u2264 fract a \u2227 fract a < 1 \u2227 \u2203 z, a - fract a = \u2191z\n[PROOFSTEP]\nexact \u27e8fract_nonneg _, fract_lt_one _, \u27e8\u230aa\u230b, sub_sub_cancel _ _\u27e9\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 (0 \u2264 b \u2227 b < 1 \u2227 \u2203 z, a - b = \u2191z) \u2192 fract a = b\n[PROOFSTEP]\nrintro \u27e8h\u2080, h\u2081, z, hz\u27e9\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nh\u2080 : 0 \u2264 b\nh\u2081 : b < 1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 fract a = b\n[PROOFSTEP]\nrw [\u2190 self_sub_floor, eq_comm, eq_sub_iff_add_eq, add_comm, \u2190 eq_sub_iff_add_eq, hz, Int.cast_inj, floor_eq_iff, \u2190 hz]\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nh\u2080 : 0 \u2264 b\nh\u2081 : b < 1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 a - b \u2264 a \u2227 a < a - b + 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nh\u2080 : 0 \u2264 b\nh\u2081 : b < 1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 a - b \u2264 a\n[PROOFSTEP]\nsimpa [sub_eq_add_neg, add_assoc]\n[GOAL]\ncase intro.intro.intro.right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nh\u2080 : 0 \u2264 b\nh\u2081 : b < 1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 a < a - b + 1\n[PROOFSTEP]\nsimpa [sub_eq_add_neg, add_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nh : fract a = fract b\n\u22a2 a - b = \u2191(\u230aa\u230b - \u230ab\u230b)\n[PROOFSTEP]\nunfold fract at h \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nh : a - \u2191\u230aa\u230b = b - \u2191\u230ab\u230b\n\u22a2 a - b = \u2191(\u230aa\u230b - \u230ab\u230b)\n[PROOFSTEP]\nrw [Int.cast_sub, sub_eq_sub_iff_sub_eq_sub.1 h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 (\u2203 z, a - b = \u2191z) \u2192 fract a = fract b\n[PROOFSTEP]\nrintro \u27e8z, hz\u27e9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 fract a = fract b\n[PROOFSTEP]\nrefine' fract_eq_iff.2 \u27e8fract_nonneg _, fract_lt_one _, z + \u230ab\u230b, _\u27e9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 a - fract b = \u2191(z + \u230ab\u230b)\n[PROOFSTEP]\nrw [eq_add_of_sub_eq hz, add_comm, Int.cast_add]\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a b : \u03b1\nz : \u2124\nhz : a - b = \u2191z\n\u22a2 b + \u2191z - fract b = \u2191z + \u2191\u230ab\u230b\n[PROOFSTEP]\nexact add_sub_sub_cancel _ _ _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 a - a = \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 fract (a + b) - fract a - fract b = \u2191(\u230aa\u230b + \u230ab\u230b - \u230aa + b\u230b)\n[PROOFSTEP]\nunfold fract\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 a + b - \u2191\u230aa + b\u230b - (a - \u2191\u230aa\u230b) - (b - \u2191\u230ab\u230b) = \u2191(\u230aa\u230b + \u230ab\u230b - \u230aa + b\u230b)\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 a + b + -\u2191\u230aa + b\u230b + (\u2191\u230aa\u230b + -a) + (\u2191\u230ab\u230b + -b) = \u2191\u230aa\u230b + \u2191\u230ab\u230b + -\u2191\u230aa + b\u230b\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 a + b + -\u2191\u230aa + b\u230b + (\u2191\u230aa\u230b + -a) + (\u2191\u230ab\u230b + -b) = \u2191\u230aa\u230b + \u2191\u230ab\u230b + -\u2191\u230aa + b\u230b\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 fract (-x) = 1 - fract x\n[PROOFSTEP]\nrw [fract_eq_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 0 \u2264 1 - fract x \u2227 1 - fract x < 1 \u2227 \u2203 z, -x - (1 - fract x) = \u2191z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 0 \u2264 1 - fract x\n[PROOFSTEP]\nrw [le_sub_iff_add_le, zero_add]\n[GOAL]\ncase left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 fract x \u2264 1\n[PROOFSTEP]\nexact (fract_lt_one x).le\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 1 - fract x < 1 \u2227 \u2203 z, -x - (1 - fract x) = \u2191z\n[PROOFSTEP]\nrefine' \u27e8sub_lt_self _ (lt_of_le_of_ne' (fract_nonneg x) hx), -\u230ax\u230b - 1, _\u27e9\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 -x - (1 - fract x) = \u2191(-\u230ax\u230b - 1)\n[PROOFSTEP]\nsimp only [sub_sub_eq_add_sub, cast_sub, cast_neg, cast_one, sub_left_inj]\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 -x + fract x = -\u2191\u230ax\u230b\n[PROOFSTEP]\nconv in -x => rw [\u2190 floor_add_fract x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n| -x\n[PROOFSTEP]\nrw [\u2190 floor_add_fract x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n| -x\n[PROOFSTEP]\nrw [\u2190 floor_add_fract x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n| -x\n[PROOFSTEP]\nrw [\u2190 floor_add_fract x]\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\nhx : fract x \u2260 0\n\u22a2 -(\u2191\u230ax\u230b + fract x) + fract x = -\u2191\u230ax\u230b\n[PROOFSTEP]\nsimp [-floor_add_fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\n\u22a2 fract (-x) = 0 \u2194 fract x = 0\n[PROOFSTEP]\nsimp only [fract_eq_iff, le_refl, zero_lt_one, tsub_zero, true_and_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\n\u22a2 (\u2203 z, -x = \u2191z) \u2194 \u2203 z, x = \u2191z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\n\u22a2 (\u2203 z, -x = \u2191z) \u2192 \u2203 z, x = \u2191z\n[PROOFSTEP]\nrintro \u27e8z, hz\u27e9\n[GOAL]\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na x : \u03b1\n\u22a2 (\u2203 z, x = \u2191z) \u2192 \u2203 z, -x = \u2191z\n[PROOFSTEP]\nrintro \u27e8z, hz\u27e9\n[GOAL]\ncase mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na x : \u03b1\nz : \u2124\nhz : -x = \u2191z\n\u22a2 \u2203 z, x = \u2191z\n[PROOFSTEP]\nuse-z\n[GOAL]\ncase mpr.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na x : \u03b1\nz : \u2124\nhz : x = \u2191z\n\u22a2 \u2203 z, -x = \u2191z\n[PROOFSTEP]\nuse-z\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na x : \u03b1\nz : \u2124\nhz : -x = \u2191z\n\u22a2 x = \u2191(-z)\n[PROOFSTEP]\nsimp [\u2190 hz]\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na x : \u03b1\nz : \u2124\nhz : x = \u2191z\n\u22a2 -x = \u2191(-z)\n[PROOFSTEP]\nsimp [\u2190 hz]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nb : \u2115\n\u22a2 \u2203 z, fract a * \u2191b - fract (a * \u2191b) = \u2191z\n[PROOFSTEP]\ninduction' b with c hc\n[GOAL]\ncase zero\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u2203 z, fract a * \u2191Nat.zero - fract (a * \u2191Nat.zero) = \u2191z\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 fract a * \u2191Nat.zero - fract (a * \u2191Nat.zero) = \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nc : \u2115\nhc : \u2203 z, fract a * \u2191c - fract (a * \u2191c) = \u2191z\n\u22a2 \u2203 z, fract a * \u2191(Nat.succ c) - fract (a * \u2191(Nat.succ c)) = \u2191z\n[PROOFSTEP]\nrcases hc with \u27e8z, hz\u27e9\n[GOAL]\ncase succ.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nc : \u2115\nz : \u2124\nhz : fract a * \u2191c - fract (a * \u2191c) = \u2191z\n\u22a2 \u2203 z, fract a * \u2191(Nat.succ c) - fract (a * \u2191(Nat.succ c)) = \u2191z\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, Nat.cast_add, mul_add, mul_add, Nat.cast_one, mul_one, mul_one]\n[GOAL]\ncase succ.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nc : \u2115\nz : \u2124\nhz : fract a * \u2191c - fract (a * \u2191c) = \u2191z\n\u22a2 \u2203 z, fract a * \u2191c + fract a - fract (a * \u2191c + a) = \u2191z\n[PROOFSTEP]\nrcases fract_add (a * c) a with \u27e8y, hy\u27e9\n[GOAL]\ncase succ.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nc : \u2115\nz : \u2124\nhz : fract a * \u2191c - fract (a * \u2191c) = \u2191z\ny : \u2124\nhy : fract (a * \u2191c + a) - fract (a * \u2191c) - fract a = \u2191y\n\u22a2 \u2203 z, fract a * \u2191c + fract a - fract (a * \u2191c + a) = \u2191z\n[PROOFSTEP]\nuse z - y\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nc : \u2115\nz : \u2124\nhz : fract a * \u2191c - fract (a * \u2191c) = \u2191z\ny : \u2124\nhy : fract (a * \u2191c + a) - fract (a * \u2191c) - fract a = \u2191y\n\u22a2 fract a * \u2191c + fract a - fract (a * \u2191c + a) = \u2191(z - y)\n[PROOFSTEP]\nrw [Int.cast_sub, \u2190 hz, \u2190 hy]\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nc : \u2115\nz : \u2124\nhz : fract a * \u2191c - fract (a * \u2191c) = \u2191z\ny : \u2124\nhy : fract (a * \u2191c + a) - fract (a * \u2191c) - fract a = \u2191y\n\u22a2 fract a * \u2191c + fract a - fract (a * \u2191c + a) =\n    fract a * \u2191c - fract (a * \u2191c) - (fract (a * \u2191c + a) - fract (a * \u2191c) - fract a)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nc : \u2115\nz : \u2124\nhz : fract a * \u2191c - fract (a * \u2191c) = \u2191z\ny : \u2124\nhy : fract (a * \u2191c + a) - fract (a * \u2191c) - fract a = \u2191y\n\u22a2 fract a * \u2191c + fract a - fract (a * \u2191c + a) =\n    fract a * \u2191c - fract (a * \u2191c) - (fract (a * \u2191c + a) - fract (a * \u2191c) - fract a)\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\n\u22a2 fract \u207b\u00b9' s = \u22c3 (m : \u2124), (fun x => x - \u2191m) \u207b\u00b9' (s \u2229 Ico 0 1)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 fract \u207b\u00b9' s \u2194 x \u2208 \u22c3 (m : \u2124), (fun x => x - \u2191m) \u207b\u00b9' (s \u2229 Ico 0 1)\n[PROOFSTEP]\nsimp only [mem_preimage, mem_iUnion, mem_inter_iff]\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 fract x \u2208 s \u2194 \u2203 i, x - \u2191i \u2208 s \u2227 x - \u2191i \u2208 Ico 0 1\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8\u230ax\u230b, h, fract_nonneg x, fract_lt_one x\u27e9, _\u27e9\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 (\u2203 i, x - \u2191i \u2208 s \u2227 x - \u2191i \u2208 Ico 0 1) \u2192 fract x \u2208 s\n[PROOFSTEP]\nrintro \u27e8m, hms, hm0, hm1\u27e9\n[GOAL]\ncase h.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\nm : \u2124\nhms : x - \u2191m \u2208 s\nhm0 : 0 \u2264 x - \u2191m\nhm1 : x - \u2191m < 1\n\u22a2 fract x \u2208 s\n[PROOFSTEP]\nobtain rfl : \u230ax\u230b = m := floor_eq_iff.2 \u27e8sub_nonneg.1 hm0, sub_lt_iff_lt_add'.1 hm1\u27e9\n[GOAL]\ncase h.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\nhms : x - \u2191\u230ax\u230b \u2208 s\nhm0 : 0 \u2264 x - \u2191\u230ax\u230b\nhm1 : x - \u2191\u230ax\u230b < 1\n\u22a2 fract x \u2208 s\n[PROOFSTEP]\nexact hms\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\n\u22a2 fract '' s = \u22c3 (m : \u2124), (fun x => x - \u2191m) '' s \u2229 Ico 0 1\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 fract '' s \u2194 x \u2208 \u22c3 (m : \u2124), (fun x => x - \u2191m) '' s \u2229 Ico 0 1\n[PROOFSTEP]\nsimp only [mem_image, mem_inter_iff, mem_iUnion]\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 (\u2203 x_1, x_1 \u2208 s \u2227 fract x_1 = x) \u2194 \u2203 i, (\u2203 x_1, x_1 \u2208 s \u2227 x_1 - \u2191i = x) \u2227 x \u2208 Ico 0 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 (\u2203 x_1, x_1 \u2208 s \u2227 fract x_1 = x) \u2192 \u2203 i, (\u2203 x_1, x_1 \u2208 s \u2227 x_1 - \u2191i = x) \u2227 x \u2208 Ico 0 1\n[PROOFSTEP]\nrintro \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\ny : \u03b1\nhy : y \u2208 s\n\u22a2 \u2203 i, (\u2203 x, x \u2208 s \u2227 x - \u2191i = fract y) \u2227 fract y \u2208 Ico 0 1\n[PROOFSTEP]\nexact \u27e8\u230ay\u230b, \u27e8y, hy, rfl\u27e9, fract_nonneg y, fract_lt_one y\u27e9\n[GOAL]\ncase h.mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 (\u2203 i, (\u2203 x_1, x_1 \u2208 s \u2227 x_1 - \u2191i = x) \u2227 x \u2208 Ico 0 1) \u2192 \u2203 x_1, x_1 \u2208 s \u2227 fract x_1 = x\n[PROOFSTEP]\nrintro \u27e8m, \u27e8y, hys, rfl\u27e9, h0, h1\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\nm : \u2124\ny : \u03b1\nhys : y \u2208 s\nh0 : 0 \u2264 y - \u2191m\nh1 : y - \u2191m < 1\n\u22a2 \u2203 x, x \u2208 s \u2227 fract x = y - \u2191m\n[PROOFSTEP]\nobtain rfl : \u230ay\u230b = m := floor_eq_iff.2 \u27e8sub_nonneg.1 h0, sub_lt_iff_lt_add'.1 h1\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\ns : Set \u03b1\ny : \u03b1\nhys : y \u2208 s\nh0 : 0 \u2264 y - \u2191\u230ay\u230b\nh1 : y - \u2191\u230ay\u230b < 1\n\u22a2 \u2203 x, x \u2208 s \u2227 fract x = y - \u2191\u230ay\u230b\n[PROOFSTEP]\nexact \u27e8y, hys, rfl\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb\u271d a b : k\nha : a \u2260 0\n\u22a2 fract (b / a) * a + \u230ab / a\u230b \u2022 a = b\n[PROOFSTEP]\nrw [zsmul_eq_mul, \u2190 add_mul, fract_add_floor, div_mul_cancel b ha]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb a : k\nhb : 0 < b\n\u22a2 a < b + \u2191\u230aa / b\u230b * b\n[PROOFSTEP]\nrw [\u2190 one_add_mul _ b, \u2190 div_lt_iff hb, add_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb a : k\nhb : 0 < b\n\u22a2 a / b < \u2191\u230aa / b\u230b + 1\n[PROOFSTEP]\nexact lt_floor_add_one _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % n) / \u2191n\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2115\n\u22a2 fract (\u2191m / \u21910) = \u2191(m % 0) / \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % n) / \u2191n\n[PROOFSTEP]\nhave hn' : 0 < (n : k) := by norm_cast\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % n) / \u2191n\n[PROOFSTEP]\nrefine fract_eq_iff.mpr \u27e8?_, ?_, m / n, ?_\u27e9\n[GOAL]\ncase inr.refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 0 \u2264 \u2191(m % n) / \u2191n\n[PROOFSTEP]\npositivity\n[GOAL]\ncase inr.refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 \u2191(m % n) / \u2191n < 1\n[PROOFSTEP]\nsimpa only [div_lt_one hn', Nat.cast_lt] using m.mod_lt hn\n[GOAL]\ncase inr.refine_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 \u2191m / \u2191n - \u2191(m % n) / \u2191n = \u2191(\u2191m / \u2191n)\n[PROOFSTEP]\nrw [sub_eq_iff_eq_add', \u2190 mul_right_inj' hn'.ne', mul_div_cancel' _ hn'.ne', mul_add, mul_div_cancel' _ hn'.ne']\n[GOAL]\ncase inr.refine_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 \u2191m = \u2191(m % n) + \u2191n * \u2191(\u2191m / \u2191n)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr.refine_3\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 \u2191m = \u2191(m % n) + \u2191(n * (m / n))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_add, Nat.mod_add_div m n]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % \u2191n) / \u2191n\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\n\u22a2 fract (\u2191m / \u21910) = \u2191(m % \u21910) / \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : n > 0\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % \u2191n) / \u2191n\n[PROOFSTEP]\nreplace hn : 0 < (n : k)\n[GOAL]\ncase hn\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : n > 0\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % \u2191n) / \u2191n\n[PROOFSTEP]\nhave : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract ((l : k) / n) = \u2191(l % n) / n :=\n  by\n  intros l hl\n  obtain \u27e8l\u2080, rfl | rfl\u27e9 := l.eq_nat_or_neg\n  \u00b7 rw [cast_ofNat, \u2190 coe_nat_mod, cast_ofNat, fract_div_natCast_eq_div_natCast_mod]\n  \u00b7 rw [Right.nonneg_neg_iff, coe_nat_nonpos_iff] at hl \n    simp [hl, zero_mod]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\n\u22a2 \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\n[PROOFSTEP]\nintros l hl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\nl : \u2124\nhl : 0 \u2264 l\n\u22a2 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\n[PROOFSTEP]\nobtain \u27e8l\u2080, rfl | rfl\u27e9 := l.eq_nat_or_neg\n[GOAL]\ncase intro.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\nl\u2080 : \u2115\nhl : 0 \u2264 \u2191l\u2080\n\u22a2 fract (\u2191\u2191l\u2080 / \u2191n) = \u2191(\u2191l\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nrw [cast_ofNat, \u2190 coe_nat_mod, cast_ofNat, fract_div_natCast_eq_div_natCast_mod]\n[GOAL]\ncase intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\nl\u2080 : \u2115\nhl : 0 \u2264 -\u2191l\u2080\n\u22a2 fract (\u2191(-\u2191l\u2080) / \u2191n) = \u2191(-\u2191l\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nrw [Right.nonneg_neg_iff, coe_nat_nonpos_iff] at hl \n[GOAL]\ncase intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\nl\u2080 : \u2115\nhl : l\u2080 = 0\n\u22a2 fract (\u2191(-\u2191l\u2080) / \u2191n) = \u2191(-\u2191l\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nsimp [hl, zero_mod]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nm : \u2124\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\n\u22a2 fract (\u2191m / \u2191n) = \u2191(m % \u2191n) / \u2191n\n[PROOFSTEP]\nobtain \u27e8m\u2080, rfl | rfl\u27e9 := m.eq_nat_or_neg\n[GOAL]\ncase inr.intro.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\n\u22a2 fract (\u2191\u2191m\u2080 / \u2191n) = \u2191(\u2191m\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nexact this (ofNat_nonneg m\u2080)\n[GOAL]\ncase inr.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\n\u22a2 fract (\u2191(-\u2191m\u2080) / \u2191n) = \u2191(-\u2191m\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nlet q := \u2308\u2191m\u2080 / (n : k)\u2309\n[GOAL]\ncase inr.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\n\u22a2 fract (\u2191(-\u2191m\u2080) / \u2191n) = \u2191(-\u2191m\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nlet m\u2081 := q * \u2191n - (\u2191m\u2080 : \u2124)\n[GOAL]\ncase inr.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\n\u22a2 fract (\u2191(-\u2191m\u2080) / \u2191n) = \u2191(-\u2191m\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\nhave hm\u2081 : 0 \u2264 m\u2081 := by simpa [\u2190 @cast_le k, \u2190 div_le_iff hn] using FloorRing.gc_ceil_coe.le_u_l _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\n\u22a2 0 \u2264 m\u2081\n[PROOFSTEP]\nsimpa [\u2190 @cast_le k, \u2190 div_le_iff hn] using FloorRing.gc_ceil_coe.le_u_l _\n[GOAL]\ncase inr.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 fract (\u2191(-\u2191m\u2080) / \u2191n) = \u2191(-\u2191m\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\ncalc\n  fract\n        ((Int.cast (-(m\u2080 : \u2124)) : k) / (n : k))\n          -- Porting note: the `rw [cast_neg, cast_ofNat]` was `push_cast` =\n      fract (-(m\u2080 : k) / n) :=\n    by rw [cast_neg, cast_ofNat]\n  _ = fract ((m\u2081 : k) / n) := ?_\n  _ = Int.cast (m\u2081 % (n : \u2124)) / Nat.cast n := (this hm\u2081)\n  _ = Int.cast (-(\u2191m\u2080 : \u2124) % \u2191n) / Nat.cast n := ?_\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 fract (\u2191(-\u2191m\u2080) / \u2191n) = fract (-\u2191m\u2080 / \u2191n)\n[PROOFSTEP]\nrw [cast_neg, cast_ofNat]\n[GOAL]\ncase inr.intro.inr.calc_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 fract (-\u2191m\u2080 / \u2191n) = fract (\u2191m\u2081 / \u2191n)\n[PROOFSTEP]\nrw [\u2190 fract_int_add q, \u2190 mul_div_cancel (q : k) (ne_of_gt hn), \u2190 add_div, \u2190 sub_eq_add_neg]\n  -- Porting note: the `simp` was `push_cast`\n[GOAL]\ncase inr.intro.inr.calc_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 fract ((\u2191q * \u2191n - \u2191m\u2080) / \u2191n) = fract (\u2191m\u2081 / \u2191n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.inr.calc_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 \u2191(m\u2081 % \u2191n) / \u2191n = \u2191(-\u2191m\u2080 % \u2191n) / \u2191n\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase inr.intro.inr.calc_2.e_a.e_a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 m\u2081 % \u2191n = -\u2191m\u2080 % \u2191n\n[PROOFSTEP]\nchange (q * \u2191n - (\u2191m\u2080 : \u2124)) % \u2191n = _\n[GOAL]\ncase inr.intro.inr.calc_2.e_a.e_a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\nz : \u2124\na : \u03b1\nk : Type u_4\ninst\u271d\u00b9 : LinearOrderedField k\ninst\u271d : FloorRing k\nb : k\nn : \u2115\nhn : 0 < \u2191n\nthis : \u2200 {l : \u2124}, 0 \u2264 l \u2192 fract (\u2191l / \u2191n) = \u2191(l % \u2191n) / \u2191n\nm\u2080 : \u2115\nq : \u2124 := \u2308\u2191m\u2080 / \u2191n\u2309\nm\u2081 : \u2124 := q * \u2191n - \u2191m\u2080\nhm\u2081 : 0 \u2264 m\u2081\n\u22a2 (q * \u2191n - \u2191m\u2080) % \u2191n = -\u2191m\u2080 % \u2191n\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_comm (q * \u2191n), add_mul_emod_self]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na : \u03b1\nz : \u2124\n\u22a2 z \u2264 \u230a-a\u230b \u2194 z \u2264 -\u2308a\u2309\n[PROOFSTEP]\nrw [le_neg, ceil_le, le_floor, Int.cast_neg, le_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na : \u03b1\nz : \u2124\n\u22a2 \u2308-a\u2309 \u2264 z \u2194 -\u230aa\u230b \u2264 z\n[PROOFSTEP]\nrw [neg_le, ceil_le, le_floor, Int.cast_neg, neg_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 z + 1 \u2264 \u2308a\u2309 \u2194 \u2191z < a\n[PROOFSTEP]\nrw [\u2190 lt_ceil, add_one_le_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 1 \u2264 \u2308a\u2309 \u2194 0 < a\n[PROOFSTEP]\nrw [\u2190 zero_add (1 : \u2124), add_one_le_ceil_iff, cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u2308a\u2309 \u2264 \u230aa\u230b + 1\n[PROOFSTEP]\nrw [ceil_le, Int.cast_add, Int.cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 a \u2264 \u2191\u230aa\u230b + 1\n[PROOFSTEP]\nexact (lt_floor_add_one a).le\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d : \u03b1\nz a : \u2124\n\u22a2 \u2308\u2191z\u2309 \u2264 a \u2194 z \u2264 a\n[PROOFSTEP]\nrw [ceil_le, Int.cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d : \u03b1\nn : \u2115\na : \u2124\n\u22a2 \u2308\u2191n\u2309 \u2264 a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nrw [ceil_le, \u2190 cast_ofNat, cast_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nz : \u2124\n\u22a2 \u2308a + \u2191z\u2309 = \u2308a\u2309 + z\n[PROOFSTEP]\nrw [\u2190 neg_inj, neg_add', \u2190 floor_neg, \u2190 floor_neg, neg_add', floor_sub_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u2308a + \u2191n\u2309 = \u2308a\u2309 + \u2191n\n[PROOFSTEP]\nrw [\u2190 Int.cast_ofNat, ceil_add_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u2308a + 1\u2309 = \u2308a\u2309 + 1\n[PROOFSTEP]\nrw [\u2190 ceil_add_int a (1 : \u2124), cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nz : \u2124\n\u22a2 \u2308a - \u2191z\u2309 = \u2308a + \u2191(-z)\u2309\n[PROOFSTEP]\nrw [Int.cast_neg, sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u2308a - \u2191n\u2309 = \u2308a\u2309 - \u2191n\n[PROOFSTEP]\nconvert ceil_sub_int a n using 1\n[GOAL]\ncase h.e'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u2308a - \u2191n\u2309 = \u2308a - \u2191\u2191n\u2309\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u2308a - 1\u2309 = \u2308a\u2309 - 1\n[PROOFSTEP]\nrw [eq_sub_iff_add_eq, \u2190 ceil_add_one, sub_add_cancel]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u2191\u2308a\u2309 < a + 1\n[PROOFSTEP]\nrw [\u2190 lt_ceil, \u2190 Int.cast_one, ceil_add_int]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 \u2308a\u2309 < \u2308a\u2309 + 1\n[PROOFSTEP]\napply lt_add_one\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2308a + b\u2309 \u2264 \u2308a\u2309 + \u2308b\u2309\n[PROOFSTEP]\nrw [ceil_le, Int.cast_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 a + b \u2264 \u2191\u2308a\u2309 + \u2191\u2308b\u2309\n[PROOFSTEP]\nexact add_le_add (le_ceil _) (le_ceil _)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2308a\u2309 + \u2308b\u2309 \u2264 \u2308a + b\u2309 + 1\n[PROOFSTEP]\nrw [\u2190 le_sub_iff_add_le, ceil_le, Int.cast_sub, Int.cast_add, Int.cast_one, le_sub_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u2308b\u2309 \u2264 \u2191\u2308a + b\u2309 + 1 - a\n[PROOFSTEP]\nrefine' (ceil_lt_add_one _).le.trans _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 b + 1 \u2264 \u2191\u2308a + b\u2309 + 1 - a\n[PROOFSTEP]\nrw [le_sub_iff_add_le', \u2190 add_assoc, add_le_add_iff_right]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 a + b \u2264 \u2191\u2308a + b\u2309\n[PROOFSTEP]\nexact le_ceil _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 0 < \u2308a\u2309 \u2194 0 < a\n[PROOFSTEP]\nrw [lt_ceil, cast_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u23080\u2309 = 0\n[PROOFSTEP]\nrw [\u2190 cast_zero, ceil_intCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u23081\u2309 = 1\n[PROOFSTEP]\nrw [\u2190 cast_one, ceil_intCast]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : 0 \u2264 a\n\u22a2 0 \u2264 \u2308a\u2309\n[PROOFSTEP]\nexact_mod_cast ha.trans (le_ceil a)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u2308a\u2309 = z \u2194 \u2191z - 1 < a \u2227 a \u2264 \u2191z\n[PROOFSTEP]\nrw [\u2190 ceil_le, \u2190 Int.cast_one, \u2190 Int.cast_sub, \u2190 lt_ceil, Int.sub_one_lt_iff, le_antisymm_iff, and_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 \u2308a\u2309 = 0 \u2194 a \u2208 Ioc (-1) 0\n[PROOFSTEP]\nsimp [ceil_eq_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d : \u03b1\nz : \u2124\na : \u03b1\nha : a \u2208 Ioc (\u2191z - 1) \u2191z\n\u22a2 \u2191\u2308a\u2309 = \u2191z\n[PROOFSTEP]\nexact_mod_cast ceil_eq_on_Ioc z a ha\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\n\u22a2 fract a = 0 \u2228 fract a = a + 1 - \u2191\u2308a\u2309\n[PROOFSTEP]\ncases' eq_or_ne (fract a) 0 with ha ha\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a = 0\n\u22a2 fract a = 0 \u2228 fract a = a + 1 - \u2191\u2308a\u2309\n[PROOFSTEP]\nexact Or.inl ha\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 fract a = 0 \u2228 fract a = a + 1 - \u2191\u2308a\u2309\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 fract a = a + 1 - \u2191\u2308a\u2309\n[PROOFSTEP]\nsuffices (\u2308a\u2309 : \u03b1) = \u230aa\u230b + 1 by\n  rw [this, \u2190 self_sub_fract]\n  abel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\nthis : \u2191\u2308a\u2309 = \u2191\u230aa\u230b + 1\n\u22a2 fract a = a + 1 - \u2191\u2308a\u2309\n[PROOFSTEP]\nrw [this, \u2190 self_sub_fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\nthis : \u2191\u2308a\u2309 = \u2191\u230aa\u230b + 1\n\u22a2 fract a = a + 1 - (a - fract a + 1)\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\nthis : \u2191\u2308a\u2309 = \u2191\u230aa\u230b + 1\n\u22a2 fract a = a + 1 - (a - fract a + 1)\n[PROOFSTEP]\nabel\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 = \u2191\u230aa\u230b + 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2308a\u2309 = \u230aa\u230b + 1\n[PROOFSTEP]\nrw [ceil_eq_iff]\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191(\u230aa\u230b + 1) - 1 < a \u2227 a \u2264 \u2191(\u230aa\u230b + 1)\n[PROOFSTEP]\nrefine' \u27e8_, _root_.le_of_lt <| by simp\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 a < \u2191(\u230aa\u230b + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191(\u230aa\u230b + 1) - 1 < a\n[PROOFSTEP]\nrw [cast_add, cast_one, add_tsub_cancel_right, \u2190 self_sub_fract a, sub_lt_self_iff]\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nha : fract a \u2260 0\n\u22a2 0 < fract a\n[PROOFSTEP]\nexact ha.symm.lt_of_le (fract_nonneg a)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 = a + 1 - fract a\n[PROOFSTEP]\nrw [(or_iff_right ha).mp (fract_eq_zero_or_add_one_sub_ceil a)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 = a + 1 - (a + 1 - \u2191\u2308a\u2309)\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 = a + 1 - (a + 1 - \u2191\u2308a\u2309)\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 - a = 1 - fract a\n[PROOFSTEP]\nrw [(or_iff_right ha).mp (fract_eq_zero_or_add_one_sub_ceil a)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 - a = 1 - (a + 1 - \u2191\u2308a\u2309)\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nha : fract a \u2260 0\n\u22a2 \u2191\u2308a\u2309 - a = 1 - (a + 1 - \u2191\u2308a\u2309)\n[PROOFSTEP]\nabel\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Ioo a b = Ioo \u230aa\u230b \u2308b\u2309\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ioo a b \u2194 x\u271d \u2208 Ioo \u230aa\u230b \u2308b\u2309\n[PROOFSTEP]\nsimp [floor_lt, lt_ceil]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Ico a b = Ico \u2308a\u2309 \u2308b\u2309\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ico a b \u2194 x\u271d \u2208 Ico \u2308a\u2309 \u2308b\u2309\n[PROOFSTEP]\nsimp [ceil_le, lt_ceil]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Ioc a b = Ioc \u230aa\u230b \u230ab\u230b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ioc a b \u2194 x\u271d \u2208 Ioc \u230aa\u230b \u230ab\u230b\n[PROOFSTEP]\nsimp [floor_lt, le_floor]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Icc a b = Icc \u2308a\u2309 \u230ab\u230b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Icc a b \u2194 x\u271d \u2208 Icc \u2308a\u2309 \u230ab\u230b\n[PROOFSTEP]\nsimp [ceil_le, le_floor]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Ioi a = Ioi \u230aa\u230b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ioi a \u2194 x\u271d \u2208 Ioi \u230aa\u230b\n[PROOFSTEP]\nsimp [floor_lt]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Ici a = Ici \u2308a\u2309\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Ici a \u2194 x\u271d \u2208 Ici \u2308a\u2309\n[PROOFSTEP]\nsimp [ceil_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Iio a = Iio \u2308a\u2309\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Iio a \u2194 x\u271d \u2208 Iio \u2308a\u2309\n[PROOFSTEP]\nsimp [lt_ceil]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Iic a = Iic \u230aa\u230b\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Iic a \u2194 x\u271d \u2208 Iic \u230aa\u230b\n[PROOFSTEP]\nsimp [le_floor]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\n\u22a2 round 0 = 0\n[PROOFSTEP]\nsimp [round]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\n\u22a2 round 1 = 1\n[PROOFSTEP]\nsimp [round]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nn : \u2115\n\u22a2 round \u2191n = \u2191n\n[PROOFSTEP]\nsimp [round]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nn : \u2124\n\u22a2 round \u2191n = n\n[PROOFSTEP]\nsimp [round]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2124\n\u22a2 round (x + \u2191y) = round x + y\n[PROOFSTEP]\nrw [round, round, Int.fract_add_int, Int.floor_add_int, Int.ceil_add_int, \u2190 apply_ite\u2082, ite_self]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\n\u22a2 round (a + 1) = round a + 1\n[PROOFSTEP]\nrw [\u2190 round_add_int a 1, cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2124\n\u22a2 round (x - \u2191y) = round x - y\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2124\n\u22a2 round (x + -\u2191y) = round x - y\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2124\n\u22a2 round (x + \u2191(-y)) = round x - y\n[PROOFSTEP]\nrw [round_add_int, sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\n\u22a2 round (a - 1) = round a - 1\n[PROOFSTEP]\nrw [\u2190 round_sub_int a 1, cast_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2115\n\u22a2 round (x + \u2191y) = round x + \u2191y\n[PROOFSTEP]\nexact_mod_cast round_add_int x y\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2115\n\u22a2 round (x - \u2191y) = round x - \u2191y\n[PROOFSTEP]\nexact_mod_cast round_sub_int x y\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2124\n\u22a2 round (\u2191y + x) = y + round x\n[PROOFSTEP]\nrw [add_comm, round_add_int, add_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\ny : \u2115\n\u22a2 round (\u2191y + x) = \u2191y + round x\n[PROOFSTEP]\nrw [add_comm, round_add_nat, add_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 |x - \u2191(round x)| = min (fract x) (1 - fract x)\n[PROOFSTEP]\nsimp_rw [round, min_def_lt, two_mul, \u2190 lt_tsub_iff_left]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 |x - \u2191(if fract x < 1 - fract x then \u230ax\u230b else \u2308x\u2309)| = if fract x < 1 - fract x then fract x else 1 - fract x\n[PROOFSTEP]\ncases' lt_or_ge (fract x) (1 - fract x) with hx hx\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 - fract x\n\u22a2 |x - \u2191(if fract x < 1 - fract x then \u230ax\u230b else \u2308x\u2309)| = if fract x < 1 - fract x then fract x else 1 - fract x\n[PROOFSTEP]\nrw [if_pos hx, if_pos hx, self_sub_floor, abs_fract]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x \u2265 1 - fract x\n\u22a2 |x - \u2191(if fract x < 1 - fract x then \u230ax\u230b else \u2308x\u2309)| = if fract x < 1 - fract x then fract x else 1 - fract x\n[PROOFSTEP]\nhave : 0 < fract x :=\n  by\n  replace hx : 0 < fract x + fract x := lt_of_lt_of_le zero_lt_one (tsub_le_iff_left.mp hx)\n  simpa only [\u2190 two_mul, zero_lt_mul_left, zero_lt_two] using hx\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x \u2265 1 - fract x\n\u22a2 0 < fract x\n[PROOFSTEP]\nreplace hx : 0 < fract x + fract x := lt_of_lt_of_le zero_lt_one (tsub_le_iff_left.mp hx)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 0 < fract x + fract x\n\u22a2 0 < fract x\n[PROOFSTEP]\nsimpa only [\u2190 two_mul, zero_lt_mul_left, zero_lt_two] using hx\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x \u2265 1 - fract x\nthis : 0 < fract x\n\u22a2 |x - \u2191(if fract x < 1 - fract x then \u230ax\u230b else \u2308x\u2309)| = if fract x < 1 - fract x then fract x else 1 - fract x\n[PROOFSTEP]\nrw [if_neg (not_lt.mpr hx), if_neg (not_lt.mpr hx), abs_sub_comm, ceil_sub_self_eq this.ne.symm, abs_one_sub_fract]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\n\u22a2 |x - \u2191(round x)| \u2264 |x - \u2191z|\n[PROOFSTEP]\nrw [abs_sub_round_eq_min, min_le_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\n\u22a2 fract x \u2264 |x - \u2191z| \u2228 1 - fract x \u2264 |x - \u2191z|\n[PROOFSTEP]\nrcases le_or_lt (z : \u03b1) x with (hx | hx) <;> [left; right]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\n\u22a2 fract x \u2264 |x - \u2191z| \u2228 1 - fract x \u2264 |x - \u2191z|\n[PROOFSTEP]\nrcases le_or_lt (z : \u03b1) x with (hx | hx)\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : \u2191z \u2264 x\n\u22a2 fract x \u2264 |x - \u2191z| \u2228 1 - fract x \u2264 |x - \u2191z|\n[PROOFSTEP]\nleft\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n\u22a2 fract x \u2264 |x - \u2191z| \u2228 1 - fract x \u2264 |x - \u2191z|\n[PROOFSTEP]\nright\n[GOAL]\ncase inl.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : \u2191z \u2264 x\n\u22a2 fract x \u2264 |x - \u2191z|\n[PROOFSTEP]\nconv_rhs => rw [abs_eq_self.mpr (sub_nonneg.mpr hx), \u2190 fract_add_floor x, add_sub_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : \u2191z \u2264 x\n| |x - \u2191z|\n[PROOFSTEP]\nrw [abs_eq_self.mpr (sub_nonneg.mpr hx), \u2190 fract_add_floor x, add_sub_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : \u2191z \u2264 x\n| |x - \u2191z|\n[PROOFSTEP]\nrw [abs_eq_self.mpr (sub_nonneg.mpr hx), \u2190 fract_add_floor x, add_sub_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : \u2191z \u2264 x\n| |x - \u2191z|\n[PROOFSTEP]\nrw [abs_eq_self.mpr (sub_nonneg.mpr hx), \u2190 fract_add_floor x, add_sub_assoc]\n[GOAL]\ncase inl.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : \u2191z \u2264 x\n\u22a2 fract x \u2264 fract x + (\u2191\u230ax\u230b - \u2191z)\n[PROOFSTEP]\nsimpa only [le_add_iff_nonneg_right, sub_nonneg, cast_le] using le_floor.mpr hx\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n\u22a2 1 - fract x \u2264 |x - \u2191z|\n[PROOFSTEP]\nrw [abs_eq_neg_self.mpr (sub_neg.mpr hx).le]\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n\u22a2 1 - fract x \u2264 -(x - \u2191z)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 fract_add_floor x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n| -(x - \u2191z)\n[PROOFSTEP]\nrw [\u2190 fract_add_floor x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n| -(x - \u2191z)\n[PROOFSTEP]\nrw [\u2190 fract_add_floor x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n| -(x - \u2191z)\n[PROOFSTEP]\nrw [\u2190 fract_add_floor x]\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n\u22a2 1 - fract x \u2264 -(fract x + \u2191\u230ax\u230b - \u2191z)\n[PROOFSTEP]\nrw [add_sub_assoc, add_comm, neg_add, neg_sub, le_add_neg_iff_add_le, sub_add_cancel, le_sub_comm]\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n\u22a2 \u2191\u230ax\u230b \u2264 \u2191z - 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr.h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nz : \u2124\nhx : x < \u2191z\n\u22a2 \u230ax\u230b \u2264 z - 1\n[PROOFSTEP]\nexact floor_le_sub_one_iff.mpr hx\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 round x = \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nsimp_rw [round, (by simp only [lt_div_iff', two_pos] : 2 * fract x < 1 \u2194 fract x < 1 / 2)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 2 * fract x < 1 \u2194 fract x < 1 / 2\n[PROOFSTEP]\nsimp only [lt_div_iff', two_pos]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 (if fract x < 1 / 2 then \u230ax\u230b else \u2308x\u2309) = \u230ax + 1 / 2\u230b\n[PROOFSTEP]\ncases' lt_or_le (fract x) (1 / 2) with hx hx\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n\u22a2 (if fract x < 1 / 2 then \u230ax\u230b else \u2308x\u2309) = \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nconv_rhs => rw [\u2190 fract_add_floor x, add_assoc, add_left_comm, floor_int_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n| \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nrw [\u2190 fract_add_floor x, add_assoc, add_left_comm, floor_int_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n| \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nrw [\u2190 fract_add_floor x, add_assoc, add_left_comm, floor_int_add]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n| \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nrw [\u2190 fract_add_floor x, add_assoc, add_left_comm, floor_int_add]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n\u22a2 (if fract x < 1 / 2 then \u230ax\u230b else \u2308x\u2309) = \u230ax\u230b + \u230afract x + 1 / 2\u230b\n[PROOFSTEP]\nrw [if_pos hx, self_eq_add_right, floor_eq_iff, cast_zero, zero_add]\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n\u22a2 0 \u2264 fract x + 1 / 2 \u2227 fract x + 1 / 2 < 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inl.left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n\u22a2 0 \u2264 fract x + 1 / 2\n[PROOFSTEP]\nlinarith [fract_nonneg x]\n[GOAL]\ncase inl.right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : fract x < 1 / 2\n\u22a2 fract x + 1 / 2 < 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 (if fract x < 1 / 2 then \u230ax\u230b else \u2308x\u2309) = \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nhave : \u230afract x + 1 / 2\u230b = 1 := by\n  rw [floor_eq_iff]\n  constructor\n  \u00b7 norm_num\n    linarith\n  \u00b7 norm_num\n    linarith [fract_lt_one x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 \u230afract x + 1 / 2\u230b = 1\n[PROOFSTEP]\nrw [floor_eq_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 \u21911 \u2264 fract x + 1 / 2 \u2227 fract x + 1 / 2 < \u21911 + 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 \u21911 \u2264 fract x + 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 1 \u2264 fract x + 1 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 fract x + 1 / 2 < \u21911 + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\n\u22a2 fract x + 1 / 2 < 2\n[PROOFSTEP]\nlinarith [fract_lt_one x]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\nthis : \u230afract x + 1 / 2\u230b = 1\n\u22a2 (if fract x < 1 / 2 then \u230ax\u230b else \u2308x\u2309) = \u230ax + 1 / 2\u230b\n[PROOFSTEP]\nrw [if_neg (not_lt.mpr hx), \u2190 fract_add_floor x, add_assoc, add_left_comm, floor_int_add, ceil_add_int, add_comm _ \u230ax\u230b,\n  add_right_inj, ceil_eq_iff, this, cast_one, sub_self]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\nthis : \u230afract x + 1 / 2\u230b = 1\n\u22a2 0 < fract x \u2227 fract x \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\nthis : \u230afract x + 1 / 2\u230b = 1\n\u22a2 0 < fract x\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr.right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nhx : 1 / 2 \u2264 fract x\nthis : \u230afract x + 1 / 2\u230b = 1\n\u22a2 fract x \u2264 1\n[PROOFSTEP]\nlinarith [fract_lt_one x]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\n\u22a2 round 2\u207b\u00b9 = 1\n[PROOFSTEP]\nsimp only [round_eq, \u2190 one_div, add_halves', floor_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\n\u22a2 round (-2\u207b\u00b9) = 0\n[PROOFSTEP]\nsimp only [round_eq, \u2190 one_div, add_left_neg, floor_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 round x = 0 \u2194 x \u2208 Ico (-(1 / 2)) (1 / 2)\n[PROOFSTEP]\nrw [round_eq, floor_eq_zero_iff, add_mem_Ico_iff_left]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 x \u2208 Ico (0 - 1 / 2) (1 - 1 / 2) \u2194 x \u2208 Ico (-(1 / 2)) (1 / 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 |x - \u2191(round x)| \u2264 1 / 2\n[PROOFSTEP]\nrw [round_eq, abs_sub_le_iff]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\n\u22a2 x - \u2191\u230ax + 1 / 2\u230b \u2264 1 / 2 \u2227 \u2191\u230ax + 1 / 2\u230b - x \u2264 1 / 2\n[PROOFSTEP]\nhave := floor_le (x + 1 / 2)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nthis : \u2191\u230ax + 1 / 2\u230b \u2264 x + 1 / 2\n\u22a2 x - \u2191\u230ax + 1 / 2\u230b \u2264 1 / 2 \u2227 \u2191\u230ax + 1 / 2\u230b - x \u2264 1 / 2\n[PROOFSTEP]\nhave := lt_floor_add_one (x + 1 / 2)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nthis\u271d : \u2191\u230ax + 1 / 2\u230b \u2264 x + 1 / 2\nthis : x + 1 / 2 < \u2191\u230ax + 1 / 2\u230b + 1\n\u22a2 x - \u2191\u230ax + 1 / 2\u230b \u2264 1 / 2 \u2227 \u2191\u230ax + 1 / 2\u230b - x \u2264 1 / 2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nthis\u271d : \u2191\u230ax + 1 / 2\u230b \u2264 x + 1 / 2\nthis : x + 1 / 2 < \u2191\u230ax + 1 / 2\u230b + 1\n\u22a2 x - \u2191\u230ax + 1 / 2\u230b \u2264 1 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase right\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nx : \u03b1\nthis\u271d : \u2191\u230ax + 1 / 2\u230b \u2264 x + 1 / 2\nthis : x + 1 / 2 < \u2191\u230ax + 1 / 2\u230b + 1\n\u22a2 \u2191\u230ax + 1 / 2\u230b - x \u2264 1 / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nm n : \u2115\n\u22a2 |\u2191m / \u2191n - \u2191(round (\u2191m / \u2191n))| = \u2191(min (m % n) (n - m % n)) / \u2191n\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nm : \u2115\n\u22a2 |\u2191m / \u21910 - \u2191(round (\u2191m / \u21910))| = \u2191(min (m % 0) (0 - m % 0)) / \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nm n : \u2115\nhn : n > 0\n\u22a2 |\u2191m / \u2191n - \u2191(round (\u2191m / \u2191n))| = \u2191(min (m % n) (n - m % n)) / \u2191n\n[PROOFSTEP]\nhave hn' : 0 < (n : \u03b1) := by norm_cast\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nm n : \u2115\nhn : n > 0\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\nm n : \u2115\nhn : n > 0\nhn' : 0 < \u2191n\n\u22a2 |\u2191m / \u2191n - \u2191(round (\u2191m / \u2191n))| = \u2191(min (m % n) (n - m % n)) / \u2191n\n[PROOFSTEP]\nrw [abs_sub_round_eq_min, Nat.cast_min, \u2190 min_div_div_right hn'.le, fract_div_natCast_eq_div_natCast_mod,\n  Nat.cast_sub (m.mod_lt hn).le, sub_div, div_self hn'.ne']\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nh : \u2200 (n : \u2115), \u2191n \u2264 a \u2194 \u2191n \u2264 b\n\u22a2 \u230aa\u230b\u208a = \u230ab\u230b\u208a\n[PROOFSTEP]\nhave h\u2080 : 0 \u2264 a \u2194 0 \u2264 b := by simpa only [cast_zero] using h 0\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nh : \u2200 (n : \u2115), \u2191n \u2264 a \u2194 \u2191n \u2264 b\n\u22a2 0 \u2264 a \u2194 0 \u2264 b\n[PROOFSTEP]\nsimpa only [cast_zero] using h 0\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nh : \u2200 (n : \u2115), \u2191n \u2264 a \u2194 \u2191n \u2264 b\nh\u2080 : 0 \u2264 a \u2194 0 \u2264 b\n\u22a2 \u230aa\u230b\u208a = \u230ab\u230b\u208a\n[PROOFSTEP]\nobtain ha | ha := lt_or_le a 0\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nh : \u2200 (n : \u2115), \u2191n \u2264 a \u2194 \u2191n \u2264 b\nh\u2080 : 0 \u2264 a \u2194 0 \u2264 b\nha : a < 0\n\u22a2 \u230aa\u230b\u208a = \u230ab\u230b\u208a\n[PROOFSTEP]\nrw [floor_of_nonpos ha.le, floor_of_nonpos (le_of_not_le <| h\u2080.not.mp ha.not_le)]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nh : \u2200 (n : \u2115), \u2191n \u2264 a \u2194 \u2191n \u2264 b\nh\u2080 : 0 \u2264 a \u2194 0 \u2264 b\nha : 0 \u2264 a\n\u22a2 \u230aa\u230b\u208a = \u230ab\u230b\u208a\n[PROOFSTEP]\nexact (le_floor <| (h _).1 <| floor_le ha).antisymm (le_floor <| (h _).2 <| floor_le <| h\u2080.1 ha)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\nn : \u2115\n\u22a2 \u2191n \u2264 \u2191f a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nrw [\u2190 map_natCast f, hf.le_iff_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedSemiring \u03b1\ninst\u271d\u00b3 : LinearOrderedSemiring \u03b2\ninst\u271d\u00b2 : FloorSemiring \u03b1\ninst\u271d\u00b9 : FloorSemiring \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\nn : \u2115\n\u22a2 \u2191f a \u2264 \u2191n \u2194 a \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 map_natCast f, hf.le_iff_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : LinearOrderedRing \u03b2\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : FloorRing \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\nn : \u2124\n\u22a2 \u2191n \u2264 \u2191f a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nrw [\u2190 map_intCast f, hf.le_iff_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : LinearOrderedRing \u03b2\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : FloorRing \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\nn : \u2124\n\u22a2 \u2191f a \u2264 \u2191n \u2194 a \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 map_intCast f, hf.le_iff_le]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : LinearOrderedRing \u03b2\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : FloorRing \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\n\u22a2 fract (\u2191f a) = \u2191f (fract a)\n[PROOFSTEP]\nsimp_rw [fract, map_sub, map_intCast, map_floor _ hf]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : LinearOrderedField \u03b2\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : FloorRing \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\n\u22a2 round (\u2191f a) = round a\n[PROOFSTEP]\nhave H : f 2 = 2 := map_natCast f 2\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedField \u03b1\ninst\u271d\u00b3 : LinearOrderedField \u03b2\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : FloorRing \u03b2\ninst\u271d : RingHomClass F \u03b1 \u03b2\na\u271d : \u03b1\nb : \u03b2\nf : F\nhf : StrictMono \u2191f\na : \u03b1\nH : \u2191f 2 = 2\n\u22a2 round (\u2191f a) = round a\n[PROOFSTEP]\nsimp_rw [round_eq, \u2190 map_floor _ hf, map_add, one_div, map_inv\u2080, H]\n  -- Porting note: was\n    -- simp_rw [round_eq, \u2190 map_floor _ hf, map_add, one_div, map_inv\u2080, map_bit0, map_one]\n    -- Would have thought that `map_natCast` would replace `map_bit0, map_one` but seems not\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\nn : \u2115\nha : 0 \u2264 a\n\u22a2 n \u2264 (fun a => toNat \u230aa\u230b) a \u2194 \u2191n \u2264 a\n[PROOFSTEP]\nrw [Int.le_toNat (Int.floor_nonneg.2 ha), Int.le_floor, Int.cast_ofNat]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\nn : \u2115\n\u22a2 (fun a => toNat \u2308a\u2309) a \u2264 n \u2194 a \u2264 \u2191n\n[PROOFSTEP]\nrw [Int.toNat_le, Int.ceil_le, Int.cast_ofNat]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u2191\u230aa\u230b\u208a = \u230aa\u230b\n[PROOFSTEP]\nrw [\u2190 Int.floor_toNat, Int.toNat_of_nonneg (Int.floor_nonneg.2 ha)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u2191\u230aa\u230b\u208a = \u2191\u230aa\u230b\n[PROOFSTEP]\nrw [\u2190 Nat.cast_floor_eq_int_floor ha, Int.cast_ofNat]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u2191\u2308a\u2309\u208a = \u2308a\u2309\n[PROOFSTEP]\nrw [\u2190 Int.ceil_toNat, Int.toNat_of_nonneg (Int.ceil_nonneg ha)]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\na : \u03b1\nha : 0 \u2264 a\n\u22a2 \u2191\u2308a\u2309\u208a = \u2191\u2308a\u2309\n[PROOFSTEP]\nrw [\u2190 Nat.cast_ceil_eq_int_ceil ha, Int.cast_ofNat]\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\n\u22a2 Subsingleton (FloorRing \u03b1)\n[PROOFSTEP]\nrefine' \u27e8fun H\u2081 H\u2082 => _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\nH\u2081 H\u2082 : FloorRing \u03b1\n\u22a2 H\u2081 = H\u2082\n[PROOFSTEP]\nhave : H\u2081.floor = H\u2082.floor := funext fun a => (H\u2081.gc_coe_floor.u_unique H\u2082.gc_coe_floor) fun _ => rfl\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\nH\u2081 H\u2082 : FloorRing \u03b1\nthis : FloorRing.floor = FloorRing.floor\n\u22a2 H\u2081 = H\u2082\n[PROOFSTEP]\nhave : H\u2081.ceil = H\u2082.ceil := funext fun a => (H\u2081.gc_ceil_coe.l_unique H\u2082.gc_ceil_coe) fun _ => rfl\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\nH\u2081 H\u2082 : FloorRing \u03b1\nthis\u271d : FloorRing.floor = FloorRing.floor\nthis : FloorRing.ceil = FloorRing.ceil\n\u22a2 H\u2081 = H\u2082\n[PROOFSTEP]\ncases H\u2081\n[GOAL]\ncase mk\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\nH\u2082 : FloorRing \u03b1\nfloor\u271d ceil\u271d : \u03b1 \u2192 \u2124\ngc_coe_floor\u271d : GaloisConnection Int.cast floor\u271d\ngc_ceil_coe\u271d : GaloisConnection ceil\u271d Int.cast\nthis\u271d : FloorRing.floor = FloorRing.floor\nthis : FloorRing.ceil = FloorRing.ceil\n\u22a2 { floor := floor\u271d, ceil := ceil\u271d, gc_coe_floor := gc_coe_floor\u271d, gc_ceil_coe := gc_ceil_coe\u271d } = H\u2082\n[PROOFSTEP]\ncases H\u2082\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\n\u03b1 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\nfloor\u271d\u00b9 ceil\u271d\u00b9 : \u03b1 \u2192 \u2124\ngc_coe_floor\u271d\u00b9 : GaloisConnection Int.cast floor\u271d\u00b9\ngc_ceil_coe\u271d\u00b9 : GaloisConnection ceil\u271d\u00b9 Int.cast\nfloor\u271d ceil\u271d : \u03b1 \u2192 \u2124\ngc_coe_floor\u271d : GaloisConnection Int.cast floor\u271d\ngc_ceil_coe\u271d : GaloisConnection ceil\u271d Int.cast\nthis\u271d : FloorRing.floor = FloorRing.floor\nthis : FloorRing.ceil = FloorRing.ceil\n\u22a2 { floor := floor\u271d\u00b9, ceil := ceil\u271d\u00b9, gc_coe_floor := gc_coe_floor\u271d\u00b9, gc_ceil_coe := gc_ceil_coe\u271d\u00b9 } =\n    { floor := floor\u271d, ceil := ceil\u271d, gc_coe_floor := gc_coe_floor\u271d, gc_ceil_coe := gc_ceil_coe\u271d }\n[PROOFSTEP]\ncongr\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Floor", "llama_tokens": 60693, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527632, "lm_q2_score": 0.6893056231680121, "lm_q1q2_score": 0.5570052741996266}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh : x \u2260 y\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n[PROOFSTEP]\nobtain h | h := h.not_le_or_not_le\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acx \u2264 y\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n[PROOFSTEP]\nexact (exists_clopen_upper_of_not_le h).imp fun _ \u21a6 And.imp_right <| And.imp_left Or.inl\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acy \u2264 x\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n[PROOFSTEP]\nobtain \u27e8U, hU, hU', hy, hx\u27e9 := exists_clopen_lower_of_not_le h\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PriestleySpace \u03b1\nx y : \u03b1\nh\u271d : x \u2260 y\nh : \u00acy \u2264 x\nU : Set \u03b1\nhU : IsClopen U\nhU' : IsLowerSet U\nhy : \u00acy \u2208 U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsClopen U \u2227 (IsUpperSet U \u2228 IsLowerSet U) \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n[PROOFSTEP]\nexact \u27e8U, hU, Or.inr hU', hx, hy\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Order.Priestley", "llama_tokens": 621, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.7371581626286834, "lm_q1q2_score": 0.5564917446601975}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u22a2 spectrum \ud835\udd5c \u2191u \u2286 Metric.sphere 0 1\n[PROOFSTEP]\nnontriviality E\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\n\u22a2 spectrum \ud835\udd5c \u2191u \u2286 Metric.sphere 0 1\n[PROOFSTEP]\nrefine' fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm _ _)\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191u\n\u22a2 \u2016k\u2016 \u2264 1\n[PROOFSTEP]\nsimpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191u\n\u22a2 1 \u2264 \u2016k\u2016\n[PROOFSTEP]\nrw [\u2190 unitary.toUnits_apply_val u] at hk \n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\n\u22a2 1 \u2264 \u2016k\u2016\n[PROOFSTEP]\nhave hnk := ne_zero_of_mem_of_unit hk\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\nhnk : k \u2260 0\n\u22a2 1 \u2264 \u2016k\u2016\n[PROOFSTEP]\nrw [\u2190 inv_inv (unitary.toUnits u), \u2190 spectrum.map_inv, Set.mem_inv] at hk \n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\n\u22a2 1 \u2264 \u2016k\u2016\n[PROOFSTEP]\nhave : \u2016k\u2016\u207b\u00b9 \u2264 \u2016(\u2191(unitary.toUnits u)\u207b\u00b9 : E)\u2016 := by simpa only [norm_inv] using norm_le_norm_of_mem hk\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\n\u22a2 \u2016k\u2016\u207b\u00b9 \u2264 \u2016\u2191(\u2191toUnits u)\u207b\u00b9\u2016\n[PROOFSTEP]\nsimpa only [norm_inv] using norm_le_norm_of_mem hk\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\nthis : \u2016k\u2016\u207b\u00b9 \u2264 \u2016\u2191(\u2191toUnits u)\u207b\u00b9\u2016\n\u22a2 1 \u2264 \u2016k\u2016\n[PROOFSTEP]\nsimpa [\u2190 Units.inv_eq_val_inv] using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\n\u22a2 spectralRadius \u2102 a = \u2191\u2016a\u2016\u208a\n[PROOFSTEP]\nhave hconst : Tendsto (fun _n : \u2115 => (\u2016a\u2016\u208a : \u211d\u22650\u221e)) atTop _ := tendsto_const_nhds\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 spectralRadius \u2102 a = \u2191\u2016a\u2016\u208a\n[PROOFSTEP]\nrefine' tendsto_nhds_unique _ hconst\n[GOAL]\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd (spectralRadius \u2102 a))\n[PROOFSTEP]\nconvert\n  (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp\n    (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two) using\n  1\n[GOAL]\ncase h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 (fun _n => \u2191\u2016a\u2016\u208a) = (fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) \u2218 fun n => 2 ^ n\n[PROOFSTEP]\nrefine' funext fun n => _\n[GOAL]\ncase h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\nn : \u2115\n\u22a2 \u2191\u2016a\u2016\u208a = ((fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) \u2218 fun n => 2 ^ n) n\n[PROOFSTEP]\nrw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, \u2190 rpow_nat_cast, \u2190 rpow_mul]\n[GOAL]\ncase h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\nn : \u2115\n\u22a2 \u2191\u2016a\u2016\u208a = \u2191\u2016a\u2016\u208a ^ (\u2191(2 ^ n) * (1 / \u2191(2 ^ n)))\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\n\u22a2 spectralRadius \u2102 a = \u2191\u2016a\u2016\u208a\n[PROOFSTEP]\nrefine' (ENNReal.pow_strictMono two_ne_zero).injective _\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\n\u22a2 spectralRadius \u2102 a ^ 2 = \u2191\u2016a\u2016\u208a ^ 2\n[PROOFSTEP]\nhave heq :\n  (fun n : \u2115 => (\u2016(a\u22c6 * a) ^ n\u2016\u208a : \u211d\u22650\u221e) ^ (1 / n : \u211d)) =\n    (fun x => x ^ 2) \u2218 fun n : \u2115 => (\u2016a ^ n\u2016\u208a : \u211d\u22650\u221e) ^ (1 / n : \u211d) :=\n  by\n  funext n\n  rw [Function.comp_apply, \u2190 rpow_nat_cast, \u2190 rpow_mul, mul_comm, rpow_mul, rpow_nat_cast, \u2190 coe_pow, sq, \u2190\n    nnnorm_star_mul_self, Commute.mul_pow (star_comm_self' a), star_pow]\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\n\u22a2 (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) = (fun x => x ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)\n[PROOFSTEP]\nfunext n\n[GOAL]\ncase h\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\nn : \u2115\n\u22a2 \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n) = ((fun x => x ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) n\n[PROOFSTEP]\nrw [Function.comp_apply, \u2190 rpow_nat_cast, \u2190 rpow_mul, mul_comm, rpow_mul, rpow_nat_cast, \u2190 coe_pow, sq, \u2190\n  nnnorm_star_mul_self, Commute.mul_pow (star_comm_self' a), star_pow]\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\nheq : (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) = (fun x => x ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)\n\u22a2 spectralRadius \u2102 a ^ 2 = \u2191\u2016a\u2016\u208a ^ 2\n[PROOFSTEP]\nhave h\u2082 :=\n  ((ENNReal.continuous_pow 2).tendsto (spectralRadius \u2102 a)).comp\n    (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a)\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\nheq : (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) = (fun x => x ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)\nh\u2082 : Tendsto ((fun a => a ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) atTop (\ud835\udcdd (spectralRadius \u2102 a ^ 2))\n\u22a2 spectralRadius \u2102 a ^ 2 = \u2191\u2016a\u2016\u208a ^ 2\n[PROOFSTEP]\nrw [\u2190 heq] at h\u2082 \n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\nheq : (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) = (fun x => x ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)\nh\u2082 : Tendsto (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) atTop (\ud835\udcdd (spectralRadius \u2102 a ^ 2))\n\u22a2 spectralRadius \u2102 a ^ 2 = \u2191\u2016a\u2016\u208a ^ 2\n[PROOFSTEP]\nconvert tendsto_nhds_unique h\u2082 (pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a\u22c6 * a))\n[GOAL]\ncase h.e'_3\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\na : A\ninst\u271d : IsStarNormal a\nheq : (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) = (fun x => x ^ 2) \u2218 fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)\nh\u2082 : Tendsto (fun n => \u2191\u2016(a\u22c6 * a) ^ n\u2016\u208a ^ (1 / \u2191n)) atTop (\ud835\udcdd (spectralRadius \u2102 a ^ 2))\n\u22a2 \u2191\u2016a\u2016\u208a ^ 2 = spectralRadius \u2102 (a\u22c6 * a)\n[PROOFSTEP]\nrw [(IsSelfAdjoint.star_mul_self a).spectralRadius_eq_nnnorm, sq, nnnorm_star_mul_self, coe_mul]\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\n\u22a2 z = \u2191z.re\n[PROOFSTEP]\nhave hu := exp_mem_unitary_of_mem_skewAdjoint \u2102 (ha.smul_mem_skewAdjoint conj_I)\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : _root_.exp \u2102 (I \u2022 a) \u2208 unitary A\n\u22a2 z = \u2191z.re\n[PROOFSTEP]\nlet Iu := Units.mk0 I I_ne_zero\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : _root_.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\n\u22a2 z = \u2191z.re\n[PROOFSTEP]\nhave : _root_.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (_root_.exp \u2102 (I \u2022 a)) := by\n  simpa only [Units.smul_def, Units.val_mk0] using spectrum.exp_mem_exp (Iu \u2022 a) (smul_mem_smul_iff.mpr hz)\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : _root_.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\n\u22a2 _root_.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (_root_.exp \u2102 (I \u2022 a))\n[PROOFSTEP]\nsimpa only [Units.smul_def, Units.val_mk0] using spectrum.exp_mem_exp (Iu \u2022 a) (smul_mem_smul_iff.mpr hz)\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : _root_.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\nthis : _root_.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (_root_.exp \u2102 (I \u2022 a))\n\u22a2 z = \u2191z.re\n[PROOFSTEP]\nexact\n  Complex.ext (ofReal_re _) <| by\n    simpa only [\u2190 Complex.exp_eq_exp_\u2102, mem_sphere_zero_iff_norm, norm_eq_abs, abs_exp, Real.exp_eq_one_iff,\n      smul_eq_mul, I_mul, neg_eq_zero] using spectrum.subset_circle_of_unitary hu this\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : _root_.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\nthis : _root_.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (_root_.exp \u2102 (I \u2022 a))\n\u22a2 z.im = (\u2191z.re).im\n[PROOFSTEP]\nsimpa only [\u2190 Complex.exp_eq_exp_\u2102, mem_sphere_zero_iff_norm, norm_eq_abs, abs_exp, Real.exp_eq_one_iff, smul_eq_mul,\n  I_mul, neg_eq_zero] using spectrum.subset_circle_of_unitary hu this\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\n\u22a2 z \u2208 ofReal' \u2218 re '' spectrum \u2102 a \u2192 z \u2208 spectrum \u2102 a\n[PROOFSTEP]\nrintro \u27e8z, hz, rfl\u27e9\n[GOAL]\ncase intro.intro\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\n\u22a2 (ofReal' \u2218 re) z \u2208 spectrum \u2102 a\n[PROOFSTEP]\nsimpa only [(ha.mem_spectrum_eq_re hz).symm, Function.comp_apply] using hz\n[GOAL]\nF : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2079 : NormedRing A\ninst\u271d\u2078 : NormedAlgebra \u2102 A\ninst\u271d\u2077 : CompleteSpace A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : NormedRing B\ninst\u271d\u00b3 : NormedAlgebra \u2102 B\ninst\u271d\u00b2 : CompleteSpace B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : CstarRing B\nhF : StarAlgHomClass F \u2102 A B\n\u03c6 : F\na : A\n\u22a2 \u2016\u2191\u03c6 a\u2016\u208a \u2264 \u2016a\u2016\u208a\n[PROOFSTEP]\nsuffices \u2200 s : A, IsSelfAdjoint s \u2192 \u2016\u03c6 s\u2016\u208a \u2264 \u2016s\u2016\u208a by\n  exact\n    nonneg_le_nonneg_of_sq_le_sq zero_le' <| by\n      simpa only [nnnorm_star_mul_self, map_star, map_mul] using this _ (IsSelfAdjoint.star_mul_self a)\n[GOAL]\nF : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2079 : NormedRing A\ninst\u271d\u2078 : NormedAlgebra \u2102 A\ninst\u271d\u2077 : CompleteSpace A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : NormedRing B\ninst\u271d\u00b3 : NormedAlgebra \u2102 B\ninst\u271d\u00b2 : CompleteSpace B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : CstarRing B\nhF : StarAlgHomClass F \u2102 A B\n\u03c6 : F\na : A\nthis : \u2200 (s : A), IsSelfAdjoint s \u2192 \u2016\u2191\u03c6 s\u2016\u208a \u2264 \u2016s\u2016\u208a\n\u22a2 \u2016\u2191\u03c6 a\u2016\u208a \u2264 \u2016a\u2016\u208a\n[PROOFSTEP]\nexact\n  nonneg_le_nonneg_of_sq_le_sq zero_le' <| by\n    simpa only [nnnorm_star_mul_self, map_star, map_mul] using this _ (IsSelfAdjoint.star_mul_self a)\n[GOAL]\nF : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2079 : NormedRing A\ninst\u271d\u2078 : NormedAlgebra \u2102 A\ninst\u271d\u2077 : CompleteSpace A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : NormedRing B\ninst\u271d\u00b3 : NormedAlgebra \u2102 B\ninst\u271d\u00b2 : CompleteSpace B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : CstarRing B\nhF : StarAlgHomClass F \u2102 A B\n\u03c6 : F\na : A\nthis : \u2200 (s : A), IsSelfAdjoint s \u2192 \u2016\u2191\u03c6 s\u2016\u208a \u2264 \u2016s\u2016\u208a\n\u22a2 \u2016\u2191\u03c6 a\u2016\u208a * \u2016\u2191\u03c6 a\u2016\u208a \u2264 \u2016a\u2016\u208a * \u2016a\u2016\u208a\n[PROOFSTEP]\nsimpa only [nnnorm_star_mul_self, map_star, map_mul] using this _ (IsSelfAdjoint.star_mul_self a)\n[GOAL]\nF : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2079 : NormedRing A\ninst\u271d\u2078 : NormedAlgebra \u2102 A\ninst\u271d\u2077 : CompleteSpace A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : NormedRing B\ninst\u271d\u00b3 : NormedAlgebra \u2102 B\ninst\u271d\u00b2 : CompleteSpace B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : CstarRing B\nhF : StarAlgHomClass F \u2102 A B\n\u03c6 : F\na : A\n\u22a2 \u2200 (s : A), IsSelfAdjoint s \u2192 \u2016\u2191\u03c6 s\u2016\u208a \u2264 \u2016s\u2016\u208a\n[PROOFSTEP]\nintro s hs\n[GOAL]\nF : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2079 : NormedRing A\ninst\u271d\u2078 : NormedAlgebra \u2102 A\ninst\u271d\u2077 : CompleteSpace A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : NormedRing B\ninst\u271d\u00b3 : NormedAlgebra \u2102 B\ninst\u271d\u00b2 : CompleteSpace B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : CstarRing B\nhF : StarAlgHomClass F \u2102 A B\n\u03c6 : F\na s : A\nhs : IsSelfAdjoint s\n\u22a2 \u2016\u2191\u03c6 s\u2016\u208a \u2264 \u2016s\u2016\u208a\n[PROOFSTEP]\nsimpa only [hs.spectralRadius_eq_nnnorm, (hs.starHom_apply \u03c6).spectralRadius_eq_nnnorm, coe_le_coe] using\n  show spectralRadius \u2102 (\u03c6 s) \u2264 spectralRadius \u2102 s from iSup_le_iSup_of_subset (AlgHom.spectrum_apply_subset \u03c6 s)\n[GOAL]\nF : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2079 : NormedRing A\ninst\u271d\u2078 : NormedAlgebra \u2102 A\ninst\u271d\u2077 : CompleteSpace A\ninst\u271d\u2076 : StarRing A\ninst\u271d\u2075 : CstarRing A\ninst\u271d\u2074 : NormedRing B\ninst\u271d\u00b3 : NormedAlgebra \u2102 B\ninst\u271d\u00b2 : CompleteSpace B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : CstarRing B\nhF : StarAlgHomClass F \u2102 A B\n\u03c6\u271d : F\nsrc\u271d : LinearMapClass F \u2102 A B := AlgHomClass.linearMapClass\n\u03c6 : F\n\u22a2 \u2200 (x : A), \u2016\u2191\u03c6 x\u2016 \u2264 1 * \u2016x\u2016\n[PROOFSTEP]\nsimpa only [one_mul] using nnnorm_apply_le \u03c6\n[GOAL]\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\n\u22a2 \u2191\u03c6 a\u22c6 = (\u2191\u03c6 a)\u22c6\n[PROOFSTEP]\nsuffices hsa : \u2200 s : selfAdjoint A, (\u03c6 s)\u22c6 = \u03c6 s\n[GOAL]\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\nhsa : \u2200 (s : { x // x \u2208 selfAdjoint A }), (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n\u22a2 \u2191\u03c6 a\u22c6 = (\u2191\u03c6 a)\u22c6\n[PROOFSTEP]\nrw [\u2190 realPart_add_I_smul_imaginaryPart a]\n[GOAL]\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\nhsa : \u2200 (s : { x // x \u2208 selfAdjoint A }), (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n\u22a2 \u2191\u03c6 (\u2191(\u2191\u211c a) + I \u2022 \u2191(\u2191\u2111 a))\u22c6 = (\u2191\u03c6 (\u2191(\u2191\u211c a) + I \u2022 \u2191(\u2191\u2111 a)))\u22c6\n[PROOFSTEP]\nsimp only [map_add, map_smul, star_add, star_smul, hsa, selfAdjoint.star_val_eq]\n[GOAL]\ncase hsa\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\n\u22a2 \u2200 (s : { x // x \u2208 selfAdjoint A }), (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n[PROOFSTEP]\nintro s\n[GOAL]\ncase hsa\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\ns : { x // x \u2208 selfAdjoint A }\n\u22a2 (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n[PROOFSTEP]\nhave := AlgHom.apply_mem_spectrum \u03c6 (s : A)\n[GOAL]\ncase hsa\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\ns : { x // x \u2208 selfAdjoint A }\nthis : \u2191\u03c6 \u2191s \u2208 spectrum \u2102 \u2191s\n\u22a2 (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n[PROOFSTEP]\nrw [selfAdjoint.val_re_map_spectrum s] at this \n[GOAL]\ncase hsa\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\ns : { x // x \u2208 selfAdjoint A }\nthis : \u2191\u03c6 \u2191s \u2208 ofReal' \u2218 re '' spectrum \u2102 \u2191s\n\u22a2 (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n[PROOFSTEP]\nrcases this with \u27e8\u27e8_, _\u27e9, _, heq\u27e9\n[GOAL]\ncase hsa.intro.mk.intro\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\ns : { x // x \u2208 selfAdjoint A }\nre\u271d im\u271d : \u211d\nleft\u271d : { re := re\u271d, im := im\u271d } \u2208 spectrum \u2102 \u2191s\nheq : (ofReal' \u2218 re) { re := re\u271d, im := im\u271d } = \u2191\u03c6 \u2191s\n\u22a2 (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n[PROOFSTEP]\nsimp only [Function.comp_apply] at heq \n[GOAL]\ncase hsa.intro.mk.intro\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\ns : { x // x \u2208 selfAdjoint A }\nre\u271d im\u271d : \u211d\nleft\u271d : { re := re\u271d, im := im\u271d } \u2208 spectrum \u2102 \u2191s\nheq : \u2191re\u271d = \u2191\u03c6 \u2191s\n\u22a2 (\u2191\u03c6 \u2191s)\u22c6 = \u2191\u03c6 \u2191s\n[PROOFSTEP]\nrw [\u2190 heq, IsROrC.star_def]\n[GOAL]\ncase hsa.intro.mk.intro\nF : Type u_1\nA : Type u_2\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\nhF : AlgHomClass F \u2102 A \u2102\n\u03c6 : F\na : A\ns : { x // x \u2208 selfAdjoint A }\nre\u271d im\u271d : \u211d\nleft\u271d : { re := re\u271d, im := im\u271d } \u2208 spectrum \u2102 \u2191s\nheq : \u2191re\u271d = \u2191\u03c6 \u2191s\n\u22a2 \u2191(starRingEnd ((fun x => \u2102) \u2191s)) \u2191re\u271d = \u2191re\u271d\n[PROOFSTEP]\nexact IsROrC.conj_ofReal _\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Star.Spectrum", "llama_tokens": 10016, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324983301568, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.5564563013277012}}
{"text": "[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q\u271d : \u03b1\u271d \u2192 Prop\n\u03b1 \u03b2 : Sort u_4\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\na : { x // p x }\nb : { y // q y }\nh : \u03b1 = \u03b2\nh' : HEq p q\n\u22a2 HEq a b \u2194 HEq \u2191a \u2191b\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\np\u271d q\u271d : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_4\np : \u03b1 \u2192 Prop\na : { x // p x }\nq : \u03b1 \u2192 Prop\nb : { y // q y }\nh' : HEq p q\n\u22a2 HEq a b \u2194 HEq \u2191a \u2191b\n[PROOFSTEP]\nsubst h'\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_4\np : \u03b1 \u2192 Prop\na b : { y // p y }\n\u22a2 HEq a b \u2194 HEq \u2191a \u2191b\n[PROOFSTEP]\nrw [heq_iff_eq, heq_iff_eq, ext_iff]\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\np q : \u03b1 \u2192 Prop\na : Subtype p\nb : \u03b1\n\u22a2 (\u2203 h, { val := b, property := h } = a) \u2194 b = \u2191a\n[PROOFSTEP]\nsimp only [@eq_comm _ b, exists_eq_subtype_mk_iff, @eq_comm _ _ a]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_5\n\u03b2 : \u03b1 \u2192 Type u_4\nf : (x : \u03b1) \u2192 \u03b2 x\np : \u03b1 \u2192 Prop\nx : Subtype p\n\u22a2 restrict p f x = f \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_5\n\u03b2 : \u03b1 \u2192 Type u_4\nne : \u2200 (a : \u03b1), Nonempty (\u03b2 a)\np : \u03b1 \u2192 Prop\n\u22a2 Surjective fun f => restrict p f\n[PROOFSTEP]\nletI := Classical.decPred p\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_5\n\u03b2 : \u03b1 \u2192 Type u_4\nne : \u2200 (a : \u03b1), Nonempty (\u03b2 a)\np : \u03b1 \u2192 Prop\nthis : DecidablePred p := Classical.decPred p\n\u22a2 Surjective fun f => restrict p f\n[PROOFSTEP]\nrefine' fun f \u21a6 \u27e8fun x \u21a6 if h : p x then f \u27e8x, h\u27e9 else Nonempty.some (ne x), funext <| _\u27e9\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_5\n\u03b2 : \u03b1 \u2192 Type u_4\nne : \u2200 (a : \u03b1), Nonempty (\u03b2 a)\np : \u03b1 \u2192 Prop\nthis : DecidablePred p := Classical.decPred p\nf : (x : Subtype p) \u2192 \u03b2 \u2191x\n\u22a2 \u2200 (x : Subtype p),\n    (fun f => restrict p f)\n        (fun x => if h : p x then f { val := x, property := h } else Nonempty.some (_ : Nonempty (\u03b2 x))) x =\n      f x\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_5\n\u03b2 : \u03b1 \u2192 Type u_4\nne : \u2200 (a : \u03b1), Nonempty (\u03b2 a)\np : \u03b1 \u2192 Prop\nthis : DecidablePred p := Classical.decPred p\nf : (x : Subtype p) \u2192 \u03b2 \u2191x\nx : \u03b1\nhx : p x\n\u22a2 (fun f => restrict p f)\n      (fun x => if h : p x then f { val := x, property := h } else Nonempty.some (_ : Nonempty (\u03b2 x)))\n      { val := x, property := hx } =\n    f { val := x, property := hx }\n[PROOFSTEP]\nexact dif_pos hx\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\np\u271d q : \u03b1\u271d \u2192 Prop\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\nf : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\nh : \u2200 (a : \u03b1), p (f a)\nhf : Injective f\nx y : \u03b1\nhxy : coind f h x = coind f h y\n\u22a2 f x = f y\n[PROOFSTEP]\napply congr_arg Subtype.val hxy\n", "meta": {"mathlib_filename": "Mathlib.Data.Subtype", "llama_tokens": 1384, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.815232489352, "lm_q2_score": 0.6825737279551495, "lm_q1q2_score": 0.5564562794071514}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semigroup R\ninst\u271d : StarSemigroup R\na : R\ns : Set R\nha : a \u2208 center R\n\u22a2 star a \u2208 center R\n[PROOFSTEP]\nsimpa only [star_mul, star_star] using fun g => congr_arg star ((Set.mem_center_iff R).mp ha <| star g).symm\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : Semigroup R\ninst\u271d : StarSemigroup R\na : R\ns : Set R\nh : \u2200 (a : R), a \u2208 s \u2192 star a \u2208 s\nha : a \u2208 centralizer s\ny : R\nhy : y \u2208 s\n\u22a2 y * star a = star a * y\n[PROOFSTEP]\nsimpa using congr_arg star (ha _ (h _ hy)).symm\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Star.Center", "llama_tokens": 227, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8031738057795402, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.5563118925412067}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nc : V\nx y : P\n\u22a2 dist (c +\u1d65 x) (c +\u1d65 y) = dist x y\n[PROOFSTEP]\nrw [NormedAddTorsor.dist_eq_norm', NormedAddTorsor.dist_eq_norm', vadd_vsub_vadd_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv\u2081 v\u2082 : V\nx : P\n\u22a2 dist (v\u2081 +\u1d65 x) (v\u2082 +\u1d65 x) = dist v\u2081 v\u2082\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V, dist_eq_norm, vadd_vsub_vadd_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv : V\nx : P\n\u22a2 dist (v +\u1d65 x) x = \u2016v\u2016\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V _ x, vadd_vsub]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv : V\nx : P\n\u22a2 dist x (v +\u1d65 x) = \u2016v\u2016\n[PROOFSTEP]\nrw [dist_comm, dist_vadd_left]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nx y z : P\n\u22a2 dist (x -\u1d65 y) (x -\u1d65 z) = dist y z\n[PROOFSTEP]\nrw [dist_eq_norm, vsub_sub_vsub_cancel_left, dist_comm, dist_eq_norm_vsub V]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv v' : V\np p' : P\n\u22a2 dist (v +\u1d65 p) (v' +\u1d65 p') \u2264 dist v v' + dist p p'\n[PROOFSTEP]\nsimpa [(dist_vadd_cancel_right)] using dist_triangle (v +\u1d65 p) (v' +\u1d65 p) (v' +\u1d65 p')\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 dist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 dist p\u2081 p\u2083 + dist p\u2082 p\u2084\n[PROOFSTEP]\nrw [dist_eq_norm, vsub_sub_vsub_comm, dist_eq_norm_vsub V, dist_eq_norm_vsub V]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 \u2016p\u2081 -\u1d65 p\u2083 - (p\u2082 -\u1d65 p\u2084)\u2016 \u2264 \u2016p\u2081 -\u1d65 p\u2083\u2016 + \u2016p\u2082 -\u1d65 p\u2084\u2016\n[PROOFSTEP]\nexact norm_sub_le _ _\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 nndist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 nndist p\u2081 p\u2083 + nndist p\u2082 p\u2084\n[PROOFSTEP]\nsimp only [\u2190 NNReal.coe_le_coe, NNReal.coe_add, \u2190 dist_nndist, (dist_vsub_vsub_le)]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv v' : V\np p' : P\n\u22a2 edist (v +\u1d65 p) (v' +\u1d65 p') \u2264 edist v v' + edist p p'\n[PROOFSTEP]\nsimp only [edist_nndist]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv v' : V\np p' : P\n\u22a2 \u2191(nndist (v +\u1d65 p) (v' +\u1d65 p')) \u2264 \u2191(nndist v v') + \u2191(nndist p p')\n[PROOFSTEP]\nnorm_cast\n  -- porting note: was apply_mod_cast\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\nv v' : V\np p' : P\n\u22a2 nndist (v +\u1d65 p) (v' +\u1d65 p') \u2264 nndist v v' + nndist p p'\n[PROOFSTEP]\napply dist_vadd_vadd_le\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 edist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 edist p\u2081 p\u2083 + edist p\u2082 p\u2084\n[PROOFSTEP]\nsimp only [edist_nndist]\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 \u2191(nndist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084)) \u2264 \u2191(nndist p\u2081 p\u2083) + \u2191(nndist p\u2082 p\u2084)\n[PROOFSTEP]\nnorm_cast\n  -- porting note: was apply_mod_cast\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 nndist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 nndist p\u2081 p\u2083 + nndist p\u2082 p\u2084\n[PROOFSTEP]\napply dist_vsub_vsub_le\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx : P\n\u22a2 dist x x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 dist x y = dist y x\n[PROOFSTEP]\nsimp only [\u2190 neg_vsub_eq_vsub_rev y x, norm_neg]\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\nchange \u2016x -\u1d65 z\u2016 \u2264 \u2016x -\u1d65 y\u2016 + \u2016y -\u1d65 z\u2016\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 \u2016x -\u1d65 z\u2016 \u2264 \u2016x -\u1d65 y\u2016 + \u2016y -\u1d65 z\u2016\n[PROOFSTEP]\nrw [\u2190 vsub_add_vsub_cancel]\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 \u2016x -\u1d65 ?p2 + (?p2 -\u1d65 z)\u2016 \u2264 \u2016x -\u1d65 y\u2016 + \u2016y -\u1d65 z\u2016\ncase p2\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 P\n[PROOFSTEP]\napply norm_add_le\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx\u271d\u00b9 x\u271d : P\n\u22a2 (fun x y => \u2191{ val := \u2016x -\u1d65 y\u2016, property := (_ : 0 \u2264 \u2016x -\u1d65 y\u2016) }) x\u271d\u00b9 x\u271d = ENNReal.ofReal (dist x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.ofReal_eq_coe_nnreal]\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx : P\n\u22a2 dist x x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y : P\n\u22a2 dist x y = dist y x\n[PROOFSTEP]\nsimp only [\u2190 neg_vsub_eq_vsub_rev y x, norm_neg]\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\nchange \u2016x -\u1d65 z\u2016 \u2264 \u2016x -\u1d65 y\u2016 + \u2016y -\u1d65 z\u2016\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 \u2016x -\u1d65 z\u2016 \u2264 \u2016x -\u1d65 y\u2016 + \u2016y -\u1d65 z\u2016\n[PROOFSTEP]\nrw [\u2190 vsub_add_vsub_cancel]\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 \u2016x -\u1d65 ?p2 + (?p2 -\u1d65 z)\u2016 \u2264 \u2016x -\u1d65 y\u2016 + \u2016y -\u1d65 z\u2016\ncase p2\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx y z : P\n\u22a2 P\n[PROOFSTEP]\napply norm_add_le\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx\u271d\u00b9 x\u271d : P\n\u22a2 (fun x y => \u2191{ val := \u2016x -\u1d65 y\u2016, property := (_ : 0 \u2264 \u2016x -\u1d65 y\u2016) }) x\u271d\u00b9 x\u271d = ENNReal.ofReal (dist x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx\u271d\u00b9 x\u271d : P\n\u22a2 \u2191{ val := \u2016x\u271d\u00b9 -\u1d65 x\u271d\u2016, property := (_ : 0 \u2264 \u2016x\u271d\u00b9 -\u1d65 x\u271d\u2016) } = ENNReal.ofReal \u2016x\u271d\u00b9 -\u1d65 x\u271d\u2016\n[PROOFSTEP]\nrw [ENNReal.ofReal_eq_coe_nnreal]\n[GOAL]\n\u03b1 : Type u_1\nV\u271d : Type u_2\nP\u271d : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2077 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2076 : PseudoMetricSpace P\u271d\ninst\u271d\u2075 : NormedAddTorsor V\u271d P\u271d\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : AddTorsor V P\nx\u271d y\u271d : P\nh : dist x\u271d y\u271d = 0\n\u22a2 x\u271d = y\u271d\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\n\u22a2 IsClosed (s +\u1d65 t)\n[PROOFSTEP]\nrefine IsSeqClosed.isClosed (fun u p husv hup \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhusv : \u2200 (n : \u2115), u n \u2208 s +\u1d65 t\nhup : Tendsto u atTop (\ud835\udcdd p)\n\u22a2 p \u2208 s +\u1d65 t\n[PROOFSTEP]\nchoose! a v hav using husv\n[GOAL]\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nv : \u2115 \u2192 P\nhav : \u2200 (n : \u2115), a n \u2208 s \u2227 v n \u2208 t \u2227 (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\n\u22a2 p \u2208 s +\u1d65 t\n[PROOFSTEP]\nrcases ht.isSeqCompact fun n \u21a6 (hav n).2.1 with \u27e8q, hqt, \u03c6, \u03c6_mono, h\u03c6q\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nv : \u2115 \u2192 P\nhav : \u2200 (n : \u2115), a n \u2208 s \u2227 v n \u2208 t \u2227 (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto ((fun n => v n) \u2218 \u03c6) atTop (\ud835\udcdd q)\n\u22a2 p \u2208 s +\u1d65 t\n[PROOFSTEP]\nrefine\n  \u27e8p -\u1d65 q, q, hs.mem_of_tendsto ((hup.comp \u03c6_mono.tendsto_atTop).vsub h\u03c6q) (eventually_of_forall fun n \u21a6 ?_), hqt,\n    vsub_vadd _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nv : \u2115 \u2192 P\nhav : \u2200 (n : \u2115), a n \u2208 s \u2227 v n \u2208 t \u2227 (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto ((fun n => v n) \u2218 \u03c6) atTop (\ud835\udcdd q)\nn : \u2115\n\u22a2 (u \u2218 \u03c6 -\u1d65 (fun n => v n) \u2218 \u03c6) n \u2208 s\n[PROOFSTEP]\nconvert (hav (\u03c6 n)).1 using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup V\ninst\u271d\u2074 : PseudoMetricSpace P\ninst\u271d\u00b3 : NormedAddTorsor V P\ninst\u271d\u00b2 : NormedAddCommGroup W\ninst\u271d\u00b9 : MetricSpace Q\ninst\u271d : NormedAddTorsor W Q\ns : Set V\nt : Set P\nhs : IsClosed s\nht : IsCompact t\nu : \u2115 \u2192 P\np : P\nhup : Tendsto u atTop (\ud835\udcdd p)\na : \u2115 \u2192 V\nv : \u2115 \u2192 P\nhav : \u2200 (n : \u2115), a n \u2208 s \u2227 v n \u2208 t \u2227 (fun x x_1 => x +\u1d65 x_1) (a n) (v n) = u n\nq : P\nhqt : q \u2208 t\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6q : Tendsto ((fun n => v n) \u2218 \u03c6) atTop (\ud835\udcdd q)\nn : \u2115\n\u22a2 (u \u2218 \u03c6 -\u1d65 (fun n => v n) \u2218 \u03c6) n = a (\u03c6 n)\n[PROOFSTEP]\nexact (eq_vadd_iff_vsub_eq _ _ _).mp (hav (\u03c6 n)).2.2.symm\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.AddTorsor", "llama_tokens": 8535, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.5563118860146916}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d : MulOneClass M\n\u22a2 1 \u2208 center M\n[PROOFSTEP]\nsimp [mem_center_iff]\n[GOAL]\nM : Type u_1\ninst\u271d : MulZeroClass M\n\u22a2 0 \u2208 center M\n[PROOFSTEP]\nsimp [mem_center_iff]\n[GOAL]\nM : Type u_1\ninst\u271d : Semigroup M\na b : M\nha : a \u2208 center M\nhb : b \u2208 center M\ng : M\n\u22a2 g * (a * b) = a * b * g\n[PROOFSTEP]\nrw [mul_assoc, \u2190 hb g, \u2190 mul_assoc, ha g, mul_assoc]\n[GOAL]\nM : Type u_1\ninst\u271d : Group M\na : M\nha : a \u2208 center M\ng : M\n\u22a2 g * a\u207b\u00b9 = a\u207b\u00b9 * g\n[PROOFSTEP]\nrw [\u2190 inv_inj, mul_inv_rev, inv_inv, \u2190 ha, mul_inv_rev, inv_inv]\n[GOAL]\nM : Type u_1\ninst\u271d : Distrib M\na b : M\nha : a \u2208 center M\nhb : b \u2208 center M\nc : M\n\u22a2 c * (a + b) = (a + b) * c\n[PROOFSTEP]\nrw [add_mul, mul_add, ha c, hb c]\n[GOAL]\nM : Type u_1\ninst\u271d : NonUnitalNonAssocRing M\na : M\nha : a \u2208 center M\nc : M\n\u22a2 c * -a = -a * c\n[PROOFSTEP]\nrw [\u2190 neg_mul_comm, ha (-c), neg_mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\u02e3\nha : a \u2208 center M\u02e3\nb : M\n\u22a2 b * \u2191a = \u2191a * b\n[PROOFSTEP]\nobtain rfl | hb := eq_or_ne b 0\n[GOAL]\ncase inl\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\u02e3\nha : a \u2208 center M\u02e3\n\u22a2 0 * \u2191a = \u2191a * 0\n[PROOFSTEP]\nrw [zero_mul, mul_zero]\n[GOAL]\ncase inr\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\u02e3\nha : a \u2208 center M\u02e3\nb : M\nhb : b \u2260 0\n\u22a2 b * \u2191a = \u2191a * b\n[PROOFSTEP]\nexact Units.ext_iff.mp (ha (Units.mk0 _ hb))\n[GOAL]\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\nha : a \u2208 center M\n\u22a2 a\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nobtain rfl | ha0 := eq_or_ne a 0\n[GOAL]\ncase inl\nM : Type u_1\ninst\u271d : GroupWithZero M\nha : 0 \u2208 center M\n\u22a2 0\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nrw [inv_zero]\n[GOAL]\ncase inl\nM : Type u_1\ninst\u271d : GroupWithZero M\nha : 0 \u2208 center M\n\u22a2 0 \u2208 center M\n[PROOFSTEP]\nexact zero_mem_center M\n[GOAL]\ncase inr\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\nha : a \u2208 center M\nha0 : a \u2260 0\n\u22a2 a\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nrcases IsUnit.mk0 _ ha0 with \u27e8a, rfl\u27e9\n[GOAL]\ncase inr.intro\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\u02e3\nha : \u2191a \u2208 center M\nha0 : \u2191a \u2260 0\n\u22a2 (\u2191a)\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nrw [\u2190 Units.val_inv_eq_inv_val]\n[GOAL]\ncase inr.intro\nM : Type u_1\ninst\u271d : GroupWithZero M\na : M\u02e3\nha : \u2191a \u2208 center M\nha0 : \u2191a \u2260 0\n\u22a2 \u2191a\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nexact center_units_subset (inv_mem_center (subset_center_units ha))\n[GOAL]\nM : Type u_1\ninst\u271d : Group M\na b : M\nha : a \u2208 center M\nhb : b \u2208 center M\n\u22a2 a / b \u2208 center M\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u_1\ninst\u271d : Group M\na b : M\nha : a \u2208 center M\nhb : b \u2208 center M\n\u22a2 a * b\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nexact mul_mem_center ha (inv_mem_center hb)\n[GOAL]\nM : Type u_1\ninst\u271d : GroupWithZero M\na b : M\nha : a \u2208 center M\nhb : b \u2208 center M\n\u22a2 a / b \u2208 center M\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u_1\ninst\u271d : GroupWithZero M\na b : M\nha : a \u2208 center M\nhb : b \u2208 center M\n\u22a2 a * b\u207b\u00b9 \u2208 center M\n[PROOFSTEP]\nexact mul_mem_center ha (inv_mem_center\u2080 hb)\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Subsemigroup.Center", "llama_tokens": 1451, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.7341195269001831, "lm_q1q2_score": 0.5563116453277608}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nx : R\n\u22a2 eval\u2082 f (\u2191f x) (X - \u2191C x) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet leval : R[X] \u2192\u2097[R] A := (aeval x).toLinearMap\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet D : \u2115 \u2192 Submodule R A := fun n => (degreeLE R n).map leval\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet M := WellFounded.min (isNoetherian_iff_wellFounded.1 H) (Set.range D) \u27e8_, \u27e80, rfl\u27e9\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave HM : M \u2208 Set.range D := WellFounded.min_mem _ _ _\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nHM : M \u2208 Set.range D\n\u22a2 IsIntegral R x\n[PROOFSTEP]\ncases' HM with N HN\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave HM : \u00acM < D (N + 1) := WellFounded.not_lt_min (isNoetherian_iff_wellFounded.1 H) (Set.range D) _ \u27e8N + 1, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acM < D (N + 1)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrw [\u2190 HN] at HM \n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acD N < D (N + 1)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave HN2 : D (N + 1) \u2264 D N :=\n  _root_.by_contradiction fun H =>\n    HM (lt_of_le_not_le (map_mono (degreeLE_mono (WithBot.coe_le_coe.2 (Nat.le_succ N)))) H)\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acD N < D (N + 1)\nHN2 : D (N + 1) \u2264 D N\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave HN3 : leval (X ^ (N + 1)) \u2208 D N := HN2 (mem_map_of_mem (mem_degreeLE.2 (degree_X_pow_le _)))\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acD N < D (N + 1)\nHN2 : D (N + 1) \u2264 D N\nHN3 : \u2191leval (X ^ (N + 1)) \u2208 D N\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrcases HN3 with \u27e8p, hdp, hpe\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acD N < D (N + 1)\nHN2 : D (N + 1) \u2264 D N\np : R[X]\nhdp : p \u2208 \u2191(degreeLE R \u2191N)\nhpe : \u2191leval p = \u2191leval (X ^ (N + 1))\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrefine' \u27e8X ^ (N + 1) - p, monic_X_pow_sub (mem_degreeLE.1 hdp), _\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acD N < D (N + 1)\nHN2 : D (N + 1) \u2264 D N\np : R[X]\nhdp : p \u2208 \u2191(degreeLE R \u2191N)\nhpe : \u2191leval p = \u2191leval (X ^ (N + 1))\n\u22a2 eval\u2082 (algebraMap R A) x (X ^ (N + 1) - p) = 0\n[PROOFSTEP]\nshow leval (X ^ (N + 1) - p) = 0\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\nf : R \u2192+* S\ninst\u271d : Algebra R A\nH : IsNoetherian R A\nx : A\nleval : R[X] \u2192\u2097[R] A := AlgHom.toLinearMap (aeval x)\nD : \u2115 \u2192 Submodule R A := fun n => Submodule.map leval (degreeLE R \u2191n)\nM : Submodule R A := WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range D) (_ : \u2203 x, x \u2208 Set.range D)\nN : \u2115\nHN : D N = M\nHM : \u00acD N < D (N + 1)\nHN2 : D (N + 1) \u2264 D N\np : R[X]\nhdp : p \u2208 \u2191(degreeLE R \u2191N)\nhpe : \u2191leval p = \u2191leval (X ^ (N + 1))\n\u22a2 \u2191leval (X ^ (N + 1) - p) = 0\n[PROOFSTEP]\nrw [LinearMap.map_sub, hpe, sub_self]\n[GOAL]\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nsuffices IsIntegral R (show S from \u27e8x, hx\u27e9) by\n  rcases this with \u27e8p, hpm, hpx\u27e9\n  replace hpx := congr_arg S.val hpx\n  refine' \u27e8p, hpm, Eq.trans _ hpx\u27e9\n  simp only [aeval_def, eval\u2082, sum_def]\n  rw [S.val.map_sum]\n  refine' Finset.sum_congr rfl fun n _hn => _\n  rw [S.val.map_mul, S.val.map_pow, S.val.commutes, S.val_apply, Subtype.coe_mk]\n[GOAL]\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\nthis :\n  IsIntegral R\n    (let_fun this := { val := x, property := hx };\n    this)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrcases this with \u27e8p, hpm, hpx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  eval\u2082 (algebraMap R { x // x \u2208 S })\n      (let_fun this := { val := x, property := hx };\n      this)\n      p =\n    0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nreplace hpx := congr_arg S.val hpx\n[GOAL]\ncase intro.intro\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrefine' \u27e8p, hpm, Eq.trans _ hpx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 eval\u2082 (algebraMap R A) x p =\n    \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p)\n[PROOFSTEP]\nsimp only [aeval_def, eval\u2082, sum_def]\n[GOAL]\ncase intro.intro\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2191(Subalgebra.val S)\n      (\u2211 n in support p, \u2191(algebraMap R { x // x \u2208 S }) (coeff p n) * { val := x, property := hx } ^ n)\n[PROOFSTEP]\nrw [S.val.map_sum]\n[GOAL]\ncase intro.intro\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2211 x_1 in support p,\n      \u2191(Subalgebra.val S) (\u2191(algebraMap R { x // x \u2208 S }) (coeff p x_1) * { val := x, property := hx } ^ x_1)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun n _hn => _\n[GOAL]\ncase intro.intro\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\nn : \u2115\n_hn : n \u2208 support p\n\u22a2 \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2191(Subalgebra.val S) (\u2191(algebraMap R { x // x \u2208 S }) (coeff p n) * { val := x, property := hx } ^ n)\n[PROOFSTEP]\nrw [S.val.map_mul, S.val.map_pow, S.val.commutes, S.val_apply, Subtype.coe_mk]\n[GOAL]\nR : Type u_1\nS\u271d : Type u_2\nA : Type u_3\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\n\u22a2 IsIntegral R\n    (let_fun this := { val := x, property := hx };\n    this)\n[PROOFSTEP]\nrefine' isIntegral_of_noetherian H \u27e8x, hx\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing A\ninst\u271d\u00b9\u00b2 : CommRing B\u271d\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nB : Type u_5\nC : Type u_6\nF : Type u_7\ninst\u271d\u2078 : Ring B\ninst\u271d\u2077 : Ring C\ninst\u271d\u2076 : Algebra R B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Algebra R C\ninst\u271d\u00b3 : IsScalarTower R A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower R A C\nb : B\ninst\u271d : AlgHomClass F A B C\nf : F\nhb : IsIntegral R b\n\u22a2 IsIntegral R (\u2191f b)\n[PROOFSTEP]\nobtain \u27e8P, hP\u27e9 := hb\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing A\ninst\u271d\u00b9\u00b2 : CommRing B\u271d\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nB : Type u_5\nC : Type u_6\nF : Type u_7\ninst\u271d\u2078 : Ring B\ninst\u271d\u2077 : Ring C\ninst\u271d\u2076 : Algebra R B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Algebra R C\ninst\u271d\u00b3 : IsScalarTower R A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower R A C\nb : B\ninst\u271d : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P \u2227 eval\u2082 (algebraMap R B) b P = 0\n\u22a2 IsIntegral R (\u2191f b)\n[PROOFSTEP]\nrefine' \u27e8P, hP.1, _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing A\ninst\u271d\u00b9\u00b2 : CommRing B\u271d\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nB : Type u_5\nC : Type u_6\nF : Type u_7\ninst\u271d\u2078 : Ring B\ninst\u271d\u2077 : Ring C\ninst\u271d\u2076 : Algebra R B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Algebra R C\ninst\u271d\u00b3 : IsScalarTower R A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower R A C\nb : B\ninst\u271d : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P \u2227 eval\u2082 (algebraMap R B) b P = 0\n\u22a2 eval\u2082 (algebraMap R ((fun x => C) b)) (\u2191f b) P = 0\n[PROOFSTEP]\nrw [\u2190 aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebraMap R A)) by simp, aeval_algHom_apply,\n  aeval_map_algebraMap, aeval_def, hP.2, _root_.map_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing A\ninst\u271d\u00b9\u00b2 : CommRing B\u271d\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R A\ninst\u271d\u2079 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nB : Type u_5\nC : Type u_6\nF : Type u_7\ninst\u271d\u2078 : Ring B\ninst\u271d\u2077 : Ring C\ninst\u271d\u2076 : Algebra R B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : Algebra R C\ninst\u271d\u00b3 : IsScalarTower R A B\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : IsScalarTower R A C\nb : B\ninst\u271d : AlgHomClass F A B C\nf : F\nP : R[X]\nhP : Monic P \u2227 eval\u2082 (algebraMap R B) b P = 0\n\u22a2 \u2191(aeval (\u2191f b)) P = \u2191(aeval (\u2191f b)) (Polynomial.map (algebraMap R A) P)\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u271d\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : CommRing S\u271d\ninst\u271d\u2077 : Algebra R\u271d A\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\nR : Type u_5\nS : Type u_6\nT : Type u_7\nU : Type u_8\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : CommRing U\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra T U\n\u03c6 : R \u2192+* T\n\u03c8 : S \u2192+* U\nh : RingHom.comp (algebraMap T U) \u03c6 = RingHom.comp \u03c8 (algebraMap R S)\na : S\nha : IsIntegral R a\n\u22a2 IsIntegral T (\u2191\u03c8 a)\n[PROOFSTEP]\nrw [IsIntegral, RingHom.IsIntegralElem] at ha \u22a2\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u271d\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : CommRing S\u271d\ninst\u271d\u2077 : Algebra R\u271d A\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\nR : Type u_5\nS : Type u_6\nT : Type u_7\nU : Type u_8\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : CommRing U\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra T U\n\u03c6 : R \u2192+* T\n\u03c8 : S \u2192+* U\nh : RingHom.comp (algebraMap T U) \u03c6 = RingHom.comp \u03c8 (algebraMap R S)\na : S\nha : \u2203 p, Monic p \u2227 eval\u2082 (algebraMap R S) a p = 0\n\u22a2 \u2203 p, Monic p \u2227 eval\u2082 (algebraMap T ((fun x => U) a)) (\u2191\u03c8 a) p = 0\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := ha\n[GOAL]\ncase intro\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u271d\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : CommRing S\u271d\ninst\u271d\u2077 : Algebra R\u271d A\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\nR : Type u_5\nS : Type u_6\nT : Type u_7\nU : Type u_8\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : CommRing U\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra T U\n\u03c6 : R \u2192+* T\n\u03c8 : S \u2192+* U\nh : RingHom.comp (algebraMap T U) \u03c6 = RingHom.comp \u03c8 (algebraMap R S)\na : S\np : R[X]\nhp : Monic p \u2227 eval\u2082 (algebraMap R S) a p = 0\n\u22a2 \u2203 p, Monic p \u2227 eval\u2082 (algebraMap T ((fun x => U) a)) (\u2191\u03c8 a) p = 0\n[PROOFSTEP]\nrefine' \u27e8p.map \u03c6, hp.left.map _, _\u27e9\n[GOAL]\ncase intro\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : CommRing R\u271d\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : CommRing S\u271d\ninst\u271d\u2077 : Algebra R\u271d A\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\nR : Type u_5\nS : Type u_6\nT : Type u_7\nU : Type u_8\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : CommRing T\ninst\u271d\u00b2 : CommRing U\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra T U\n\u03c6 : R \u2192+* T\n\u03c8 : S \u2192+* U\nh : RingHom.comp (algebraMap T U) \u03c6 = RingHom.comp \u03c8 (algebraMap R S)\na : S\np : R[X]\nhp : Monic p \u2227 eval\u2082 (algebraMap R S) a p = 0\n\u22a2 eval\u2082 (algebraMap T ((fun x => U) a)) (\u2191\u03c8 a) (Polynomial.map \u03c6 p) = 0\n[PROOFSTEP]\nrw [\u2190 eval_map, map_map, h, \u2190 map_map, eval_map, eval\u2082_at_apply, eval_map, hp.right, RingHom.map_zero]\n[GOAL]\nR : Type u_1\nA\u271d : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing A\u271d\ninst\u271d\u2077 : CommRing B\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nA : Type u_5\nB : Type u_6\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nx : A\n\u22a2 IsIntegral R (\u2191f x) \u2194 IsIntegral R x\n[PROOFSTEP]\nrefine' \u27e8_, map_isIntegral f\u27e9\n[GOAL]\nR : Type u_1\nA\u271d : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing A\u271d\ninst\u271d\u2077 : CommRing B\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nA : Type u_5\nB : Type u_6\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nx : A\n\u22a2 IsIntegral R (\u2191f x) \u2192 IsIntegral R x\n[PROOFSTEP]\nrintro \u27e8p, hp, hx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA\u271d : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing A\u271d\ninst\u271d\u2077 : CommRing B\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nA : Type u_5\nB : Type u_6\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nx : A\np : R[X]\nhp : Monic p\nhx : eval\u2082 (algebraMap R ((fun x => B) x)) (\u2191f x) p = 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nuse p, hp\n[GOAL]\ncase right\nR : Type u_1\nA\u271d : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing A\u271d\ninst\u271d\u2077 : CommRing B\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nA : Type u_5\nB : Type u_6\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nx : A\np : R[X]\nhp : Monic p\nhx : eval\u2082 (algebraMap R ((fun x => B) x)) (\u2191f x) p = 0\n\u22a2 eval\u2082 (algebraMap R A) x p = 0\n[PROOFSTEP]\nrwa [\u2190 f.comp_algebraMap, \u2190 AlgHom.coe_toRingHom, \u2190 Polynomial.hom_eval\u2082, AlgHom.coe_toRingHom, map_eq_zero_iff f hf] at\n  hx \n[GOAL]\nR : Type u_1\nA\u271d : Type u_2\nB\u271d : Type u_3\nS : Type u_4\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing A\u271d\ninst\u271d\u2077 : CommRing B\u271d\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Algebra R A\u271d\ninst\u271d\u2074 : Algebra R B\u271d\nf\u271d : R \u2192+* S\nA : Type u_5\nB : Type u_6\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2243\u2090[R] B\nx : A\nh : IsIntegral R (\u2191f x)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nsimpa using map_isIntegral f.symm.toAlgHom h\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : B\nhx : IsIntegral R x\np : R[X]\nhp : Monic p\nhpx : eval\u2082 (algebraMap R B) x p = 0\n\u22a2 eval\u2082 (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0\n[PROOFSTEP]\nrw [\u2190 aeval_def, aeval_map_algebraMap, aeval_def, hpx]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : A\nh : IsIntegral R x\n\u22a2 IsIntegral R (\u2191(_root_.algebraMap A B) x)\n[PROOFSTEP]\nrcases h with \u27e8f, hf, hx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R B\nf\u271d : R \u2192+* S\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : A\nf : R[X]\nhf : Monic f\nhx : eval\u2082 (_root_.algebraMap R A) x f = 0\n\u22a2 IsIntegral R (\u2191(_root_.algebraMap A B) x)\n[PROOFSTEP]\nuse f, hf\n[GOAL]\ncase right\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R B\nf\u271d : R \u2192+* S\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nx : A\nf : R[X]\nhf : Monic f\nhx : eval\u2082 (_root_.algebraMap R A) x f = 0\n\u22a2 eval\u2082 (_root_.algebraMap R ((fun x => B) x)) (\u2191(_root_.algebraMap A B) x) f = 0\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq R A B, \u2190 hom_eval\u2082, hx, RingHom.map_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\n\u22a2 IsIntegral R r \u2194 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\n\u22a2 IsIntegral R r \u2192 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n[PROOFSTEP]\nintro hr\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\n\u22a2 (\u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r) \u2192 IsIntegral R r\n[PROOFSTEP]\nintro hr\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\nhr : IsIntegral R r\n\u22a2 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n[PROOFSTEP]\nrcases hr with \u27e8p, hmp, hpr\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\np : R[X]\nhmp : Monic p\nhpr : eval\u2082 (algebraMap R A) r p = 0\n\u22a2 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n[PROOFSTEP]\nrefine' \u27e8_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, _\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\np : R[X]\nhmp : Monic p\nhpr : eval\u2082 (algebraMap R A) r p = 0\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 Subring.closure \u2191(frange p) } A) r (restriction p) = 0\n[PROOFSTEP]\nrw [\u2190 aeval_def, \u2190 aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\nhr : \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n\u22a2 IsIntegral R r\n[PROOFSTEP]\nrcases hr with \u27e8s, _, hsr\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\ns : Set R\nleft\u271d : Set.Finite s\nhsr : IsIntegral { x // x \u2208 Subring.closure s } r\n\u22a2 IsIntegral R r\n[PROOFSTEP]\nexact isIntegral_ofSubring _ hsr\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nhx : IsIntegral R x\n\u22a2 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nrcases hx with \u27e8f, hfm, hfx\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\n\u22a2 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nexists Finset.image ((\u00b7 ^ \u00b7) x) (Finset.range (natDegree f + 1))\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\n\u22a2 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))) =\n    \u2191Subalgebra.toSubmodule (Algebra.adjoin R {x})\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\n\u22a2 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))) \u2264\n    \u2191Subalgebra.toSubmodule (Algebra.adjoin R {x})\n[PROOFSTEP]\nrw [span_le]\n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\n\u22a2 \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))) \u2286\n    \u2191(\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\ns : A\nhs : s \u2208 \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n\u22a2 s \u2208 \u2191(\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nrw [Finset.mem_coe] at hs \n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\ns : A\nhs : s \u2208 Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n\u22a2 s \u2208 \u2191(\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nrcases Finset.mem_image.1 hs with \u27e8k, hk, rfl\u27e9\n[GOAL]\ncase intro.intro.a.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\nk : \u2115\nhk : k \u2208 Finset.range (natDegree f + 1)\nhs : (fun x x_1 => x ^ x_1) x k \u2208 Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n\u22a2 (fun x x_1 => x ^ x_1) x k \u2208 \u2191(\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nclear hk\n[GOAL]\ncase intro.intro.a.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\nk : \u2115\nhs : (fun x x_1 => x ^ x_1) x k \u2208 Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n\u22a2 (fun x x_1 => x ^ x_1) x k \u2208 \u2191(\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}))\n[PROOFSTEP]\nexact (Algebra.adjoin R { x }).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k\n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\n\u22a2 \u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}) \u2264\n    span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\nr : A\nhr : r \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin R {x})\n\u22a2 r \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nchange r \u2208 Algebra.adjoin R ({ x } : Set A) at hr \n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\nr : A\nhr : r \u2208 Algebra.adjoin R {x}\n\u22a2 r \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [Algebra.adjoin_singleton_eq_range_aeval] at hr \n[GOAL]\ncase intro.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\nr : A\nhr : r \u2208 AlgHom.range (aeval x)\n\u22a2 r \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrcases(aeval x).mem_range.mp hr with \u27e8p, rfl\u27e9\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\n\u22a2 \u2191(aeval x) p \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [\u2190 modByMonic_add_div p hfm]\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : eval\u2082 (algebraMap R A) x f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\n\u22a2 \u2191(aeval x) (p %\u2098 f + f * (p /\u2098 f)) \u2208\n    span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [\u2190 aeval_def] at hfx \n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\n\u22a2 \u2191(aeval x) (p %\u2098 f + f * (p /\u2098 f)) \u2208\n    span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [AlgHom.map_add, AlgHom.map_mul, hfx, zero_mul, add_zero]\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\n\u22a2 \u2191(aeval x) (p %\u2098 f) \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nhave : degree (p %\u2098 f) \u2264 degree f := degree_modByMonic_le p hfm\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nthis : degree (p %\u2098 f) \u2264 degree f\n\u22a2 \u2191(aeval x) (p %\u2098 f) \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\ngeneralize p %\u2098 f = q at this \u22a2\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\n\u22a2 \u2191(aeval x) q \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [\u2190 sum_C_mul_X_pow_eq q, aeval_def, eval\u2082_sum, sum_def]\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\n\u22a2 \u2211 n in support q, eval\u2082 (algebraMap R A) x (\u2191C (coeff q n) * X ^ n) \u2208\n    span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrefine' sum_mem fun k hkq => _\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\n\u22a2 eval\u2082 (algebraMap R A) x (\u2191C (coeff q k) * X ^ k) \u2208\n    span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [eval\u2082_mul, eval\u2082_C, eval\u2082_pow, eval\u2082_X, \u2190 Algebra.smul_def]\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\n\u22a2 coeff q k \u2022 x ^ k \u2208 span R \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrefine' smul_mem _ _ (subset_span _)\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\n\u22a2 x ^ k \u2208 \u2191(Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1)))\n[PROOFSTEP]\nrw [Finset.mem_coe]\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\n\u22a2 x ^ k \u2208 Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (natDegree f + 1))\n[PROOFSTEP]\nrefine' Finset.mem_image.2 \u27e8_, _, rfl\u27e9\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\n\u22a2 k \u2208 Finset.range (natDegree f + 1)\n[PROOFSTEP]\nrw [Finset.mem_range, Nat.lt_succ_iff]\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\n\u22a2 k \u2264 natDegree f\n[PROOFSTEP]\nrefine' le_of_not_lt fun hk => _\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : degree q \u2264 degree f\nk : \u2115\nhkq : k \u2208 support q\nhk : natDegree f < k\n\u22a2 False\n[PROOFSTEP]\nrw [degree_le_iff_coeff_zero] at this \n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : \u2200 (m : \u2115), degree f < \u2191m \u2192 coeff q m = 0\nk : \u2115\nhkq : k \u2208 support q\nhk : natDegree f < k\n\u22a2 False\n[PROOFSTEP]\nrw [mem_support_iff] at hkq \n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : \u2200 (m : \u2115), degree f < \u2191m \u2192 coeff q m = 0\nk : \u2115\nhkq : coeff q k \u2260 0\nhk : natDegree f < k\n\u22a2 False\n[PROOFSTEP]\napply hkq\n[GOAL]\ncase intro.intro.a.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : \u2200 (m : \u2115), degree f < \u2191m \u2192 coeff q m = 0\nk : \u2115\nhkq : coeff q k \u2260 0\nhk : natDegree f < k\n\u22a2 coeff q k = 0\n[PROOFSTEP]\napply this\n[GOAL]\ncase intro.intro.a.intro.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nx : A\nf : R[X]\nhfm : Monic f\nhfx : \u2191(aeval x) f = 0\np : R[X]\nhr : \u2191(aeval x) p \u2208 AlgHom.range (aeval x)\nq : R[X]\nthis : \u2200 (m : \u2115), degree f < \u2191m \u2192 coeff q m = 0\nk : \u2115\nhkq : coeff q k \u2260 0\nhk : natDegree f < k\n\u22a2 degree f < \u2191k\n[PROOFSTEP]\nexact lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 hk)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\ns : Set A\nhfs : Set.Finite s\nhis : \u2200 (x : A), x \u2208 s \u2192 IsIntegral R x\nx\u271d : \u2200 (x : A), x \u2208 \u2205 \u2192 IsIntegral R x\nx : A\n\u22a2 x \u2208 span R \u2191{1} \u2194 x \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin R \u2205)\n[PROOFSTEP]\nerw [Algebra.adjoin_empty, Finset.coe_singleton, \u2190 one_eq_span, one_eq_range, LinearMap.mem_range, Algebra.mem_bot]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\ns : Set A\nhfs : Set.Finite s\nhis : \u2200 (x : A), x \u2208 s \u2192 IsIntegral R x\nx\u271d : \u2200 (x : A), x \u2208 \u2205 \u2192 IsIntegral R x\nx : A\n\u22a2 (\u2203 y, \u2191(Algebra.linearMap R A) y = x) \u2194 x \u2208 Set.range \u2191(algebraMap R A)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\ns\u271d : Set A\nhfs : Set.Finite s\u271d\nhis\u271d : \u2200 (x : A), x \u2208 s\u271d \u2192 IsIntegral R x\na : A\ns : Set A\nx\u271d\u00b9 : \u00aca \u2208 s\nx\u271d : Set.Finite s\nih : (\u2200 (x : A), x \u2208 s \u2192 IsIntegral R x) \u2192 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : \u2200 (x : A), x \u2208 insert a s \u2192 IsIntegral R x\n\u22a2 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R (insert a s)))\n[PROOFSTEP]\nrw [\u2190 Set.union_singleton, Algebra.adjoin_union_coe_submodule]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\ns\u271d : Set A\nhfs : Set.Finite s\u271d\nhis\u271d : \u2200 (x : A), x \u2208 s\u271d \u2192 IsIntegral R x\na : A\ns : Set A\nx\u271d\u00b9 : \u00aca \u2208 s\nx\u271d : Set.Finite s\nih : (\u2200 (x : A), x \u2208 s \u2192 IsIntegral R x) \u2192 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R s))\nhis : \u2200 (x : A), x \u2208 insert a s \u2192 IsIntegral R x\n\u22a2 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R s) * \u2191Subalgebra.toSubmodule (Algebra.adjoin R {a}))\n[PROOFSTEP]\nexact\n  FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)\n    (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nHS : FG (\u2191Subalgebra.toSubmodule S)\nx : A\nhx : x \u2208 S\n\u22a2 IsIntegral R x\n[PROOFSTEP]\ncases' HS with y hy\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nobtain \u27e8lx, hlx1, hlx2\u27e9 : \u2203 (l : A \u2192\u2080 R), l \u2208 Finsupp.supported R R \u2191y \u2227 (Finsupp.total A A R id) l = x := by\n  rwa [\u2190 @Finsupp.mem_span_image_iff_total A A R _ _ _ id (\u2191y) x, Set.image_id (y : Set A), hy]\n    -- Note that `y \u2286 S`.\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\n\u22a2 \u2203 l, l \u2208 Finsupp.supported R R \u2191y \u2227 \u2191(Finsupp.total A A R id) l = x\n[PROOFSTEP]\nrwa [\u2190 @Finsupp.mem_span_image_iff_total A A R _ _ _ id (\u2191y) x, Set.image_id (y : Set A), hy]\n  -- Note that `y \u2286 S`.\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave hyS : \u2200 {p}, p \u2208 y \u2192 p \u2208 S := fun {p} hp =>\n  show p \u2208 Subalgebra.toSubmodule S by\n    rw [\u2190 hy]\n    exact subset_span hp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\np : A\nhp : p \u2208 y\n\u22a2 p \u2208 \u2191Subalgebra.toSubmodule S\n[PROOFSTEP]\nrw [\u2190 hy]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\np : A\nhp : p \u2208 y\n\u22a2 p \u2208 span R \u2191y\n[PROOFSTEP]\nexact subset_span hp\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave : \u2200 jk : (y \u00d7\u02e2 y : Finset (A \u00d7 A)), jk.1.1 * jk.1.2 \u2208 (Subalgebra.toSubmodule S) := fun jk =>\n  S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nthis : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), (\u2191jk).fst * (\u2191jk).snd \u2208 \u2191Subalgebra.toSubmodule S\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrw [\u2190 hy, \u2190 Set.image_id (y : Set A)] at this \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nthis : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), (\u2191jk).fst * (\u2191jk).snd \u2208 span R (id '' \u2191y)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nsimp only [Finsupp.mem_span_image_iff_total] at this \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nthis :\n  \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2203 l, l \u2208 Finsupp.supported R R \u2191y \u2227 \u2191(Finsupp.total A A R id) l = (\u2191jk).fst * (\u2191jk).snd\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nchoose ly hly1 hly2 using this\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet S\u2080 : Subring R :=\n  Subring.closure\n    \u2191(lx.frange \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\n        -- It suffices to prove that `x` is integral over `S\u2080`.\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrefine' isIntegral_ofSubring S\u2080 _\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nletI : CommRing S\u2080 := SubringClass.toCommRing S\u2080\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nletI : Algebra S\u2080 A := Algebra.ofSubring S\u2080\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nhave : span S\u2080 (insert 1 \u2191y : Set A) * span S\u2080 (insert 1 \u2191y : Set A) \u2264 span S\u2080 (insert 1 \u2191y : Set A) :=\n  by\n  rw [span_mul_span]\n  refine' span_le.2 fun z hz => _\n  rcases Set.mem_mul.1 hz with \u27e8p, q, rfl | hp, hq, rfl\u27e9\n  \u00b7 rw [one_mul]\n    exact subset_span hq\n  rcases hq with (rfl | hq)\n  \u00b7 rw [mul_one]\n    exact subset_span (Or.inr hp)\n  erw [\u2190 hly2 \u27e8(p, q), Finset.mem_product.2 \u27e8hp, hq\u27e9\u27e9]\n  rw [Finsupp.total_apply, Finsupp.sum]\n  refine' (span S\u2080 (insert 1 \u2191y : Set A)).sum_mem fun t ht => _\n  have : ly \u27e8(p, q), Finset.mem_product.2 \u27e8hp, hq\u27e9\u27e9 t \u2208 S\u2080 :=\n    Subring.subset_closure\n      (Finset.mem_union_right _ <|\n        Finset.mem_biUnion.2\n          \u27e8\u27e8(p, q), Finset.mem_product.2 \u27e8hp, hq\u27e9\u27e9, Finset.mem_univ _,\n            Finsupp.mem_frange.2 \u27e8Finsupp.mem_support_iff.1 ht, _, rfl\u27e9\u27e9)\n  change (\u27e8_, this\u27e9 : S\u2080) \u2022 t \u2208 _\n  exact\n    smul_mem _ _\n      (subset_span <| Or.inr <| hly1 _ ht)\n        -- Hence this span is a subring. Call this subring `S\u2081`.\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\n\u22a2 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nrw [span_mul_span]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\n\u22a2 span { x // x \u2208 S\u2080 } (insert 1 \u2191y * insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nrefine' span_le.2 fun z hz => _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nz : A\nhz : z \u2208 insert 1 \u2191y * insert 1 \u2191y\n\u22a2 z \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nrcases Set.mem_mul.1 hz with \u27e8p, q, rfl | hp, hq, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.inl.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nq : A\nhq : q \u2208 insert 1 \u2191y\nhz : 1 * q \u2208 insert 1 \u2191y * insert 1 \u2191y\n\u22a2 1 * q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\ncase intro.intro.intro.inl.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nq : A\nhq : q \u2208 insert 1 \u2191y\nhz : 1 * q \u2208 insert 1 \u2191y * insert 1 \u2191y\n\u22a2 q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nexact subset_span hq\n[GOAL]\ncase intro.intro.intro.inr.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhq : q \u2208 insert 1 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\n\u22a2 p * q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nrcases hq with (rfl | hq)\n[GOAL]\ncase intro.intro.intro.inr.intro.inl\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np : A\nhp : p \u2208 \u2191y\nhz : p * 1 \u2208 insert 1 \u2191y * insert 1 \u2191y\n\u22a2 p * 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase intro.intro.intro.inr.intro.inl\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np : A\nhp : p \u2208 \u2191y\nhz : p * 1 \u2208 insert 1 \u2191y * insert 1 \u2191y\n\u22a2 p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nexact subset_span (Or.inr hp)\n[GOAL]\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\nhq : q \u2208 \u2191y\n\u22a2 p * q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nerw [\u2190 hly2 \u27e8(p, q), Finset.mem_product.2 \u27e8hp, hq\u27e9\u27e9]\n[GOAL]\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\nhq : q \u2208 \u2191y\n\u22a2 \u2191(Finsupp.total A A R id) (ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) \u2208\n    \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nrw [Finsupp.total_apply, Finsupp.sum]\n[GOAL]\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\nhq : q \u2208 \u2191y\n\u22a2 \u2211 a in (ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }).support,\n      \u2191(ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) a \u2022 id a \u2208\n    \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))\n[PROOFSTEP]\nrefine' (span S\u2080 (insert 1 \u2191y : Set A)).sum_mem fun t ht => _\n[GOAL]\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\nhq : q \u2208 \u2191y\nt : A\nht : t \u2208 (ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }).support\n\u22a2 \u2191(ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) t \u2022 id t \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nhave : ly \u27e8(p, q), Finset.mem_product.2 \u27e8hp, hq\u27e9\u27e9 t \u2208 S\u2080 :=\n  Subring.subset_closure\n    (Finset.mem_union_right _ <|\n      Finset.mem_biUnion.2\n        \u27e8\u27e8(p, q), Finset.mem_product.2 \u27e8hp, hq\u27e9\u27e9, Finset.mem_univ _,\n          Finsupp.mem_frange.2 \u27e8Finsupp.mem_support_iff.1 ht, _, rfl\u27e9\u27e9)\n[GOAL]\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\nhq : q \u2208 \u2191y\nt : A\nht : t \u2208 (ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }).support\nthis : \u2191(ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) t \u2208 S\u2080\n\u22a2 \u2191(ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) t \u2022 id t \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nchange (\u27e8_, this\u27e9 : S\u2080) \u2022 t \u2208 _\n[GOAL]\ncase intro.intro.intro.inr.intro.inr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\np q : A\nhp : p \u2208 \u2191y\nhz : p * q \u2208 insert 1 \u2191y * insert 1 \u2191y\nhq : q \u2208 \u2191y\nt : A\nht : t \u2208 (ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }).support\nthis : \u2191(ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) t \u2208 S\u2080\n\u22a2 { val := \u2191(ly { val := (p, q), property := (_ : (p, q) \u2208 y \u00d7\u02e2 y) }) t, property := this } \u2022 t \u2208\n    span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nexact\n  smul_mem _ _\n    (subset_span <| Or.inr <| hly1 _ ht)\n      -- Hence this span is a subring. Call this subring `S\u2081`.\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nlet S\u2081 : Subring A :=\n  { carrier := span S\u2080 (insert 1 \u2191y : Set A)\n    one_mem' := subset_span <| Or.inl rfl\n    mul_mem' := fun {p q} hp hq => this <| mul_mem_mul hp hq\n    zero_mem' := (span S\u2080 (insert 1 \u2191y : Set A)).zero_mem\n    add_mem' := fun {_ _} => (span S\u2080 (insert 1 \u2191y : Set A)).add_mem\n    neg_mem' := fun {_} => (span S\u2080 (insert 1 \u2191y : Set A)).neg_mem }\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nhave : S\u2081 = Subalgebra.toSubring (Algebra.adjoin S\u2080 (\u2191y : Set A)) :=\n  by\n  ext z\n  suffices z \u2208 span (\u21a5S\u2080) (insert 1 \u2191y : Set A) \u2194 z \u2208 Subalgebra.toSubmodule (Algebra.adjoin (\u21a5S\u2080) (y : Set A)) by simpa\n  constructor <;> intro hz\n  \u00b7 exact (span_le.2 (Set.insert_subset_iff.2 \u27e8(Algebra.adjoin S\u2080 (y : Set A)).one_mem, Algebra.subset_adjoin\u27e9)) hz\n  \u00b7 rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz \n    suffices Subring.closure (Set.range (algebraMap (\u21a5S\u2080) A) \u222a \u2191y) \u2264 S\u2081 by exact this hz\n    refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)\n    rw [Set.range_subset_iff]\n    intro y'\n    rw [Algebra.algebraMap_eq_smul_one]\n    exact smul_mem (span S\u2080 (insert (1 : A) (y : Set A))) y' (subset_span (Or.inl rfl))\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\n\u22a2 S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\next z\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\n\u22a2 z \u2208 S\u2081 \u2194 z \u2208 Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nsuffices z \u2208 span (\u21a5S\u2080) (insert 1 \u2191y : Set A) \u2194 z \u2208 Subalgebra.toSubmodule (Algebra.adjoin (\u21a5S\u2080) (y : Set A)) by simpa\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b2 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b9 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nthis : z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2194 z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n\u22a2 z \u2208 S\u2081 \u2194 z \u2208 Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\n\u22a2 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2194 z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\n\u22a2 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nintro hz\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\n\u22a2 z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y) \u2192 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nintro hz\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n\u22a2 z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nexact (span_le.2 (Set.insert_subset_iff.2 \u27e8(Algebra.adjoin S\u2080 (y : Set A)).one_mem, Algebra.subset_adjoin\u27e9)) hz\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n\u22a2 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nrw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz \n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\n\u22a2 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nsuffices Subring.closure (Set.range (algebraMap (\u21a5S\u2080) A) \u222a \u2191y) \u2264 S\u2081 by exact this hz\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b2 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b9 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\nthis : Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y) \u2264 S\u2081\n\u22a2 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\n[PROOFSTEP]\nexact this hz\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\n\u22a2 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y) \u2264 S\u2081\n[PROOFSTEP]\nrefine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\n\u22a2 Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u2286 \u2191S\u2081\n[PROOFSTEP]\nrw [Set.range_subset_iff]\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\n\u22a2 \u2200 (y : { x // x \u2208 S\u2080 }), \u2191(algebraMap { x // x \u2208 S\u2080 } A) y \u2208 \u2191S\u2081\n[PROOFSTEP]\nintro y'\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\ny' : { x // x \u2208 S\u2080 }\n\u22a2 \u2191(algebraMap { x // x \u2208 S\u2080 } A) y' \u2208 \u2191S\u2081\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one]\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b9 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nz : A\nhz : z \u2208 Subring.closure (Set.range \u2191(algebraMap { x // x \u2208 S\u2080 } A) \u222a \u2191y)\ny' : { x // x \u2208 S\u2080 }\n\u22a2 y' \u2022 1 \u2208 \u2191S\u2081\n[PROOFSTEP]\nexact smul_mem (span S\u2080 (insert (1 : A) (y : Set A))) y' (subset_span (Or.inl rfl))\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b2 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b9 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nhave foo : \u2200 z, z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin (\u21a5S\u2080) (y : Set A)\n[GOAL]\ncase foo\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b2 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b9 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n\u22a2 \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b2 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b9 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b2 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b9 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nhaveI : IsNoetherianRing S\u2080 := is_noetherian_subring_closure _ (Finset.finite_toSet _)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\n\u22a2 IsIntegral { x // x \u2208 S\u2080 } x\n[PROOFSTEP]\nrefine'\n  isIntegral_of_submodule_noetherian (Algebra.adjoin S\u2080 \u2191y)\n    (isNoetherian_of_fg_of_noetherian _\n      \u27e8insert 1 y, by\n        rw [Finset.coe_insert]\n        ext z\n        simp only [Finset.coe_sort_coe, Finset.univ_eq_attach, Finset.mem_coe, Subalgebra.mem_toSubmodule]\n        convert foo z\u27e9)\n    _ _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\n\u22a2 span { x // x \u2208 S\u2080 } \u2191(insert 1 y) = \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nrw [Finset.coe_insert]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\n\u22a2 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) = \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\next z\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\nz : A\n\u22a2 z \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2194 z \u2208 \u2191Subalgebra.toSubmodule (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\n[PROOFSTEP]\nsimp only [Finset.coe_sort_coe, Finset.univ_eq_attach, Finset.mem_coe, Subalgebra.mem_toSubmodule]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\nz : A\n\u22a2 z \u2208\n      span\n        { x //\n          x \u2208 Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion (Finset.attach (y \u00d7\u02e2 y)) (Finsupp.frange \u2218 ly)) }\n        (insert 1 \u2191y) \u2194\n    z \u2208\n      Algebra.adjoin\n        { x //\n          x \u2208 Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion (Finset.attach (y \u00d7\u02e2 y)) (Finsupp.frange \u2218 ly)) }\n        \u2191y\n[PROOFSTEP]\nconvert foo z\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\n\u22a2 x \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n[PROOFSTEP]\nrw [\u2190 hlx2, Finsupp.total_apply, Finsupp.sum]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\n\u22a2 \u2211 a in lx.support, \u2191lx a \u2022 id a \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n[PROOFSTEP]\nrefine' Subalgebra.sum_mem _ fun r hr => _\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u00b3 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b2 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b9 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis : IsNoetherianRing { x // x \u2208 S\u2080 }\nr : A\nhr : r \u2208 lx.support\n\u22a2 \u2191lx r \u2022 id r \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n[PROOFSTEP]\nhave : lx r \u2208 S\u2080 := Subring.subset_closure (Finset.mem_union_left _ (Finset.mem_image_of_mem _ hr))\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u2074 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b3 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b2 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d\u00b9 : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis\u271d : IsNoetherianRing { x // x \u2208 S\u2080 }\nr : A\nhr : r \u2208 lx.support\nthis : \u2191lx r \u2208 S\u2080\n\u22a2 \u2191lx r \u2022 id r \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n[PROOFSTEP]\nchange (\u27e8_, this\u27e9 : S\u2080) \u2022 r \u2208 _\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : lx \u2208 Finsupp.supported R R \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u2074 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b3 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b2 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d\u00b9 : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis\u271d : IsNoetherianRing { x // x \u2208 S\u2080 }\nr : A\nhr : r \u2208 lx.support\nthis : \u2191lx r \u2208 S\u2080\n\u22a2 { val := \u2191lx r, property := this } \u2022 r \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n[PROOFSTEP]\nrw [Finsupp.mem_supported] at hlx1 \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS : Subalgebra R A\nx : A\nhx : x \u2208 S\ny : Finset A\nhy : span R \u2191y = \u2191Subalgebra.toSubmodule S\nlx : A \u2192\u2080 R\nhlx1 : \u2191lx.support \u2286 \u2191y\nhlx2 : \u2191(Finsupp.total A A R id) lx = x\nhyS : \u2200 {p : A}, p \u2208 y \u2192 p \u2208 S\nly : { x // x \u2208 y \u00d7\u02e2 y } \u2192 A \u2192\u2080 R\nhly1 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), ly jk \u2208 Finsupp.supported R R \u2191y\nhly2 : \u2200 (jk : { x // x \u2208 y \u00d7\u02e2 y }), \u2191(Finsupp.total A A R id) (ly jk) = (\u2191jk).fst * (\u2191jk).snd\nS\u2080 : Subring R := Subring.closure \u2191(Finsupp.frange lx \u222a Finset.biUnion Finset.univ (Finsupp.frange \u2218 ly))\nthis\u271d\u2074 : CommRing { x // x \u2208 S\u2080 } := SubringClass.toCommRing S\u2080\nthis\u271d\u00b3 : Algebra { x // x \u2208 S\u2080 } A := Algebra.ofSubring S\u2080\nthis\u271d\u00b2 : span { x // x \u2208 S\u2080 } (insert 1 \u2191y) * span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2264 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)\nS\u2081 : Subring A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n                mul_mem' :=\n                  (_ :\n                    \u2200 {p q : A},\n                      p \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192\n                        q \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) \u2192 p * q \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n            one_mem' := (_ : 1 \u2208 \u2191(span { x // x \u2208 S\u2080 } (insert 1 \u2191y))) },\n        add_mem' :=\n          (_ :\n            \u2200 {x x_1 : A},\n              x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192\n                x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 x + x_1 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)),\n        zero_mem' := (_ : 0 \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) },\n    neg_mem' := (_ : \u2200 {x : A}, x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y) \u2192 -x \u2208 span { x // x \u2208 S\u2080 } (insert 1 \u2191y)) }\nthis\u271d\u00b9 : S\u2081 = Subalgebra.toSubring (Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y)\nfoo : \u2200 (z : A), z \u2208 S\u2081 \u2194 z \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\nthis\u271d : IsNoetherianRing { x // x \u2208 S\u2080 }\nr : A\nhr : r \u2208 lx.support\nthis : \u2191lx r \u2208 S\u2080\n\u22a2 { val := \u2191lx r, property := this } \u2022 r \u2208 Algebra.adjoin { x // x \u2208 S\u2080 } \u2191y\n[PROOFSTEP]\nexact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet A' : Subalgebra R A :=\n  { carrier := {x | \u2200 n \u2208 N, x \u2022 n \u2208 N}\n    mul_mem' := fun {a b} ha hb n hn => smul_smul a b n \u25b8 ha _ (hb _ hn)\n    one_mem' := fun n hn => (one_smul A n).symm \u25b8 hn\n    add_mem' := fun {a b} ha hb n hn => (add_smul a b n).symm \u25b8 N.add_mem (ha _ hn) (hb _ hn)\n    zero_mem' := fun n _hn => (zero_smul A n).symm \u25b8 N.zero_mem\n    algebraMap_mem' := fun r n hn => (algebraMap_smul A r n).symm \u25b8 N.smul_mem r hn }\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet f : A' \u2192\u2090[R] Module.End R N :=\n  AlgHom.ofLinearMap\n    { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict x.prop\n      map_add' := by intros x y; ext;\n        exact\n          add_smul _ _\n            _\n              -- porting note: was\n                              --  `fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n`\n      map_smul' := by intros r s; ext; apply smul_assoc }\n      -- porting note: the next two lines were\n            --`(LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>`\n            --`LinearMap.ext fun n => Subtype.ext <| mul_smul x y n` (by ext; apply one_smul)\n    (by intros x y; ext; apply mul_smul)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\n\u22a2 \u2200 (x y : { x // x \u2208 A' }),\n    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y\n[PROOFSTEP]\nintros x y\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx\u271d : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x\u271d \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nx y : { x // x \u2208 A' }\n\u22a2 (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx\u271d\u00b9 : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x\u271d\u00b9 \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nx y : { x // x \u2208 A' }\nx\u271d : { x // x \u2208 N }\n\u22a2 \u2191(\u2191((fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y)) x\u271d) =\n    \u2191(\u2191((fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y)\n        x\u271d)\n[PROOFSTEP]\nexact\n  add_smul _ _\n    _\n      -- porting note: was\n                      --  `fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n`\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\n\u22a2 \u2200 (r : R) (x : { x // x \u2208 A' }),\n    AddHom.toFun\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n        (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022\n        AddHom.toFun\n          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n            map_add' :=\n              (_ :\n                \u2200 (x y : { x // x \u2208 A' }),\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n          x\n[PROOFSTEP]\nintros r s\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nr : R\ns : { x // x \u2208 A' }\n\u22a2 AddHom.toFun\n      { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n        map_add' :=\n          (_ :\n            \u2200 (x y : { x // x \u2208 A' }),\n              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n      (r \u2022 s) =\n    \u2191(RingHom.id R) r \u2022\n      AddHom.toFun\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n        s\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nr : R\ns : { x // x \u2208 A' }\nx\u271d : { x // x \u2208 N }\n\u22a2 \u2191(\u2191(AddHom.toFun\n            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : { x // x \u2208 A' }),\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n            (r \u2022 s))\n        x\u271d) =\n    \u2191(\u2191(\u2191(RingHom.id R) r \u2022\n            AddHom.toFun\n              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : { x // x \u2208 A' }),\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n              s)\n        x\u271d)\n[PROOFSTEP]\napply smul_assoc\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\n\u22a2 \u2191{\n          toAddHom :=\n            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : { x // x \u2208 A' }),\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (s : { x // x \u2208 A' }),\n                AddHom.toFun\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                    (r \u2022 s) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 A' }),\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                      s) }\n      1 =\n    1\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nx\u271d : { x // x \u2208 N }\n\u22a2 \u2191(\u2191(\u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            1)\n        x\u271d) =\n    \u2191(\u21911 x\u271d)\n[PROOFSTEP]\napply one_smul\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\n\u22a2 \u2200 (x y : { x // x \u2208 A' }),\n    \u2191{\n            toAddHom :=\n              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : { x // x \u2208 A' }),\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (s : { x // x \u2208 A' }),\n                  AddHom.toFun\n                      { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 A' }),\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                      (r \u2022 s) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        s) }\n        (x * y) =\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          x *\n        \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          y\n[PROOFSTEP]\nintros x y\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx\u271d : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x\u271d \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nx y : { x // x \u2208 A' }\n\u22a2 \u2191{\n          toAddHom :=\n            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : { x // x \u2208 A' }),\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (s : { x // x \u2208 A' }),\n                AddHom.toFun\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                    (r \u2022 s) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 A' }),\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                      s) }\n      (x * y) =\n    \u2191{\n            toAddHom :=\n              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : { x // x \u2208 A' }),\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (s : { x // x \u2208 A' }),\n                  AddHom.toFun\n                      { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 A' }),\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                      (r \u2022 s) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        s) }\n        x *\n      \u2191{\n            toAddHom :=\n              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : { x // x \u2208 A' }),\n                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (s : { x // x \u2208 A' }),\n                  AddHom.toFun\n                      { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 A' }),\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                      (r \u2022 s) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        s) }\n        y\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx\u271d\u00b9 : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x\u271d\u00b9 \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nx y : { x // x \u2208 A' }\nx\u271d : { x // x \u2208 N }\n\u22a2 \u2191(\u2191(\u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y))\n        x\u271d) =\n    \u2191(\u2191(\u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\n        x\u271d)\n[PROOFSTEP]\napply mul_smul\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nobtain \u27e8a, ha\u2081, ha\u2082\u27e9 : \u2203 a \u2208 N, a \u2260 (0 : M) := by\n  by_contra h'\n  push_neg at h' \n  apply hN\n  rwa [eq_bot_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\n\u22a2 \u2203 a, a \u2208 N \u2227 a \u2260 0\n[PROOFSTEP]\nby_contra h'\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\nh' : \u00ac\u2203 a, a \u2208 N \u2227 a \u2260 0\n\u22a2 False\n[PROOFSTEP]\npush_neg at h' \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\nh' : \u2200 (a : M), a \u2208 N \u2192 a = 0\n\u22a2 False\n[PROOFSTEP]\napply hN\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\nh' : \u2200 (a : M), a \u2208 N \u2192 a = 0\n\u22a2 N = \u22a5\n[PROOFSTEP]\nrwa [eq_bot_iff]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nhave : Function.Injective f := by\n  show Function.Injective f.toLinearMap\n  rw [\u2190 LinearMap.ker_eq_bot, eq_bot_iff]\n  intro s hs\n  have : s.1 \u2022 a = 0 := congr_arg Subtype.val (LinearMap.congr_fun hs \u27e8a, ha\u2081\u27e9)\n  exact Subtype.ext ((eq_zero_or_eq_zero_of_smul_eq_zero this).resolve_right ha\u2082)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\n\u22a2 Function.Injective \u2191f\n[PROOFSTEP]\nshow Function.Injective f.toLinearMap\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\n\u22a2 Function.Injective \u2191(AlgHom.toLinearMap f)\n[PROOFSTEP]\nrw [\u2190 LinearMap.ker_eq_bot, eq_bot_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\n\u22a2 LinearMap.ker (AlgHom.toLinearMap f) \u2264 \u22a5\n[PROOFSTEP]\nintro s hs\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\ns : { x // x \u2208 A' }\nhs : s \u2208 LinearMap.ker (AlgHom.toLinearMap f)\n\u22a2 s \u2208 \u22a5\n[PROOFSTEP]\nhave : s.1 \u2022 a = 0 := congr_arg Subtype.val (LinearMap.congr_fun hs \u27e8a, ha\u2081\u27e9)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\ns : { x // x \u2208 A' }\nhs : s \u2208 LinearMap.ker (AlgHom.toLinearMap f)\nthis : \u2191s \u2022 a = 0\n\u22a2 s \u2208 \u22a5\n[PROOFSTEP]\nexact Subtype.ext ((eq_zero_or_eq_zero_of_smul_eq_zero this).resolve_right ha\u2082)\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\nthis : Function.Injective \u2191f\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nshow IsIntegral R (A'.val \u27e8x, hx\u27e9)\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\nthis : Function.Injective \u2191f\n\u22a2 IsIntegral R (\u2191(Subalgebra.val A') { val := x, property := hx })\n[PROOFSTEP]\nrw [isIntegral_algHom_iff A'.val Subtype.val_injective, \u2190 isIntegral_algHom_iff f this]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\nthis : Function.Injective \u2191f\n\u22a2 IsIntegral R (\u2191f { val := x, property := hx })\n[PROOFSTEP]\nhaveI : Module.Finite R N := by rwa [Module.finite_def, Submodule.fg_top]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\nthis : Function.Injective \u2191f\n\u22a2 Module.Finite R { x // x \u2208 N }\n[PROOFSTEP]\nrwa [Module.finite_def, Submodule.fg_top]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R A\ninst\u271d\u2075 : Algebra R B\nf\u271d : R \u2192+* S\nM : Type u_5\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : IsScalarTower R A M\ninst\u271d : NoZeroSMulDivisors A M\nN : Submodule R M\nhN : N \u2260 \u22a5\nhN' : FG N\nx : A\nhx : \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N\nA' : Subalgebra R A :=\n  {\n    toSubsemiring :=\n      {\n        toSubmonoid :=\n          {\n            toSubsemigroup :=\n              { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                mul_mem' :=\n                  (_ :\n                    \u2200 {a b : A},\n                      a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                        b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n            one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) },\n        add_mem' :=\n          (_ :\n            \u2200 {a b : A},\n              a \u2208\n                  {\n                        toSubsemigroup :=\n                          { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                            mul_mem' :=\n                              (_ :\n                                \u2200 {a b : A},\n                                  a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                    b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                        one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                b \u2208\n                    {\n                          toSubsemigroup :=\n                            { carrier := {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N},\n                              mul_mem' :=\n                                (_ :\n                                  \u2200 {a b : A},\n                                    a \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192\n                                      b \u2208 {x | \u2200 (n : M), n \u2208 N \u2192 x \u2022 n \u2208 N} \u2192 \u2200 (n : M), n \u2208 N \u2192 (a * b) \u2022 n \u2208 N) },\n                          one_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 1 \u2022 n \u2208 N) }.toSubsemigroup.carrier \u2192\n                  \u2200 (n : M), n \u2208 N \u2192 (a + b) \u2022 n \u2208 N),\n        zero_mem' := (_ : \u2200 (n : M), n \u2208 N \u2192 0 \u2022 n \u2208 N) },\n    algebraMap_mem' := (_ : \u2200 (r : R) (n : M), n \u2208 N \u2192 \u2191(algebraMap R A) r \u2022 n \u2208 N) }\nf : { x // x \u2208 A' } \u2192\u2090[R] Module.End R { x // x \u2208 N } :=\n  AlgHom.ofLinearMap\n    {\n      toAddHom :=\n        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n          map_add' :=\n            (_ :\n              \u2200 (x y : { x // x \u2208 A' }),\n                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n      map_smul' :=\n        (_ :\n          \u2200 (r : R) (s : { x // x \u2208 A' }),\n            AddHom.toFun\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                (r \u2022 s) =\n              \u2191(RingHom.id R) r \u2022\n                AddHom.toFun\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) }\n                  s) }\n    (_ :\n      \u2191{\n              toAddHom :=\n                { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : { x // x \u2208 A' }),\n                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (s : { x // x \u2208 A' }),\n                    AddHom.toFun\n                        { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : { x // x \u2208 A' }),\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                    (x + y) =\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      y) }\n                        (r \u2022 s) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          s) }\n          1 =\n        1)\n    (_ :\n      \u2200 (x y : { x // x \u2208 A' }),\n        \u2191{\n                toAddHom :=\n                  { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : { x // x \u2208 A' }),\n                          (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (s : { x // x \u2208 A' }),\n                      AddHom.toFun\n                          { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : { x // x \u2208 A' }),\n                                  (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                      (x + y) =\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        y) }\n                          (r \u2022 s) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            s) }\n            (x * y) =\n          \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              x *\n            \u2191{\n                  toAddHom :=\n                    { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 A' }),\n                            (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) (x + y) =\n                              (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) x +\n                                (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A')) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (s : { x // x \u2208 A' }),\n                        AddHom.toFun\n                            { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : { x // x \u2208 A' }),\n                                    (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                        (x + y) =\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          x +\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          y) }\n                            (r \u2022 s) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : { x // x \u2208 A' }),\n                                      (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                          (x + y) =\n                                        (fun x => LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            x +\n                                          (fun x =>\n                                              LinearMap.restrict (DistribMulAction.toLinearMap R M x) (_ : \u2191x \u2208 A'))\n                                            y) }\n                              s) }\n              y)\na : M\nha\u2081 : a \u2208 N\nha\u2082 : a \u2260 0\nthis\u271d : Function.Injective \u2191f\nthis : Module.Finite R { x // x \u2208 N }\n\u22a2 IsIntegral R (\u2191f { val := x, property := hx })\n[PROOFSTEP]\napply Module.End.isIntegral\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : IsIntegral f\nh' : FiniteType f\n\u22a2 Finite f\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : IsIntegral f\nh' : FiniteType f\nthis : Algebra R S := toAlgebra f\n\u22a2 Finite f\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := h'\n[GOAL]\ncase mk.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : IsIntegral f\nthis : Algebra R S := toAlgebra f\ns : Finset S\nhs : Algebra.adjoin R \u2191s = \u22a4\n\u22a2 Finite f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.intro.out\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : IsIntegral f\nthis : Algebra R S := toAlgebra f\ns : Finset S\nhs : Algebra.adjoin R \u2191s = \u22a4\n\u22a2 FG \u22a4\n[PROOFSTEP]\nchange (\u22a4 : Subalgebra R S).toSubmodule.FG\n[GOAL]\ncase mk.intro.out\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : IsIntegral f\nthis : Algebra R S := toAlgebra f\ns : Finset S\nhs : Algebra.adjoin R \u2191s = \u22a4\n\u22a2 FG (\u2191Subalgebra.toSubmodule \u22a4)\n[PROOFSTEP]\nrw [\u2190 hs]\n[GOAL]\ncase mk.intro.out\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : IsIntegral f\nthis : Algebra R S := toAlgebra f\ns : Finset S\nhs : Algebra.adjoin R \u2191s = \u22a4\n\u22a2 FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R \u2191s))\n[PROOFSTEP]\nexact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\n\u22a2 Module.Finite R A\n[PROOFSTEP]\nhave :=\n  h.to_finite\n    (by\n      rw [RingHom.FiniteType]\n      convert h'\n      refine IsScalarTower.Algebra.ext (algebraMap R A).toAlgebra _ fun r x => ?_\n      exact (Algebra.smul_def _ _).symm)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\n\u22a2 RingHom.FiniteType (algebraMap R A)\n[PROOFSTEP]\nrw [RingHom.FiniteType]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\n\u22a2 FiniteType R A\n[PROOFSTEP]\nconvert h'\n[GOAL]\ncase h.e'_5\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\n\u22a2 RingHom.toAlgebra (algebraMap R A) = inst\u271d\u00b9\n[PROOFSTEP]\nrefine IsScalarTower.Algebra.ext (algebraMap R A).toAlgebra _ fun r x => ?_\n[GOAL]\ncase h.e'_5\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\nr : R\nx : A\n\u22a2 (let_fun I := RingHom.toAlgebra (algebraMap R A);\n    r \u2022 x) =\n    r \u2022 x\n[PROOFSTEP]\nexact (Algebra.smul_def _ _).symm\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\nthis : RingHom.Finite (algebraMap R A)\n\u22a2 Module.Finite R A\n[PROOFSTEP]\nrw [RingHom.Finite] at this \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\nthis : Module.Finite R A\n\u22a2 Module.Finite R A\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_5.h.e'_5\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\nthis : Module.Finite R A\n\u22a2 inst\u271d\u00b9 = RingHom.toAlgebra (algebraMap R A)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h.e'_5.h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Algebra.IsIntegral R A\nh' : FiniteType R A\nthis : Module.Finite R A\nr\u271d : R\nx\u271d : A\n\u22a2 (let_fun I := inst\u271d\u00b9;\n    r\u271d \u2022 x\u271d) =\n    r\u271d \u2022 x\u271d\n[PROOFSTEP]\nexact Algebra.smul_def _ _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Module.Finite R A\n\u22a2 Algebra.IsIntegral R A\n[PROOFSTEP]\napply RingHom.Finite.to_isIntegral\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Module.Finite R A\n\u22a2 RingHom.Finite (algebraMap R A)\n[PROOFSTEP]\nrw [RingHom.Finite]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Module.Finite R A\n\u22a2 Module.Finite R A\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_5.h.e'_5\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Module.Finite R A\n\u22a2 RingHom.toAlgebra (algebraMap R A) = inst\u271d\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h.e'_5.h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nh : Module.Finite R A\nr\u271d : R\nx\u271d : A\n\u22a2 (let_fun I := RingHom.toAlgebra (algebraMap R A);\n    r\u271d \u2022 x\u271d) =\n    r\u271d \u2022 x\u271d\n[PROOFSTEP]\nexact (Algebra.smul_def _ _).symm\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\n\u22a2 IsIntegralElem f z\n[PROOFSTEP]\nletI : Algebra R S := f.toAlgebra\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis : Algebra R S := toAlgebra f\n\u22a2 IsIntegralElem f z\n[PROOFSTEP]\nhave := (FG_adjoin_singleton_of_integral x hx).mul (FG_adjoin_singleton_of_integral y hy)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis\u271d : Algebra R S := toAlgebra f\nthis : FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}) * \u2191Subalgebra.toSubmodule (Algebra.adjoin R {y}))\n\u22a2 IsIntegralElem f z\n[PROOFSTEP]\nrw [\u2190 Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis\u271d : Algebra R S := toAlgebra f\nthis : FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x, y}))\n\u22a2 IsIntegralElem f z\n[PROOFSTEP]\nexact\n  isIntegral_of_mem_of_FG (Algebra.adjoin R { x, y }) this z\n    (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\n\u22a2 IsIntegralElem f (x - y)\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\nf : R \u2192+* S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsScalarTower R S A\nx : A\nr : R\nhx : IsIntegral S x\n\u22a2 IsIntegral S (r \u2022 x)\n[PROOFSTEP]\nrw [Algebra.smul_def, IsScalarTower.algebraMap_apply R S A]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\nf : R \u2192+* S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : IsScalarTower R S A\nx : A\nr : R\nhx : IsIntegral S x\n\u22a2 IsIntegral S (\u2191(algebraMap S A) (\u2191(algebraMap R S) r) * x)\n[PROOFSTEP]\nexact isIntegral_mul isIntegral_algebraMap hx\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nn : \u2115\nhn : 0 < n\nhx : IsIntegral R (x ^ n)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrcases hx with \u27e8p, \u27e8hmonic, heval\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nn : \u2115\nhn : 0 < n\np : R[X]\nhmonic : Monic p\nheval : eval\u2082 (algebraMap R A) (x ^ n) p = 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nexact \u27e8expand R n p, Monic.expand hn hmonic, by rwa [eval\u2082_eq_eval_map, map_expand, expand_eval, \u2190 eval\u2082_eq_eval_map]\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nn : \u2115\nhn : 0 < n\np : R[X]\nhmonic : Monic p\nheval : eval\u2082 (algebraMap R A) (x ^ n) p = 0\n\u22a2 eval\u2082 (algebraMap R A) x (\u2191(expand R n) p) = 0\n[PROOFSTEP]\nrwa [eval\u2082_eq_eval_map, map_expand, expand_eval, \u2190 eval\u2082_eq_eval_map]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nhx : IsIntegral R x\n\u22a2 Algebra.adjoin R {x} \u2264 integralClosure R A\n[PROOFSTEP]\nrw [Algebra.adjoin_le_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nhx : IsIntegral R x\n\u22a2 {x} \u2286 \u2191(integralClosure R A)\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, Set.singleton_subset_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\nhx : IsIntegral R x\n\u22a2 x \u2208 integralClosure R A\n[PROOFSTEP]\nexact hx\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\u271d\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\nS T : Subalgebra R A\n\u22a2 Algebra.IsIntegral R { x // x \u2208 S \u2294 T } \u2194 Algebra.IsIntegral R { x // x \u2208 S } \u2227 Algebra.IsIntegral R { x // x \u2208 T }\n[PROOFSTEP]\nsimp only [\u2190 le_integralClosure_iff_isIntegral, sup_le_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\n\u22a2 Subalgebra.map (\u2191f) (integralClosure R A) = integralClosure R B\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\ny : B\n\u22a2 y \u2208 Subalgebra.map (\u2191f) (integralClosure R A) \u2194 y \u2208 integralClosure R B\n[PROOFSTEP]\nrw [Subalgebra.mem_map]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\ny : B\n\u22a2 (\u2203 x, x \u2208 integralClosure R A \u2227 \u2191\u2191f x = y) \u2194 y \u2208 integralClosure R B\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\ny : B\n\u22a2 (\u2203 x, x \u2208 integralClosure R A \u2227 \u2191\u2191f x = y) \u2192 y \u2208 integralClosure R B\n[PROOFSTEP]\nrintro \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\nx : A\nhx : x \u2208 integralClosure R A\n\u22a2 \u2191\u2191f x \u2208 integralClosure R B\n[PROOFSTEP]\nexact map_isIntegral f hx\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\ny : B\n\u22a2 y \u2208 integralClosure R B \u2192 \u2203 x, x \u2208 integralClosure R A \u2227 \u2191\u2191f x = y\n[PROOFSTEP]\nintro hy\n[GOAL]\ncase h.mpr\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\ny : B\nhy : y \u2208 integralClosure R B\n\u22a2 \u2203 x, x \u2208 integralClosure R A \u2227 \u2191\u2191f x = y\n[PROOFSTEP]\nuse f.symm y, map_isIntegral (f.symm : B \u2192\u2090[R] A) hy\n[GOAL]\ncase right\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\nf : A \u2243\u2090[R] B\ny : B\nhy : y \u2208 integralClosure R B\n\u22a2 \u2191\u2191f (\u2191(AlgEquiv.symm f) y) = y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : { x // x \u2208 integralClosure R A }\np : R[X]\nhpm : Monic p\nhpx : eval\u2082 (algebraMap R A) (\u2191x) p = 0\n\u22a2 \u2191(eval\u2082 (algebraMap R { x // x \u2208 integralClosure R A }) x p) = \u21910\n[PROOFSTEP]\nrwa [\u2190 aeval_def, \u2190 Subalgebra.val_apply, aeval_algHom_apply] at hpx \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y : S\nr : R\nhr : \u2191f r * y = 1\nhx : IsIntegralElem f (x * y)\n\u22a2 IsIntegralElem f x\n[PROOFSTEP]\nobtain \u27e8p, \u27e8p_monic, hp\u27e9\u27e9 := hx\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y : S\nr : R\nhr : \u2191f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 f (x * y) p = 0\n\u22a2 IsIntegralElem f x\n[PROOFSTEP]\nrefine' \u27e8scaleRoots p r, \u27e8(monic_scaleRoots_iff r).2 p_monic, _\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y : S\nr : R\nhr : \u2191f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 f (x * y) p = 0\n\u22a2 eval\u2082 f x (scaleRoots p r) = 0\n[PROOFSTEP]\nconvert scaleRoots_eval\u2082_eq_zero f hp\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx y : S\nr : R\nhr : \u2191f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 f (x * y) p = 0\n\u22a2 x = \u2191f r * (x * y)\n[PROOFSTEP]\nrw [mul_comm x y, \u2190 mul_assoc, hr, one_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n A\nh : \u2200 (i j : n), IsIntegral R (M i j)\n\u22a2 IsIntegral R (Matrix.det M)\n[PROOFSTEP]\nrw [Matrix.det_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : CommRing S\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n A\nh : \u2200 (i j : n), IsIntegral R (M i j)\n\u22a2 IsIntegral R (\u2211 \u03c3 : Equiv.Perm n, \u2191Equiv.Perm.sign \u03c3 \u2022 \u220f i : n, M (\u2191\u03c3 i) i)\n[PROOFSTEP]\nexact IsIntegral.sum _ fun \u03c3 _h\u03c3 => IsIntegral.zsmul (IsIntegral.prod _ fun i _hi => h _ _) _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\nh : IsIntegral R y\n\u22a2 IsIntegral A (x \u2297\u209c[R] y)\n[PROOFSTEP]\nobtain \u27e8p, hp, hp'\u27e9 := h\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 IsIntegral A (x \u2297\u209c[R] y)\n[PROOFSTEP]\nrefine' \u27e8(p.map (_root_.algebraMap R A)).scaleRoots x, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 Monic (scaleRoots (Polynomial.map (_root_.algebraMap R A) p) x)\n[PROOFSTEP]\nrw [Polynomial.monic_scaleRoots_iff]\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 Monic (Polynomial.map (_root_.algebraMap R A) p)\n[PROOFSTEP]\nexact hp.map _\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 eval\u2082 (_root_.algebraMap A (A \u2297[R] B)) (x \u2297\u209c[R] y) (scaleRoots (Polynomial.map (_root_.algebraMap R A) p) x) = 0\n[PROOFSTEP]\nconvert Polynomial.scaleRoots_eval\u2082_mul (R := A \u2297[R] B) (S := A) Algebra.TensorProduct.includeLeftRingHom (?_) x\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 x \u2297\u209c[R] y = \u2191Algebra.TensorProduct.includeLeftRingHom x * ?intro.intro.refine'_2.convert_2\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 0 =\n    \u2191Algebra.TensorProduct.includeLeftRingHom x ^ natDegree (Polynomial.map (_root_.algebraMap R A) p) *\n      eval\u2082 Algebra.TensorProduct.includeLeftRingHom ?intro.intro.refine'_2.convert_2\n        (Polynomial.map (_root_.algebraMap R A) p)\ncase intro.intro.refine'_2.convert_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 A \u2297[R] B\n[PROOFSTEP]\nany_goals exact 1 \u2297\u209c y\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 x \u2297\u209c[R] y = \u2191Algebra.TensorProduct.includeLeftRingHom x * ?intro.intro.refine'_2.convert_2\n[PROOFSTEP]\nexact 1 \u2297\u209c y\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 0 =\n    \u2191Algebra.TensorProduct.includeLeftRingHom x ^ natDegree (Polynomial.map (_root_.algebraMap R A) p) *\n      eval\u2082 Algebra.TensorProduct.includeLeftRingHom ?intro.intro.refine'_2.convert_2\n        (Polynomial.map (_root_.algebraMap R A) p)\n[PROOFSTEP]\nexact 1 \u2297\u209c y\n[GOAL]\ncase intro.intro.refine'_2.convert_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 A \u2297[R] B\n[PROOFSTEP]\nexact 1 \u2297\u209c y\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 x \u2297\u209c[R] y = \u2191Algebra.TensorProduct.includeLeftRingHom x * 1 \u2297\u209c[R] y\n[PROOFSTEP]\nsimp only [Algebra.TensorProduct.includeLeftRingHom_apply, Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 0 =\n    \u2191Algebra.TensorProduct.includeLeftRingHom x ^ natDegree (Polynomial.map (_root_.algebraMap R A) p) *\n      eval\u2082 Algebra.TensorProduct.includeLeftRingHom (1 \u2297\u209c[R] y) (Polynomial.map (_root_.algebraMap R A) p)\n[PROOFSTEP]\nsimp only [Algebra.TensorProduct.includeLeftRingHom_apply, Algebra.TensorProduct.tmul_pow, one_pow]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 0 =\n    (x ^ natDegree (Polynomial.map (_root_.algebraMap R A) p)) \u2297\u209c[R] 1 *\n      eval\u2082 Algebra.TensorProduct.includeLeftRingHom (1 \u2297\u209c[R] y) (Polynomial.map (_root_.algebraMap R A) p)\n[PROOFSTEP]\nconvert (mul_zero (M\u2080 := A \u2297[R] B) _).symm\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 eval\u2082 Algebra.TensorProduct.includeLeftRingHom (1 \u2297\u209c[R] y) (Polynomial.map (_root_.algebraMap R A) p) = 0\n[PROOFSTEP]\nerw [Polynomial.eval\u2082_map, Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap, \u2190 Polynomial.eval\u2082_map]\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 eval\u2082 (\u2191Algebra.TensorProduct.includeRight) (1 \u2297\u209c[R] y) (Polynomial.map (_root_.algebraMap R B) p) = 0\n[PROOFSTEP]\nconvert Polynomial.eval\u2082_at_apply (Algebra.TensorProduct.includeRight : B \u2192\u2090[R] A \u2297[R] B).toRingHom y\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nx : A\ny : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (_root_.algebraMap R B) y p = 0\n\u22a2 0 = \u2191\u2191Algebra.TensorProduct.includeRight (eval y (Polynomial.map (_root_.algebraMap R B) p))\n[PROOFSTEP]\nrw [Polynomial.eval_map, hp', _root_.map_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\n\u22a2 coeff (normalizeScaleRoots p) i * leadingCoeff p ^ i = coeff p i * leadingCoeff p ^ (natDegree p - 1)\n[PROOFSTEP]\nsimp only [normalizeScaleRoots, finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', one_mul, zero_mul,\n  mem_support_iff, ite_mul, Ne.def, ite_not]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\n\u22a2 (if coeff p i = 0 then 0\n    else\n      if i = natDegree p then leadingCoeff p ^ i\n      else coeff p i * leadingCoeff p ^ (natDegree p - 1 - i) * leadingCoeff p ^ i) =\n    coeff p i * leadingCoeff p ^ (natDegree p - 1)\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\nh\u2081 : coeff p i = 0\n\u22a2 0 = coeff p i * leadingCoeff p ^ (natDegree p - 1)\n[PROOFSTEP]\nsimp [h\u2081]\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\nh\u2081 : \u00accoeff p i = 0\nh\u2082 : i = natDegree p\n\u22a2 leadingCoeff p ^ i = coeff p i * leadingCoeff p ^ (natDegree p - 1)\n[PROOFSTEP]\nrw [h\u2082, leadingCoeff, \u2190 pow_succ, tsub_add_cancel_of_le hp]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\nh\u2081 : \u00accoeff p i = 0\nh\u2082 : \u00aci = natDegree p\n\u22a2 coeff p i * leadingCoeff p ^ (natDegree p - 1 - i) * leadingCoeff p ^ i =\n    coeff p i * leadingCoeff p ^ (natDegree p - 1)\n[PROOFSTEP]\nrw [mul_assoc, \u2190 pow_add, tsub_add_cancel_of_le]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\nh\u2081 : \u00accoeff p i = 0\nh\u2082 : \u00aci = natDegree p\n\u22a2 i \u2264 natDegree p - 1\n[PROOFSTEP]\napply Nat.le_pred_of_lt\n[GOAL]\ncase neg.h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\nh\u2081 : \u00accoeff p i = 0\nh\u2082 : \u00aci = natDegree p\n\u22a2 i < natDegree p\n[PROOFSTEP]\nrw [lt_iff_le_and_ne]\n[GOAL]\ncase neg.h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ni : \u2115\nhp : 1 \u2264 natDegree p\nh\u2081 : \u00accoeff p i = 0\nh\u2082 : \u00aci = natDegree p\n\u22a2 i \u2264 natDegree p \u2227 i \u2260 natDegree p\n[PROOFSTEP]\nexact \u27e8le_natDegree_of_ne_zero h\u2081, h\u2082\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\n\u22a2 leadingCoeff p \u2022 normalizeScaleRoots p = scaleRoots p (leadingCoeff p)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\n\u22a2 coeff (leadingCoeff p \u2022 normalizeScaleRoots p) n\u271d = coeff (scaleRoots p (leadingCoeff p)) n\u271d\n[PROOFSTEP]\nsimp only [coeff_scaleRoots, normalizeScaleRoots, coeff_monomial, coeff_smul, Finset.smul_sum, Ne.def,\n  Finset.sum_ite_eq', finset_sum_coeff, smul_ite, smul_zero, mem_support_iff]\n  -- porting note: added the following `simp only`\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\n\u22a2 (\u2211 x in support p,\n      leadingCoeff p \u2022\n        if x = n\u271d then if x = natDegree p then 1 else coeff p x * leadingCoeff p ^ (natDegree p - 1 - x) else 0) =\n    coeff p n\u271d * leadingCoeff p ^ (natDegree p - n\u271d)\n[PROOFSTEP]\nsimp only [ge_iff_le, tsub_le_iff_right, smul_eq_mul, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', mem_support_iff,\n  ne_eq, ite_not]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\n\u22a2 (if coeff p n\u271d = 0 then 0\n    else\n      if n\u271d = natDegree p then leadingCoeff p\n      else leadingCoeff p * (coeff p n\u271d * leadingCoeff p ^ (natDegree p - 1 - n\u271d))) =\n    coeff p n\u271d * leadingCoeff p ^ (natDegree p - n\u271d)\n[PROOFSTEP]\nsplit_ifs with h\u2081 h\u2082\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\nh\u2081 : coeff p n\u271d = 0\n\u22a2 0 = coeff p n\u271d * leadingCoeff p ^ (natDegree p - n\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\nh\u2081 : \u00accoeff p n\u271d = 0\nh\u2082 : n\u271d = natDegree p\n\u22a2 leadingCoeff p = coeff p n\u271d * leadingCoeff p ^ (natDegree p - n\u271d)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\nh\u2081 : \u00accoeff p n\u271d = 0\nh\u2082 : \u00acn\u271d = natDegree p\n\u22a2 leadingCoeff p * (coeff p n\u271d * leadingCoeff p ^ (natDegree p - 1 - n\u271d)) =\n    coeff p n\u271d * leadingCoeff p ^ (natDegree p - n\u271d)\n[PROOFSTEP]\nrw [mul_comm, mul_assoc, \u2190 pow_succ', tsub_right_comm, tsub_add_cancel_of_le]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\nh\u2081 : \u00accoeff p n\u271d = 0\nh\u2082 : \u00acn\u271d = natDegree p\n\u22a2 1 \u2264 natDegree p - n\u271d\n[PROOFSTEP]\nrw [Nat.succ_le_iff]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np\u271d : R[X]\nx : S\np : R[X]\nn\u271d : \u2115\nh\u2081 : \u00accoeff p n\u271d = 0\nh\u2082 : \u00acn\u271d = natDegree p\n\u22a2 0 < natDegree p - n\u271d\n[PROOFSTEP]\nexact tsub_pos_of_lt (lt_of_le_of_ne (le_natDegree_of_ne_zero h\u2081) h\u2082)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\n\u22a2 support (normalizeScaleRoots p) \u2264 support p\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx\u271d : S\nx : \u2115\n\u22a2 x \u2208 support (normalizeScaleRoots p) \u2192 x \u2208 support p\n[PROOFSTEP]\ncontrapose\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx\u271d : S\nx : \u2115\n\u22a2 \u00acx \u2208 support p \u2192 \u00acx \u2208 support (normalizeScaleRoots p)\n[PROOFSTEP]\nsimp only [not_mem_support_iff, normalizeScaleRoots, finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq',\n  mem_support_iff, Ne.def, Classical.not_not, ite_eq_right_iff]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx\u271d : S\nx : \u2115\n\u22a2 coeff p x = 0 \u2192\n    \u00accoeff p x = 0 \u2192 (if x = natDegree p then 1 else coeff p x * leadingCoeff p ^ (natDegree p - 1 - x)) = 0\n[PROOFSTEP]\nintro h\u2081 h\u2082\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx\u271d : S\nx : \u2115\nh\u2081 : coeff p x = 0\nh\u2082 : \u00accoeff p x = 0\n\u22a2 (if x = natDegree p then 1 else coeff p x * leadingCoeff p ^ (natDegree p - 1 - x)) = 0\n[PROOFSTEP]\nexact (h\u2082 h\u2081).elim\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\n\u22a2 degree (normalizeScaleRoots p) = degree p\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\n\u22a2 degree (normalizeScaleRoots p) \u2264 degree p\n[PROOFSTEP]\nexact Finset.sup_mono (normalizeScaleRoots_support p)\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\n\u22a2 degree p \u2264 degree (normalizeScaleRoots p)\n[PROOFSTEP]\nrw [\u2190 degree_scaleRoots, \u2190 leadingCoeff_smul_normalizeScaleRoots]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\n\u22a2 degree (leadingCoeff p \u2022 normalizeScaleRoots p) \u2264 degree (normalizeScaleRoots p)\n[PROOFSTEP]\nexact degree_smul_le _ _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\np : R[X]\nx\u271d : S\nh : 1 \u2264 natDegree p\nf : R \u2192+* S\nx : S\n\u22a2 eval\u2082 f (\u2191f (leadingCoeff p) * x) (normalizeScaleRoots p) = \u2191f (leadingCoeff p) ^ (natDegree p - 1) * eval\u2082 f x p\n[PROOFSTEP]\nrw [eval\u2082_eq_sum_range, eval\u2082_eq_sum_range, Finset.mul_sum]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\np : R[X]\nx\u271d : S\nh : 1 \u2264 natDegree p\nf : R \u2192+* S\nx : S\n\u22a2 \u2211 i in Finset.range (natDegree (normalizeScaleRoots p) + 1),\n      \u2191f (coeff (normalizeScaleRoots p) i) * (\u2191f (leadingCoeff p) * x) ^ i =\n    \u2211 x_1 in Finset.range (natDegree p + 1), \u2191f (leadingCoeff p) ^ (natDegree p - 1) * (\u2191f (coeff p x_1) * x ^ x_1)\n[PROOFSTEP]\napply Finset.sum_congr\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\np : R[X]\nx\u271d : S\nh : 1 \u2264 natDegree p\nf : R \u2192+* S\nx : S\n\u22a2 Finset.range (natDegree (normalizeScaleRoots p) + 1) = Finset.range (natDegree p + 1)\n[PROOFSTEP]\nrw [natDegree_eq_of_degree_eq (normalizeScaleRoots_degree p)]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\np : R[X]\nx\u271d : S\nh : 1 \u2264 natDegree p\nf : R \u2192+* S\nx : S\n\u22a2 \u2200 (x_1 : \u2115),\n    x_1 \u2208 Finset.range (natDegree p + 1) \u2192\n      \u2191f (coeff (normalizeScaleRoots p) x_1) * (\u2191f (leadingCoeff p) * x) ^ x_1 =\n        \u2191f (leadingCoeff p) ^ (natDegree p - 1) * (\u2191f (coeff p x_1) * x ^ x_1)\n[PROOFSTEP]\nintro n _hn\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\np : R[X]\nx\u271d : S\nh : 1 \u2264 natDegree p\nf : R \u2192+* S\nx : S\nn : \u2115\n_hn : n \u2208 Finset.range (natDegree p + 1)\n\u22a2 \u2191f (coeff (normalizeScaleRoots p) n) * (\u2191f (leadingCoeff p) * x) ^ n =\n    \u2191f (leadingCoeff p) ^ (natDegree p - 1) * (\u2191f (coeff p n) * x ^ n)\n[PROOFSTEP]\nrw [mul_pow, \u2190 mul_assoc, \u2190 f.map_pow, \u2190 f.map_mul, normalizeScaleRoots_coeff_mul_leadingCoeff_pow _ _ h, f.map_mul,\n  f.map_pow]\n[GOAL]\ncase a\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf\u271d : R \u2192+* S\np : R[X]\nx\u271d : S\nh : 1 \u2264 natDegree p\nf : R \u2192+* S\nx : S\nn : \u2115\n_hn : n \u2208 Finset.range (natDegree p + 1)\n\u22a2 \u2191f (coeff p n) * \u2191f (leadingCoeff p) ^ (natDegree p - 1) * x ^ n =\n    \u2191f (leadingCoeff p) ^ (natDegree p - 1) * (\u2191f (coeff p n) * x ^ n)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : p \u2260 0\n\u22a2 Monic (normalizeScaleRoots p)\n[PROOFSTEP]\ndelta Monic leadingCoeff\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : p \u2260 0\n\u22a2 coeff (normalizeScaleRoots p) (natDegree (normalizeScaleRoots p)) = 1\n[PROOFSTEP]\nrw [natDegree_eq_of_degree_eq (normalizeScaleRoots_degree p)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : p \u2260 0\n\u22a2 coeff (normalizeScaleRoots p) (natDegree p) = 1\n[PROOFSTEP]\nsuffices p = 0 \u2192 (0 : R) = 1 by simpa [normalizeScaleRoots, coeff_monomial]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : p \u2260 0\nthis : p = 0 \u2192 0 = 1\n\u22a2 coeff (normalizeScaleRoots p) (natDegree p) = 1\n[PROOFSTEP]\nsimpa [normalizeScaleRoots, coeff_monomial]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : p \u2260 0\n\u22a2 p = 0 \u2192 0 = 1\n[PROOFSTEP]\nexact fun h' => (h h').elim\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nby_cases h' : 1 \u2264 p.natDegree\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : 1 \u2264 natDegree p\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nuse normalizeScaleRoots p\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : 1 \u2264 natDegree p\n\u22a2 Monic (normalizeScaleRoots p) \u2227 eval\u2082 f (\u2191f (leadingCoeff p) * x) (normalizeScaleRoots p) = 0\n[PROOFSTEP]\nhave : p \u2260 0 := fun h'' => by\n  rw [h'', natDegree_zero] at h' \n  exact Nat.not_succ_le_zero 0 h'\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : 1 \u2264 natDegree p\nh'' : p = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h'', natDegree_zero] at h' \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : 1 \u2264 0\nh'' : p = 0\n\u22a2 False\n[PROOFSTEP]\nexact Nat.not_succ_le_zero 0 h'\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : 1 \u2264 natDegree p\nthis : p \u2260 0\n\u22a2 Monic (normalizeScaleRoots p) \u2227 eval\u2082 f (\u2191f (leadingCoeff p) * x) (normalizeScaleRoots p) = 0\n[PROOFSTEP]\nuse normalizeScaleRoots_monic p this\n[GOAL]\ncase right\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : 1 \u2264 natDegree p\nthis : p \u2260 0\n\u22a2 eval\u2082 f (\u2191f (leadingCoeff p) * x) (normalizeScaleRoots p) = 0\n[PROOFSTEP]\nrw [normalizeScaleRoots_eval\u2082_leadingCoeff_mul p h' f x, h, mul_zero]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : \u00ac1 \u2264 natDegree p\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nby_cases hp : p.map f = 0\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : \u00ac1 \u2264 natDegree p\nhp : Polynomial.map f p = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\napply_fun fun q => coeff q p.natDegree at hp \n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : \u00ac1 \u2264 natDegree p\nhp : coeff (Polynomial.map f p) (natDegree p) = coeff 0 (natDegree p)\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nrw [coeff_map, coeff_zero, coeff_natDegree] at hp \n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : \u00ac1 \u2264 natDegree p\nhp : \u2191f (leadingCoeff p) = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nrw [hp, zero_mul]\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : \u00ac1 \u2264 natDegree p\nhp : \u2191f (leadingCoeff p) = 0\n\u22a2 IsIntegralElem f 0\n[PROOFSTEP]\nexact f.is_integral_zero\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : \u00ac1 \u2264 natDegree p\nhp : \u00acPolynomial.map f p = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nrw [Nat.one_le_iff_ne_zero, Classical.not_not] at h' \n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : eval\u2082 f x p = 0\nh' : natDegree p = 0\nhp : \u00acPolynomial.map f p = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nrw [eq_C_of_natDegree_eq_zero h', eval\u2082_C] at h \n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : \u2191f (coeff p 0) = 0\nh' : natDegree p = 0\nhp : \u00acPolynomial.map f p = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nsuffices p.map f = 0 by exact (hp this).elim\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : \u2191f (coeff p 0) = 0\nh' : natDegree p = 0\nhp : \u00acPolynomial.map f p = 0\nthis : Polynomial.map f p = 0\n\u22a2 IsIntegralElem f (\u2191f (leadingCoeff p) * x)\n[PROOFSTEP]\nexact (hp this).elim\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\nh : \u2191f (coeff p 0) = 0\nh' : natDegree p = 0\nhp : \u00acPolynomial.map f p = 0\n\u22a2 Polynomial.map f p = 0\n[PROOFSTEP]\nrw [eq_C_of_natDegree_eq_zero h', map_C, h, C_eq_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ninst\u271d : Algebra R S\nh : \u2191(aeval x) p = 0\n\u22a2 IsIntegral R (leadingCoeff p \u2022 x)\n[PROOFSTEP]\nrw [aeval_def] at h \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ninst\u271d : Algebra R S\nh : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 IsIntegral R (leadingCoeff p \u2022 x)\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R B\nf : R \u2192+* S\np : R[X]\nx : S\ninst\u271d : Algebra R S\nh : eval\u2082 (algebraMap R S) x p = 0\n\u22a2 IsIntegral R (\u2191(algebraMap R S) (leadingCoeff p) * x)\n[PROOFSTEP]\nexact (algebraMap R S).isIntegralElem_leadingCoeff_mul p x h\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nx : A\n\u22a2 (\u2203 y, \u2191(algebraMap { x // x \u2208 integralClosure R A } A) y = x) \u2192 IsIntegral R x\n[PROOFSTEP]\nrintro \u27e8\u27e8_, h\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nval\u271d : A\nh : val\u271d \u2208 integralClosure R A\n\u22a2 IsIntegral R (\u2191(algebraMap { x // x \u2208 integralClosure R A } A) { val := val\u271d, property := h })\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra R B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsIntegralClosure A R B\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : NoZeroSMulDivisors R B\n\u22a2 NoZeroSMulDivisors R A\n[PROOFSTEP]\nrefine' Function.Injective.noZeroSMulDivisors _ (IsIntegralClosure.algebraMap_injective A R B) (map_zero _) fun _ _ => _\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra R B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsIntegralClosure A R B\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : NoZeroSMulDivisors R B\nx\u271d\u00b9 : R\nx\u271d : A\n\u22a2 \u2191(algebraMap A B) (x\u271d\u00b9 \u2022 x\u271d) = x\u271d\u00b9 \u2022 \u2191(algebraMap A B) x\u271d\n[PROOFSTEP]\nsimp only [Algebra.algebraMap_eq_smul_one, IsScalarTower.smul_assoc]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsIntegralClosure A R B\nh : optParam (IsIntegral R 1) (_ : IsIntegral R 1)\n\u22a2 \u2191(algebraMap A B) (mk' A 1 h) = \u2191(algebraMap A B) 1\n[PROOFSTEP]\nrw [algebraMap_mk', RingHom.map_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsIntegralClosure A R B\nh : optParam (IsIntegral R 0) (_ : IsIntegral R 0)\n\u22a2 \u2191(algebraMap A B) (mk' A 0 h) = \u2191(algebraMap A B) 0\n[PROOFSTEP]\nrw [algebraMap_mk', RingHom.map_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsIntegralClosure A R B\nx y : B\nhx : IsIntegral R x\nhy : IsIntegral R y\n\u22a2 \u2191(algebraMap A B) (mk' A (x + y) (_ : IsIntegral R (x + y))) = \u2191(algebraMap A B) (mk' A x hx + mk' A y hy)\n[PROOFSTEP]\nsimp only [algebraMap_mk', RingHom.map_add]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsIntegralClosure A R B\nx y : B\nhx : IsIntegral R x\nhy : IsIntegral R y\n\u22a2 \u2191(algebraMap A B) (mk' A (x * y) (_ : IsIntegral R (x * y))) = \u2191(algebraMap A B) (mk' A x hx * mk' A y hy)\n[PROOFSTEP]\nsimp only [algebraMap_mk', RingHom.map_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsIntegralClosure A R B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nx : R\nh : optParam (IsIntegral R (\u2191(algebraMap R B) x)) (_ : IsIntegral R (\u2191(algebraMap R ((fun x => B) x)) x))\n\u22a2 \u2191(algebraMap A B) (mk' A (\u2191(algebraMap R B) x) h) = \u2191(algebraMap A B) (\u2191(algebraMap R A) x)\n[PROOFSTEP]\nrw [algebraMap_mk', \u2190 IsScalarTower.algebraMap_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : Algebra R B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsIntegralClosure A R B\nS : Type u_4\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S B\ninst\u271d\u00b2 : IsScalarTower R S B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nh : Algebra.IsIntegral R S\n\u22a2 (fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x))) 1 = 1\n[PROOFSTEP]\nsimp only [RingHom.map_one, mk'_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : Algebra R B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsIntegralClosure A R B\nS : Type u_4\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S B\ninst\u271d\u00b2 : IsScalarTower R S B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nh : Algebra.IsIntegral R S\nx y : S\n\u22a2 OneHom.toFun\n      { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n        map_one' := (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n      (x * y) =\n    OneHom.toFun\n        { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n          map_one' := (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n          map_one' := (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n        y\n[PROOFSTEP]\nsimp_rw [\u2190 mk'_mul, RingHom.map_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : Algebra R B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsIntegralClosure A R B\nS : Type u_4\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S B\ninst\u271d\u00b2 : IsScalarTower R S B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nh : Algebra.IsIntegral R S\n\u22a2 OneHom.toFun\n      (\u2191{\n          toOneHom :=\n            { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n              map_one' :=\n                (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : S),\n                OneHom.toFun\n                    {\n                      toFun := fun x =>\n                        mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                      map_one' :=\n                        (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                    (x * y) =\n                  OneHom.toFun\n                      {\n                        toFun := fun x =>\n                          mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                        map_one' :=\n                          (_ :\n                            mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                      x *\n                    OneHom.toFun\n                      {\n                        toFun := fun x =>\n                          mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                        map_one' :=\n                          (_ :\n                            mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                      y) })\n      0 =\n    0\n[PROOFSTEP]\nsimp only [RingHom.map_zero, mk'_zero]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : Algebra R B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsIntegralClosure A R B\nS : Type u_4\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S B\ninst\u271d\u00b2 : IsScalarTower R S B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nh : Algebra.IsIntegral R S\nx y : S\n\u22a2 OneHom.toFun\n      (\u2191{\n          toOneHom :=\n            { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n              map_one' :=\n                (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : S),\n                OneHom.toFun\n                    {\n                      toFun := fun x =>\n                        mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                      map_one' :=\n                        (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                    (x * y) =\n                  OneHom.toFun\n                      {\n                        toFun := fun x =>\n                          mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                        map_one' :=\n                          (_ :\n                            mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                      x *\n                    OneHom.toFun\n                      {\n                        toFun := fun x =>\n                          mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                        map_one' :=\n                          (_ :\n                            mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                      y) })\n      (x + y) =\n    OneHom.toFun\n        (\u2191{\n            toOneHom :=\n              { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                map_one' :=\n                  (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : S),\n                  OneHom.toFun\n                      {\n                        toFun := fun x =>\n                          mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                        map_one' :=\n                          (_ :\n                            mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                      (x * y) =\n                    OneHom.toFun\n                        {\n                          toFun := fun x =>\n                            mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                          map_one' :=\n                            (_ :\n                              mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                        x *\n                      OneHom.toFun\n                        {\n                          toFun := fun x =>\n                            mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                          map_one' :=\n                            (_ :\n                              mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                        y) })\n        x +\n      OneHom.toFun\n        (\u2191{\n            toOneHom :=\n              { toFun := fun x => mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                map_one' :=\n                  (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : S),\n                  OneHom.toFun\n                      {\n                        toFun := fun x =>\n                          mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                        map_one' :=\n                          (_ :\n                            mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                      (x * y) =\n                    OneHom.toFun\n                        {\n                          toFun := fun x =>\n                            mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                          map_one' :=\n                            (_ :\n                              mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                        x *\n                      OneHom.toFun\n                        {\n                          toFun := fun x =>\n                            mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                          map_one' :=\n                            (_ :\n                              mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) }\n                        y) })\n        y\n[PROOFSTEP]\nsimp_rw [\u2190 mk'_add, RingHom.map_add]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : Algebra R B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsIntegralClosure A R B\nS : Type u_4\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S B\ninst\u271d\u00b2 : IsScalarTower R S B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nh : Algebra.IsIntegral R S\nx : R\n\u22a2 OneHom.toFun\n      (\u2191\u2191{\n            toMonoidHom :=\n              {\n                toOneHom :=\n                  {\n                    toFun := fun x =>\n                      mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                    map_one' :=\n                      (_ : mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : S),\n                      OneHom.toFun\n                          {\n                            toFun := fun x =>\n                              mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                            map_one' :=\n                              (_ :\n                                mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                  1) }\n                          (x * y) =\n                        OneHom.toFun\n                            {\n                              toFun := fun x =>\n                                mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                              map_one' :=\n                                (_ :\n                                  mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                    1) }\n                            x *\n                          OneHom.toFun\n                            {\n                              toFun := fun x =>\n                                mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                              map_one' :=\n                                (_ :\n                                  mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                    1) }\n                            y) },\n            map_zero' := (_ : mk' A (\u2191(algebraMap S B) 0) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 0)) 0)) = 0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : S),\n                  OneHom.toFun\n                      (\u2191{\n                          toOneHom :=\n                            {\n                              toFun := fun x =>\n                                mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                              map_one' :=\n                                (_ :\n                                  mk' A (\u2191(algebraMap S B) 1) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                    1) },\n                          map_mul' :=\n                            (_ :\n                              \u2200 (x y : S),\n                                OneHom.toFun\n                                    {\n                                      toFun := fun x =>\n                                        mk' A (\u2191(algebraMap S B) x)\n                                          (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                      map_one' :=\n                                        (_ :\n                                          mk' A (\u2191(algebraMap S B) 1)\n                                              (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                            1) }\n                                    (x * y) =\n                                  OneHom.toFun\n                                      {\n                                        toFun := fun x =>\n                                          mk' A (\u2191(algebraMap S B) x)\n                                            (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                        map_one' :=\n                                          (_ :\n                                            mk' A (\u2191(algebraMap S B) 1)\n                                                (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                              1) }\n                                      x *\n                                    OneHom.toFun\n                                      {\n                                        toFun := fun x =>\n                                          mk' A (\u2191(algebraMap S B) x)\n                                            (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                        map_one' :=\n                                          (_ :\n                                            mk' A (\u2191(algebraMap S B) 1)\n                                                (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                              1) }\n                                      y) })\n                      (x + y) =\n                    OneHom.toFun\n                        (\u2191{\n                            toOneHom :=\n                              {\n                                toFun := fun x =>\n                                  mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                map_one' :=\n                                  (_ :\n                                    mk' A (\u2191(algebraMap S B) 1)\n                                        (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                      1) },\n                            map_mul' :=\n                              (_ :\n                                \u2200 (x y : S),\n                                  OneHom.toFun\n                                      {\n                                        toFun := fun x =>\n                                          mk' A (\u2191(algebraMap S B) x)\n                                            (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                        map_one' :=\n                                          (_ :\n                                            mk' A (\u2191(algebraMap S B) 1)\n                                                (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                              1) }\n                                      (x * y) =\n                                    OneHom.toFun\n                                        {\n                                          toFun := fun x =>\n                                            mk' A (\u2191(algebraMap S B) x)\n                                              (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                          map_one' :=\n                                            (_ :\n                                              mk' A (\u2191(algebraMap S B) 1)\n                                                  (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                                1) }\n                                        x *\n                                      OneHom.toFun\n                                        {\n                                          toFun := fun x =>\n                                            mk' A (\u2191(algebraMap S B) x)\n                                              (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                          map_one' :=\n                                            (_ :\n                                              mk' A (\u2191(algebraMap S B) 1)\n                                                  (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                                1) }\n                                        y) })\n                        x +\n                      OneHom.toFun\n                        (\u2191{\n                            toOneHom :=\n                              {\n                                toFun := fun x =>\n                                  mk' A (\u2191(algebraMap S B) x) (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                map_one' :=\n                                  (_ :\n                                    mk' A (\u2191(algebraMap S B) 1)\n                                        (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                      1) },\n                            map_mul' :=\n                              (_ :\n                                \u2200 (x y : S),\n                                  OneHom.toFun\n                                      {\n                                        toFun := fun x =>\n                                          mk' A (\u2191(algebraMap S B) x)\n                                            (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                        map_one' :=\n                                          (_ :\n                                            mk' A (\u2191(algebraMap S B) 1)\n                                                (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                              1) }\n                                      (x * y) =\n                                    OneHom.toFun\n                                        {\n                                          toFun := fun x =>\n                                            mk' A (\u2191(algebraMap S B) x)\n                                              (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                          map_one' :=\n                                            (_ :\n                                              mk' A (\u2191(algebraMap S B) 1)\n                                                  (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                                1) }\n                                        x *\n                                      OneHom.toFun\n                                        {\n                                          toFun := fun x =>\n                                            mk' A (\u2191(algebraMap S B) x)\n                                              (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) x)) x)),\n                                          map_one' :=\n                                            (_ :\n                                              mk' A (\u2191(algebraMap S B) 1)\n                                                  (_ : IsIntegral R (\u2191(algebraMap S ((fun x => B) 1)) 1)) =\n                                                1) }\n                                        y) })\n                        y) })\n      (\u2191(algebraMap R S) x) =\n    \u2191(algebraMap R A) x\n[PROOFSTEP]\nsimp_rw [\u2190 IsScalarTower.algebraMap_apply, mk'_algebraMap]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra R B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : IsIntegralClosure A R B\nA' : Type u_4\ninst\u271d\u2076 : CommRing A'\ninst\u271d\u2075 : Algebra A' B\ninst\u271d\u2074 : IsIntegralClosure A' R B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R A'\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsScalarTower R A' B\n\u22a2 AlgHom.comp (lift A' B (_ : Algebra.IsIntegral R A)) (lift A B (_ : Algebra.IsIntegral R A')) = AlgHom.id R A'\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra R B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : IsIntegralClosure A R B\nA' : Type u_4\ninst\u271d\u2076 : CommRing A'\ninst\u271d\u2075 : Algebra A' B\ninst\u271d\u2074 : IsIntegralClosure A' R B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R A'\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsScalarTower R A' B\nx : A'\n\u22a2 \u2191(AlgHom.comp (lift A' B (_ : Algebra.IsIntegral R A)) (lift A B (_ : Algebra.IsIntegral R A'))) x =\n    \u2191(AlgHom.id R A') x\n[PROOFSTEP]\napply algebraMap_injective A' R B\n[GOAL]\ncase H.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra R B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : IsIntegralClosure A R B\nA' : Type u_4\ninst\u271d\u2076 : CommRing A'\ninst\u271d\u2075 : Algebra A' B\ninst\u271d\u2074 : IsIntegralClosure A' R B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R A'\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsScalarTower R A' B\nx : A'\n\u22a2 \u2191(algebraMap A' B)\n      (\u2191(AlgHom.comp (lift A' B (_ : Algebra.IsIntegral R A)) (lift A B (_ : Algebra.IsIntegral R A'))) x) =\n    \u2191(algebraMap A' B) (\u2191(AlgHom.id R A') x)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra R B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : IsIntegralClosure A R B\nA' : Type u_4\ninst\u271d\u2076 : CommRing A'\ninst\u271d\u2075 : Algebra A' B\ninst\u271d\u2074 : IsIntegralClosure A' R B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R A'\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsScalarTower R A' B\n\u22a2 AlgHom.comp (lift A B (_ : Algebra.IsIntegral R A')) (lift A' B (_ : Algebra.IsIntegral R A)) = AlgHom.id R A\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra R B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : IsIntegralClosure A R B\nA' : Type u_4\ninst\u271d\u2076 : CommRing A'\ninst\u271d\u2075 : Algebra A' B\ninst\u271d\u2074 : IsIntegralClosure A' R B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R A'\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsScalarTower R A' B\nx : A\n\u22a2 \u2191(AlgHom.comp (lift A B (_ : Algebra.IsIntegral R A')) (lift A' B (_ : Algebra.IsIntegral R A))) x =\n    \u2191(AlgHom.id R A) x\n[PROOFSTEP]\napply algebraMap_injective A R B\n[GOAL]\ncase H.a\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra R B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : IsIntegralClosure A R B\nA' : Type u_4\ninst\u271d\u2076 : CommRing A'\ninst\u271d\u2075 : Algebra A' B\ninst\u271d\u2074 : IsIntegralClosure A' R B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra R A'\ninst\u271d\u00b9 : IsScalarTower R A B\ninst\u271d : IsScalarTower R A' B\nx : A\n\u22a2 \u2191(algebraMap A B)\n      (\u2191(AlgHom.comp (lift A B (_ : Algebra.IsIntegral R A')) (lift A' B (_ : Algebra.IsIntegral R A))) x) =\n    \u2191(algebraMap A B) (\u2191(AlgHom.id R A) x)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\n\u22a2 IsIntegral { x // x \u2208 adjoin R \u2191(frange (Polynomial.map (algebraMap A B) p)) } x\n[PROOFSTEP]\ngeneralize hS : (\u2191(p.map <| algebraMap A B).frange : Set B) = S\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\n\u22a2 IsIntegral { x // x \u2208 adjoin R S } x\n[PROOFSTEP]\nhave coeffs_mem : \u2200 i, (p.map <| algebraMap A B).coeff i \u2208 adjoin R S :=\n  by\n  intro i\n  by_cases hi : (p.map <| algebraMap A B).coeff i = 0\n  \u00b7 rw [hi]\n    exact Subalgebra.zero_mem _\n  rw [\u2190 hS]\n  exact subset_adjoin (coeff_mem_frange _ _ hi)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\n\u22a2 \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ni : \u2115\n\u22a2 coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n[PROOFSTEP]\nby_cases hi : (p.map <| algebraMap A B).coeff i = 0\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ni : \u2115\nhi : coeff (Polynomial.map (algebraMap A B) p) i = 0\n\u22a2 coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n[PROOFSTEP]\nrw [hi]\n[GOAL]\ncase pos\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ni : \u2115\nhi : coeff (Polynomial.map (algebraMap A B) p) i = 0\n\u22a2 0 \u2208 adjoin R S\n[PROOFSTEP]\nexact Subalgebra.zero_mem _\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ni : \u2115\nhi : \u00accoeff (Polynomial.map (algebraMap A B) p) i = 0\n\u22a2 coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n[PROOFSTEP]\nrw [\u2190 hS]\n[GOAL]\ncase neg\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ni : \u2115\nhi : \u00accoeff (Polynomial.map (algebraMap A B) p) i = 0\n\u22a2 coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R \u2191(frange (Polynomial.map (algebraMap A B) p))\n[PROOFSTEP]\nexact subset_adjoin (coeff_mem_frange _ _ hi)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n\u22a2 IsIntegral { x // x \u2208 adjoin R S } x\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 : \u2203 q : (adjoin R S)[X], q.map (algebraMap (adjoin R S) B) = (p.map <| algebraMap A B) :=\n  by\n  rw [\u2190 Set.mem_range]\n  exact (Polynomial.mem_map_range _).2 fun i => \u27e8\u27e8_, coeffs_mem i\u27e9, rfl\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n\u22a2 \u2203 q, Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n[PROOFSTEP]\nrw [\u2190 Set.mem_range]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\n\u22a2 Polynomial.map (algebraMap A B) p \u2208 Set.range fun q => Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q\n[PROOFSTEP]\nexact (Polynomial.mem_map_range _).2 fun i => \u27e8\u27e8_, coeffs_mem i\u27e9, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 IsIntegral { x // x \u2208 adjoin R S } x\n[PROOFSTEP]\nuse q\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 Monic q \u2227 eval\u2082 (algebraMap { x // x \u2208 adjoin R S } B) x q = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 Monic q\n[PROOFSTEP]\nsuffices h : (q.map (algebraMap (adjoin R S) B)).Monic\n[GOAL]\ncase h.left\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\nh : Monic (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q)\n\u22a2 Monic q\n[PROOFSTEP]\nrefine' monic_of_injective _ h\n[GOAL]\ncase h.left\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\nh : Monic (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q)\n\u22a2 Function.Injective \u2191(algebraMap { x // x \u2208 adjoin R S } B)\n[PROOFSTEP]\nexact Subtype.val_injective\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 Monic (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q)\n[PROOFSTEP]\nrw [hq]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 Monic (Polynomial.map (algebraMap A B) p)\n[PROOFSTEP]\nexact pmonic.map _\n[GOAL]\ncase h.right\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 adjoin R S } B) x q = 0\n[PROOFSTEP]\nconvert hp using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q = Polynomial.map (algebraMap A B) p\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 adjoin R S } B) x q = \u2191(aeval x) p\n[PROOFSTEP]\nreplace hq := congr_arg (eval x) hq\n[GOAL]\ncase h.e'_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q) = eval x (Polynomial.map (algebraMap A B) p)\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 adjoin R S } B) x q = \u2191(aeval x) p\n[PROOFSTEP]\nconvert hq using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q) = eval x (Polynomial.map (algebraMap A B) p)\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 adjoin R S } B) x q = eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q) = eval x (Polynomial.map (algebraMap A B) p)\n\u22a2 \u2191(aeval x) p = eval x (Polynomial.map (algebraMap A B) p)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q) = eval x (Polynomial.map (algebraMap A B) p)\n\u22a2 eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q) = eval\u2082 (algebraMap { x // x \u2208 adjoin R S } B) x q\n[PROOFSTEP]\napply eval_map\n[GOAL]\ncase h.e'_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\nx : B\np : A[X]\npmonic : Monic p\nhp : \u2191(aeval x) p = 0\nS : Set B\nhS : \u2191(frange (Polynomial.map (algebraMap A B) p)) = S\ncoeffs_mem : \u2200 (i : \u2115), coeff (Polynomial.map (algebraMap A B) p) i \u2208 adjoin R S\nq : { x // x \u2208 adjoin R S }[X]\nhq : eval x (Polynomial.map (algebraMap { x // x \u2208 adjoin R S } B) q) = eval x (Polynomial.map (algebraMap A B) p)\n\u22a2 eval x (Polynomial.map (algebraMap A B) p) = \u2191(aeval x) p\n[PROOFSTEP]\napply eval_map\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\nhx : IsIntegral A x\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrcases hx with \u27e8p, pmonic, hp\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nlet S : Set B := \u2191(p.map <| algebraMap A B).frange\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrefine' isIntegral_of_mem_of_FG (adjoin R (S \u222a { x })) _ _ (subset_adjoin <| Or.inr rfl)\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\n\u22a2 FG (\u2191Subalgebra.toSubmodule (adjoin R (S \u222a {x})))\n[PROOFSTEP]\nrefine' fg_trans (FG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx\u271d : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x\u271d p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\nx : B\nhx : x \u2208 S\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrw [Finset.mem_coe, frange, Finset.mem_image] at hx \n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx\u271d : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x\u271d p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\nx : B\nhx : \u2203 a, a \u2208 support (Polynomial.map (algebraMap A B) p) \u2227 coeff (Polynomial.map (algebraMap A B) p) a = x\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrcases hx with \u27e8i, _, rfl\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\ni : \u2115\nleft\u271d : i \u2208 support (Polynomial.map (algebraMap A B) p)\n\u22a2 IsIntegral R (coeff (Polynomial.map (algebraMap A B) p) i)\n[PROOFSTEP]\nrw [coeff_map]\n[GOAL]\ncase intro.intro.refine'_1.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\ni : \u2115\nleft\u271d : i \u2208 support (Polynomial.map (algebraMap A B) p)\n\u22a2 IsIntegral R (\u2191(algebraMap A B) (coeff p i))\n[PROOFSTEP]\nexact map_isIntegral (IsScalarTower.toAlgHom R A B) (A_int _)\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\n\u22a2 FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R S } {x}))\n[PROOFSTEP]\napply FG_adjoin_singleton_of_integral\n[GOAL]\ncase intro.intro.refine'_2.hx\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\u271d\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nA_int : Algebra.IsIntegral R A\nx : B\np : A[X]\npmonic : Monic p\nhp : eval\u2082 (algebraMap A B) x p = 0\nS : Set B := \u2191(frange (Polynomial.map (algebraMap A B) p))\n\u22a2 IsIntegral { x // x \u2208 adjoin R S } x\n[PROOFSTEP]\nexact isIntegral_trans_aux _ pmonic hp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nH : Function.Injective \u2191(algebraMap A B)\nx : A\nh : IsIntegral R (\u2191(algebraMap A B) x)\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrcases h with \u27e8p, \u27e8hp, hp'\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nH : Function.Injective \u2191(algebraMap A B)\nx : A\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (algebraMap R ((fun x => B) x)) (\u2191(algebraMap A B) x) p = 0\n\u22a2 IsIntegral R x\n[PROOFSTEP]\nrefine' \u27e8p, \u27e8hp, _\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nH : Function.Injective \u2191(algebraMap A B)\nx : A\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (algebraMap R ((fun x => B) x)) (\u2191(algebraMap A B) x) p = 0\n\u22a2 eval\u2082 (algebraMap R A) x p = 0\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq R A B, \u2190 eval\u2082_map, eval\u2082_hom, \u2190 RingHom.map_zero (algebraMap A B)] at hp' \n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nH : Function.Injective \u2191(algebraMap A B)\nx : A\np : R[X]\nhp : Monic p\nhp' : \u2191(algebraMap A B) (eval x (Polynomial.map (algebraMap R A) p)) = \u2191(algebraMap A B) 0\n\u22a2 eval\u2082 (algebraMap R A) x p = 0\n[PROOFSTEP]\nrw [eval\u2082_eq_eval_map]\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nH : Function.Injective \u2191(algebraMap A B)\nx : A\np : R[X]\nhp : Monic p\nhp' : \u2191(algebraMap A B) (eval x (Polynomial.map (algebraMap R A) p)) = \u2191(algebraMap A B) 0\n\u22a2 eval x (Polynomial.map (algebraMap R A) p) = 0\n[PROOFSTEP]\nexact H hp'\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nx : T\nh : IsIntegralElem (comp g f) x\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (comp g f) x p = 0\n\u22a2 eval\u2082 g x (Polynomial.map f p) = 0\n[PROOFSTEP]\nrwa [\u2190 eval\u2082_map] at hp' \n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nx : B\nh : IsIntegral R x\n\u22a2 IsIntegral A x\n[PROOFSTEP]\nrcases h with \u27e8p, \u27e8hp, hp'\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (algebraMap R B) x p = 0\n\u22a2 IsIntegral A x\n[PROOFSTEP]\nrefine' \u27e8p.map (algebraMap R A), \u27e8hp.map (algebraMap R A), _\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (algebraMap R B) x p = 0\n\u22a2 eval\u2082 (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0\n[PROOFSTEP]\nrw [IsScalarTower.algebraMap_eq R A B, \u2190 eval\u2082_map] at hp' \n[GOAL]\ncase intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nx : B\np : R[X]\nhp : Monic p\nhp' : eval\u2082 (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0\n\u22a2 eval\u2082 (algebraMap A B) x (Polynomial.map (algebraMap R A) p) = 0\n[PROOFSTEP]\nexact hp'\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\nhf : IsIntegral f\n\u22a2 IsIntegral (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I))\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\nhf : IsIntegral f\nx\u271d : S \u29f8 I\nx : S\n\u22a2 IsIntegralElem (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) (Quot.mk Setoid.r x)\n[PROOFSTEP]\nobtain \u27e8p, \u27e8p_monic, hpx\u27e9\u27e9 := hf x\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\nhf : IsIntegral f\nx\u271d : S \u29f8 I\nx : S\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 f x p = 0\n\u22a2 IsIntegralElem (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) (Quot.mk Setoid.r x)\n[PROOFSTEP]\nrefine' \u27e8p.map (Ideal.Quotient.mk _), \u27e8p_monic.map _, _\u27e9\u27e9\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\nhf : IsIntegral f\nx\u271d : S \u29f8 I\nx : S\np : R[X]\np_monic : Monic p\nhpx : eval\u2082 f x p = 0\n\u22a2 eval\u2082 (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) (Quot.mk Setoid.r x)\n      (Polynomial.map (Ideal.Quotient.mk (Ideal.comap f I)) p) =\n    0\n[PROOFSTEP]\nsimpa only [hom_eval\u2082, eval\u2082_map] using congr_arg (Ideal.Quotient.mk I) hpx\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\n\u22a2 RingHom.IsIntegral (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) \u2194\n    RingHom.IsIntegral (RingHom.comp (Ideal.Quotient.mk I) f)\n[PROOFSTEP]\nlet g :=\n  Ideal.Quotient.mk\n    (I.comap f)\n      -- Porting note: added type ascription\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng\u271d : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\ng : R \u2192+* R \u29f8 Ideal.comap f I := Ideal.Quotient.mk (Ideal.comap f I)\n\u22a2 RingHom.IsIntegral (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) \u2194\n    RingHom.IsIntegral (RingHom.comp (Ideal.Quotient.mk I) f)\n[PROOFSTEP]\nhave : (Ideal.quotientMap I f le_rfl).comp g = (Ideal.Quotient.mk I).comp f := Ideal.quotientMap_comp_mk le_rfl\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng\u271d : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\ng : R \u2192+* R \u29f8 Ideal.comap f I := Ideal.Quotient.mk (Ideal.comap f I)\nthis :\n  RingHom.comp (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) g = RingHom.comp (Ideal.Quotient.mk I) f\n\u22a2 RingHom.IsIntegral (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) \u2194\n    RingHom.IsIntegral (RingHom.comp (Ideal.Quotient.mk I) f)\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => RingHom.isIntegral_tower_top_of_isIntegral g _ (this \u25b8 h)\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng\u271d : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\ng : R \u2192+* R \u29f8 Ideal.comap f I := Ideal.Quotient.mk (Ideal.comap f I)\nthis :\n  RingHom.comp (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) g = RingHom.comp (Ideal.Quotient.mk I) f\nh : RingHom.IsIntegral (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I))\n\u22a2 RingHom.IsIntegral (RingHom.comp (Ideal.Quotient.mk I) f)\n[PROOFSTEP]\nrefine' this \u25b8 RingHom.isIntegral_trans g (Ideal.quotientMap I f le_rfl) _ h\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : CommRing T\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : Algebra R B\nf : R \u2192+* S\ng\u271d : S \u2192+* T\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R A B\nI : Ideal S\ng : R \u2192+* R \u29f8 Ideal.comap f I := Ideal.Quotient.mk (Ideal.comap f I)\nthis :\n  RingHom.comp (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I)) g = RingHom.comp (Ideal.Quotient.mk I) f\nh : RingHom.IsIntegral (Ideal.quotientMap I f (_ : Ideal.comap f I \u2264 Ideal.comap f I))\n\u22a2 RingHom.IsIntegral g\n[PROOFSTEP]\nexact RingHom.isIntegral_of_surjective g Ideal.Quotient.mk_surjective\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\n\u22a2 IsField R\n[PROOFSTEP]\nrefine'\n  \u27e8\u27e80, 1, zero_ne_one\u27e9, mul_comm, fun {a} ha => _\u27e9\n    -- Let `a_inv` be the inverse of `algebraMap R S a`,\n      -- then we need to show that `a_inv` is of the form `algebraMap R S b`.\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\n\u22a2 \u2203 b, a * b = 1\n[PROOFSTEP]\nobtain \u27e8a_inv, ha_inv\u27e9 :=\n  hS.mul_inv_cancel fun h =>\n    ha\n      (hRS (_root_.trans h (RingHom.map_zero _).symm))\n        -- Let `p : R[X]` be monic with root `a_inv`,\n          -- and `q` be `p` with coefficients reversed (so `q(a) = q'(a) * a + 1`).\n          -- We claim that `q(a) = 0`, so `-q'(a)` is the inverse of `a`.\n[GOAL]\ncase intro\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\n\u22a2 \u2203 b, a * b = 1\n[PROOFSTEP]\nobtain \u27e8p, p_monic, hp\u27e9 := H a_inv\n[GOAL]\ncase intro.intro.intro\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\n\u22a2 \u2203 b, a * b = 1\n[PROOFSTEP]\nuse-\u2211 i : \u2115 in Finset.range p.natDegree,\n      p.coeff i *\n        a ^\n          (p.natDegree - i - 1)\n            -- `q(a) = 0`, because multiplying everything with `a_inv^n` gives `p(a_inv) = 0`.\n              -- TODO: this could be a lemma for `Polynomial.reverse`.\n[GOAL]\ncase h\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\n\u22a2 a * -\u2211 i in Finset.range (natDegree p), coeff p i * a ^ (natDegree p - i - 1) = 1\n[PROOFSTEP]\nhave hq : (\u2211 i : \u2115 in Finset.range (p.natDegree + 1), p.coeff i * a ^ (p.natDegree - i)) = 0 :=\n  by\n  apply (injective_iff_map_eq_zero (algebraMap R S)).mp hRS\n  have a_inv_ne_zero : a_inv \u2260 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)\n  refine' (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.natDegree a_inv_ne_zero)\n  rw [eval\u2082_eq_sum_range] at hp \n  rw [map_sum, Finset.sum_mul]\n  refine' (Finset.sum_congr rfl fun i hi => _).trans hp\n  rw [RingHom.map_mul, mul_assoc]\n  congr\n  have : a_inv ^ p.natDegree = a_inv ^ (p.natDegree - i) * a_inv ^ i := by\n    rw [\u2190 pow_add a_inv, tsub_add_cancel_of_le (Nat.le_of_lt_succ (Finset.mem_range.mp hi))]\n  rw [RingHom.map_pow, this, \u2190 mul_assoc, \u2190 mul_pow, ha_inv, one_pow, one_mul]\n    -- Since `q(a) = 0` and `q(a) = q'(a) * a + 1`, we have `a * -q'(a) = 1`.\n      -- TODO: we could use a lemma for `Polynomial.divX` here.\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\n\u22a2 \u2211 i in Finset.range (natDegree p + 1), coeff p i * a ^ (natDegree p - i) = 0\n[PROOFSTEP]\napply (injective_iff_map_eq_zero (algebraMap R S)).mp hRS\n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\n\u22a2 \u2191(algebraMap R S) (\u2211 i in Finset.range (natDegree p + 1), coeff p i * a ^ (natDegree p - i)) = 0\n[PROOFSTEP]\nhave a_inv_ne_zero : a_inv \u2260 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)\n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\na_inv_ne_zero : a_inv \u2260 0\n\u22a2 \u2191(algebraMap R S) (\u2211 i in Finset.range (natDegree p + 1), coeff p i * a ^ (natDegree p - i)) = 0\n[PROOFSTEP]\nrefine' (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.natDegree a_inv_ne_zero)\n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\na_inv_ne_zero : a_inv \u2260 0\n\u22a2 \u2191(algebraMap R S) (\u2211 i in Finset.range (natDegree p + 1), coeff p i * a ^ (natDegree p - i)) * a_inv ^ natDegree p = 0\n[PROOFSTEP]\nrw [eval\u2082_eq_sum_range] at hp \n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\n\u22a2 \u2191(algebraMap R S) (\u2211 i in Finset.range (natDegree p + 1), coeff p i * a ^ (natDegree p - i)) * a_inv ^ natDegree p = 0\n[PROOFSTEP]\nrw [map_sum, Finset.sum_mul]\n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\n\u22a2 \u2211 x in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p x * a ^ (natDegree p - x)) * a_inv ^ natDegree p = 0\n[PROOFSTEP]\nrefine' (Finset.sum_congr rfl fun i hi => _).trans hp\n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p + 1)\n\u22a2 \u2191(algebraMap R S) (coeff p i * a ^ (natDegree p - i)) * a_inv ^ natDegree p =\n    \u2191(algebraMap R S) (coeff p i) * a_inv ^ i\n[PROOFSTEP]\nrw [RingHom.map_mul, mul_assoc]\n[GOAL]\ncase a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p + 1)\n\u22a2 \u2191(algebraMap R S) (coeff p i) * (\u2191(algebraMap R S) (a ^ (natDegree p - i)) * a_inv ^ natDegree p) =\n    \u2191(algebraMap R S) (coeff p i) * a_inv ^ i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.e_a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p + 1)\n\u22a2 \u2191(algebraMap R S) (a ^ (natDegree p - i)) * a_inv ^ natDegree p = a_inv ^ i\n[PROOFSTEP]\nhave : a_inv ^ p.natDegree = a_inv ^ (p.natDegree - i) * a_inv ^ i := by\n  rw [\u2190 pow_add a_inv, tsub_add_cancel_of_le (Nat.le_of_lt_succ (Finset.mem_range.mp hi))]\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p + 1)\n\u22a2 a_inv ^ natDegree p = a_inv ^ (natDegree p - i) * a_inv ^ i\n[PROOFSTEP]\nrw [\u2190 pow_add a_inv, tsub_add_cancel_of_le (Nat.le_of_lt_succ (Finset.mem_range.mp hi))]\n[GOAL]\ncase a.e_a\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : \u2211 i in Finset.range (natDegree p + 1), \u2191(algebraMap R S) (coeff p i) * a_inv ^ i = 0\na_inv_ne_zero : a_inv \u2260 0\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p + 1)\nthis : a_inv ^ natDegree p = a_inv ^ (natDegree p - i) * a_inv ^ i\n\u22a2 \u2191(algebraMap R S) (a ^ (natDegree p - i)) * a_inv ^ natDegree p = a_inv ^ i\n[PROOFSTEP]\nrw [RingHom.map_pow, this, \u2190 mul_assoc, \u2190 mul_pow, ha_inv, one_pow, one_mul]\n  -- Since `q(a) = 0` and `q(a) = q'(a) * a + 1`, we have `a * -q'(a) = 1`.\n    -- TODO: we could use a lemma for `Polynomial.divX` here.\n[GOAL]\ncase h\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\nhq : \u2211 i in Finset.range (natDegree p + 1), coeff p i * a ^ (natDegree p - i) = 0\n\u22a2 a * -\u2211 i in Finset.range (natDegree p), coeff p i * a ^ (natDegree p - i - 1) = 1\n[PROOFSTEP]\nrw [Finset.sum_range_succ_comm, p_monic.coeff_natDegree, one_mul, tsub_self, pow_zero, add_eq_zero_iff_eq_neg,\n  eq_comm] at hq \n[GOAL]\ncase h\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\nhq\u271d : a ^ (natDegree p - natDegree p) + \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 0\nhq : -\u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 1\n\u22a2 a * -\u2211 i in Finset.range (natDegree p), coeff p i * a ^ (natDegree p - i - 1) = 1\n[PROOFSTEP]\nrw [mul_comm, neg_mul, Finset.sum_mul]\n[GOAL]\ncase h\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\nhq\u271d : a ^ (natDegree p - natDegree p) + \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 0\nhq : -\u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 1\n\u22a2 -\u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x - 1) * a = 1\n[PROOFSTEP]\nconvert hq using 2\n[GOAL]\ncase h.e'_2.h.e'_3\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\nhq\u271d : a ^ (natDegree p - natDegree p) + \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 0\nhq : -\u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 1\n\u22a2 \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x - 1) * a =\n    \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\ncase h.e'_2.h.e'_3\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\nhq\u271d : a ^ (natDegree p - natDegree p) + \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 0\nhq : -\u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 1\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p)\n\u22a2 coeff p i * a ^ (natDegree p - i - 1) * a = coeff p i * a ^ (natDegree p - i)\n[PROOFSTEP]\nhave : 1 \u2264 p.natDegree - i := le_tsub_of_add_le_left (Finset.mem_range.mp hi)\n[GOAL]\ncase h.e'_2.h.e'_3\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b3 : CommRing R\u271d\ninst\u271d\u00b9\u00b2 : CommRing A\ninst\u271d\u00b9\u00b9 : CommRing B\ninst\u271d\u00b9\u2070 : CommRing S\u271d\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2076 : Algebra R\u271d A\ninst\u271d\u2075 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Nontrivial R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhRS : Function.Injective \u2191(algebraMap R S)\nhS : IsField S\na : R\nha : a \u2260 0\na_inv : S\nha_inv : \u2191(algebraMap R S) a * a_inv = 1\np : R[X]\np_monic : Monic p\nhp : eval\u2082 (algebraMap R S) a_inv p = 0\nhq\u271d : a ^ (natDegree p - natDegree p) + \u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 0\nhq : -\u2211 x in Finset.range (natDegree p), coeff p x * a ^ (natDegree p - x) = 1\ni : \u2115\nhi : i \u2208 Finset.range (natDegree p)\nthis : 1 \u2264 natDegree p - i\n\u22a2 coeff p i * a ^ (natDegree p - i - 1) * a = coeff p i * a ^ (natDegree p - i)\n[PROOFSTEP]\nrw [mul_assoc, \u2190 pow_succ', tsub_add_cancel_of_le this]\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\ninst\u271d\u2074 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\n\u22a2 IsField S\n[PROOFSTEP]\nletI := hR.toField\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\ninst\u271d\u2074 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\n\u22a2 IsField S\n[PROOFSTEP]\nrefine' \u27e8\u27e80, 1, zero_ne_one\u27e9, mul_comm, fun {x} hx => _\u27e9\n[GOAL]\nR\u271d : Type u_1\nA : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\ninst\u271d\u2074 : IsScalarTower R\u271d A B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\nx : S\nhx : x \u2260 0\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nlet A := Algebra.adjoin R ({ x } : Set S)\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\u271d\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A\u271d B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\u271d\ninst\u271d\u2074 : IsScalarTower R\u271d A\u271d B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\nx : S\nhx : x \u2260 0\nA : Subalgebra R S := adjoin R {x}\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nhaveI : IsNoetherian R A :=\n  isNoetherian_of_fg_of_noetherian (Subalgebra.toSubmodule A) (FG_adjoin_singleton_of_integral x (H x))\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\u271d\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A\u271d B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\u271d\ninst\u271d\u2074 : IsScalarTower R\u271d A\u271d B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis\u271d : Field R := IsField.toField hR\nx : S\nhx : x \u2260 0\nA : Subalgebra R S := adjoin R {x}\nthis : IsNoetherian R { x // x \u2208 A }\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nhaveI : Module.Finite R A := Module.IsNoetherian.finite R A\n[GOAL]\nR\u271d : Type u_1\nA\u271d : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\u271d\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A\u271d B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\u271d\ninst\u271d\u2074 : IsScalarTower R\u271d A\u271d B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis\u271d\u00b9 : Field R := IsField.toField hR\nx : S\nhx : x \u2260 0\nA : Subalgebra R S := adjoin R {x}\nthis\u271d : IsNoetherian R { x // x \u2208 A }\nthis : Module.Finite R { x // x \u2208 A }\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 :=\n  LinearMap.surjective_of_injective\n    (@LinearMap.mulLeft_injective R A _ _ _ _ \u27e8x, subset_adjoin (Set.mem_singleton x)\u27e9 fun h =>\n      hx (Subtype.ext_iff.mp h))\n    1\n[GOAL]\ncase intro\nR\u271d : Type u_1\nA\u271d : Type u_2\nB : Type u_3\nS\u271d : Type u_4\nT : Type u_5\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\ninst\u271d\u00b9\u00b9 : CommRing A\u271d\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : CommRing S\u271d\ninst\u271d\u2078 : CommRing T\ninst\u271d\u2077 : Algebra A\u271d B\ninst\u271d\u2076 : Algebra R\u271d B\nf : R\u271d \u2192+* S\u271d\ng : S\u271d \u2192+* T\ninst\u271d\u2075 : Algebra R\u271d A\u271d\ninst\u271d\u2074 : IsScalarTower R\u271d A\u271d B\nR : Type u_6\nS : Type u_7\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis\u271d\u00b9 : Field R := IsField.toField hR\nx : S\nhx : x \u2260 0\nA : Subalgebra R S := adjoin R {x}\nthis\u271d : IsNoetherian R { x // x \u2208 A }\nthis : Module.Finite R { x // x \u2208 A }\ny : { x // x \u2208 A }\nhy : \u2191(LinearMap.mulLeft R { val := x, property := (_ : x \u2208 \u2191(adjoin R {x})) }) y = 1\n\u22a2 \u2203 b, x * b = 1\n[PROOFSTEP]\nexact \u27e8y, Subtype.ext_iff.mp hy\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.IntegralClosure", "llama_tokens": 224906, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117855317474, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.5558245394981012}}
{"text": "[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 IsUnit a \u2194 a = 1\n[PROOFSTEP]\nrefine'\n  \u27e8fun h => _, by\n    rintro rfl\n    exact isUnit_one\u27e9\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 a = 1 \u2192 IsUnit a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nb : Cardinal.{u}\nn m : \u2115\n\u22a2 IsUnit 1\n[PROOFSTEP]\nexact isUnit_one\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\nh : IsUnit a\n\u22a2 a = 1\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\nb : Cardinal.{u}\nn m : \u2115\nh : IsUnit 0\n\u22a2 0 = 1\n[PROOFSTEP]\nexact (not_isUnit_zero h).elim\n[GOAL]\ncase inr\na b : Cardinal.{u}\nn m : \u2115\nh : IsUnit a\nha : a \u2260 0\n\u22a2 a = 1\n[PROOFSTEP]\nrw [isUnit_iff_forall_dvd] at h \n[GOAL]\ncase inr\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\n\u22a2 a = 1\n[PROOFSTEP]\ncases' h 1 with t ht\n[GOAL]\ncase inr.intro\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : 1 = a * t\n\u22a2 a = 1\n[PROOFSTEP]\nrw [eq_comm, mul_eq_one_iff'] at ht \n[GOAL]\ncase inr.intro\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : a = 1 \u2227 t = 1\n\u22a2 a = 1\n[PROOFSTEP]\nexact ht.1\n[GOAL]\ncase inr.intro.ha\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : a * t = 1\n\u22a2 1 \u2264 a\n[PROOFSTEP]\nexact one_le_iff_ne_zero.mpr ha\n[GOAL]\ncase inr.intro.hb\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : a * t = 1\n\u22a2 1 \u2264 t\n[PROOFSTEP]\napply one_le_iff_ne_zero.mpr\n[GOAL]\ncase inr.intro.hb\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : a * t = 1\n\u22a2 t \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase inr.intro.hb\na b : Cardinal.{u}\nn m : \u2115\nh\u271d : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : a * t = 1\nh : t = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h, mul_zero] at ht \n[GOAL]\ncase inr.intro.hb\na b : Cardinal.{u}\nn m : \u2115\nh\u271d : \u2200 (y : Cardinal.{u}), a \u2223 y\nha : a \u2260 0\nt : Cardinal.{u}\nht : 0 = 1\nh : t = 0\n\u22a2 False\n[PROOFSTEP]\nexact zero_ne_one ht\n[GOAL]\na\u271d b\u271d : Cardinal.{u}\nn m : \u2115\na x : Cardinal.{u_1}\nb0 : x \u2260 0\nb : Cardinal.{u_1}\nhab : x = a * b\n\u22a2 a \u2264 x\n[PROOFSTEP]\nsimpa only [hab, mul_one] using\n  mul_le_mul_left' (one_le_iff_ne_zero.2 fun h : b = 0 => b0 (by rwa [h, mul_zero] at hab )) a\n[GOAL]\na\u271d b\u271d : Cardinal.{u}\nn m : \u2115\na x : Cardinal.{u_1}\nb0 : x \u2260 0\nb : Cardinal.{u_1}\nhab : x = a * b\nh : b = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrwa [h, mul_zero] at hab \n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\n\u22a2 Prime a\n[PROOFSTEP]\nrefine' \u27e8(aleph0_pos.trans_le ha).ne', _, fun b c hbc => _\u27e9\n[GOAL]\ncase refine'_1\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\n\u22a2 \u00acIsUnit a\n[PROOFSTEP]\nrw [isUnit_iff]\n[GOAL]\ncase refine'_1\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\n\u22a2 \u00aca = 1\n[PROOFSTEP]\nexact (one_lt_aleph0.trans_le ha).ne'\n[GOAL]\ncase refine'_2\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\ncases' eq_or_ne (b * c) 0 with hz hz\n[GOAL]\ncase refine'_2.inl\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c = 0\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\nrcases mul_eq_zero.mp hz with (rfl | rfl)\n[GOAL]\ncase refine'_2.inl.inl\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nc : Cardinal.{u}\nhbc : a \u2223 0 * c\nhz : 0 * c = 0\n\u22a2 a \u2223 0 \u2228 a \u2223 c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.inl.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb : Cardinal.{u}\nhbc : a \u2223 b * 0\nhz : b * 0 = 0\n\u22a2 a \u2223 b \u2228 a \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\nwlog h : c \u2264 b\n[GOAL]\ncase refine'_2.inr.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\ncases le_total c b <;> [skip; rw [or_comm]]\n[GOAL]\ncase refine'_2.inr.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\ncases le_total c b\n[GOAL]\ncase refine'_2.inr.inr.inl\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_2.inr.inr.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\nrw [or_comm]\n[GOAL]\ncase refine'_2.inr.inr.inl\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase refine'_2.inr.inr.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\n\u22a2 a \u2223 c \u2228 a \u2223 b\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase refine'_2.inr.inr.inl.b\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 Cardinal.{u}\ncase refine'_2.inr.inr.inl.n\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 \u2115\ncase refine'_2.inr.inr.inl.m\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 \u2115\ncase refine'_2.inr.inr.inl.ha\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 \u2135\u2080 \u2264 a\ncase refine'_2.inr.inr.inl.hbc\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 a \u2223 b * c\ncase refine'_2.inr.inr.inl.hz\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 b * c \u2260 0\ncase refine'_2.inr.inr.inl.h\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : c \u2264 b\nhz_symm : 0 \u2260 b * c\n\u22a2 c \u2264 b\ncase refine'_2.inr.inr.inr.b\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 Cardinal.{u}\ncase refine'_2.inr.inr.inr.n\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 \u2115\ncase refine'_2.inr.inr.inr.m\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 \u2115\ncase refine'_2.inr.inr.inr.ha\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 \u2135\u2080 \u2264 a\ncase refine'_2.inr.inr.inr.hbc\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 a \u2223 c * b\ncase refine'_2.inr.inr.inr.hz\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 c * b \u2260 0\ncase refine'_2.inr.inr.inr.h\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 b \u2264 c\n[PROOFSTEP]\nassumption'\n[GOAL]\ncase refine'_2.inr.inr.inr.hbc\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 a \u2223 c * b\ncase refine'_2.inr.inr.inr.hz\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 c * b \u2260 0\n[PROOFSTEP]\nall_goals rwa [mul_comm]\n[GOAL]\ncase refine'_2.inr.inr.inr.hbc\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 a \u2223 c * b\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\ncase refine'_2.inr.inr.inr.hz\na b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nthis :\n  \u2200 {a : Cardinal.{u}} {b : Cardinal.{u}} {n m : \u2115},\n    \u2135\u2080 \u2264 a \u2192 \u2200 (b c : Cardinal.{u}), a \u2223 b * c \u2192 b * c \u2260 0 \u2192 c \u2264 b \u2192 a \u2223 b \u2228 a \u2223 c\nh : \u00acc \u2264 b\nh\u271d : b \u2264 c\nhz_symm : 0 \u2260 b * c\n\u22a2 c * b \u2260 0\n[PROOFSTEP]\nrwa [mul_comm]\n[GOAL]\na\u271d a b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nh : c \u2264 b\n\u22a2 a \u2223 b \u2228 a \u2223 c\n[PROOFSTEP]\nleft\n[GOAL]\ncase h\na\u271d a b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nh : c \u2264 b\n\u22a2 a \u2223 b\n[PROOFSTEP]\nhave habc := le_of_dvd hz hbc\n[GOAL]\ncase h\na\u271d a b\u271d : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nb c : Cardinal.{u}\nhbc : a \u2223 b * c\nhz : b * c \u2260 0\nh : c \u2264 b\nhabc : a \u2264 b * c\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrwa [mul_eq_max' <| ha.trans <| habc, max_def', if_pos h] at hbc \n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\n\u22a2 \u00acIrreducible a\n[PROOFSTEP]\nrw [irreducible_iff, not_and_or]\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\n\u22a2 \u00ac\u00acIsUnit a \u2228 \u00ac\u2200 (a_1 b : Cardinal.{u}), a = a_1 * b \u2192 IsUnit a_1 \u2228 IsUnit b\n[PROOFSTEP]\nrefine' Or.inr fun h => _\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\nha : \u2135\u2080 \u2264 a\nh : \u2200 (a_1 b : Cardinal.{u}), a = a_1 * b \u2192 IsUnit a_1 \u2228 IsUnit b\n\u22a2 False\n[PROOFSTEP]\nsimpa [mul_aleph0_eq ha, isUnit_iff, (one_lt_aleph0.trans_le ha).ne', one_lt_aleph0.ne'] using h a \u2135\u2080\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 \u2191n \u2223 \u2191m \u2194 n \u2223 m\n[PROOFSTEP]\nrefine' \u27e8_, fun \u27e8h, ht\u27e9 => \u27e8h, by exact_mod_cast ht\u27e9\u27e9\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\nx\u271d : n \u2223 m\nh : \u2115\nht : m = n * h\n\u22a2 \u2191m = \u2191n * \u2191h\n[PROOFSTEP]\nexact_mod_cast ht\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 \u2191n \u2223 \u2191m \u2192 n \u2223 m\n[PROOFSTEP]\nrintro \u27e8k, hk\u27e9\n[GOAL]\ncase intro\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\n\u22a2 n \u2223 m\n[PROOFSTEP]\nhave : \u2191m < \u2135\u2080 := nat_lt_aleph0 m\n[GOAL]\ncase intro\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nthis : \u2191m < \u2135\u2080\n\u22a2 n \u2223 m\n[PROOFSTEP]\nrw [hk, mul_lt_aleph0_iff] at this \n[GOAL]\ncase intro\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nthis : \u2191n = 0 \u2228 k = 0 \u2228 \u2191n < \u2135\u2080 \u2227 k < \u2135\u2080\n\u22a2 n \u2223 m\n[PROOFSTEP]\nrcases this with (h | h | \u27e8-, hk'\u27e9)\n[GOAL]\ncase intro.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nh : \u2191n = 0\n\u22a2 n \u2223 m\ncase intro.inr.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nh : k = 0\n\u22a2 n \u2223 m\ncase intro.inr.inr.intro a b : Cardinal.{u} n m : \u2115 k : Cardinal.{u_1} hk : \u2191m = \u2191n * k hk' : k < \u2135\u2080 \u22a2 n \u2223 m\n[PROOFSTEP]\niterate 2 simp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk ; simp [hk]\n[GOAL]\ncase intro.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nh : \u2191n = 0\n\u22a2 n \u2223 m\ncase intro.inr.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nh : k = 0\n\u22a2 n \u2223 m\ncase intro.inr.inr.intro a b : Cardinal.{u} n m : \u2115 k : Cardinal.{u_1} hk : \u2191m = \u2191n * k hk' : k < \u2135\u2080 \u22a2 n \u2223 m\n[PROOFSTEP]\nsimp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk \n[GOAL]\ncase intro.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nh : \u2191n = 0\nhk : m = 0\n\u22a2 n \u2223 m\ncase intro.inr.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nh : k = 0\n\u22a2 n \u2223 m\ncase intro.inr.inr.intro a b : Cardinal.{u} n m : \u2115 k : Cardinal.{u_1} hk : \u2191m = \u2191n * k hk' : k < \u2135\u2080 \u22a2 n \u2223 m\n[PROOFSTEP]\nsimp [hk]\n[GOAL]\ncase intro.inr.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nh : k = 0\n\u22a2 n \u2223 m\ncase intro.inr.inr.intro a b : Cardinal.{u} n m : \u2115 k : Cardinal.{u_1} hk : \u2191m = \u2191n * k hk' : k < \u2135\u2080 \u22a2 n \u2223 m\n[PROOFSTEP]\nsimp only [h, mul_zero, zero_mul, Nat.cast_eq_zero] at hk \n[GOAL]\ncase intro.inr.inl\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nh : k = 0\nhk : m = 0\n\u22a2 n \u2223 m\ncase intro.inr.inr.intro a b : Cardinal.{u} n m : \u2115 k : Cardinal.{u_1} hk : \u2191m = \u2191n * k hk' : k < \u2135\u2080 \u22a2 n \u2223 m\n[PROOFSTEP]\nsimp [hk]\n[GOAL]\ncase intro.inr.inr.intro\na b : Cardinal.{u}\nn m : \u2115\nk : Cardinal.{u_1}\nhk : \u2191m = \u2191n * k\nhk' : k < \u2135\u2080\n\u22a2 n \u2223 m\n[PROOFSTEP]\nlift k to \u2115 using hk'\n[GOAL]\ncase intro.inr.inr.intro.intro\na b : Cardinal.{u}\nn m k : \u2115\nhk : \u2191m = \u2191n * \u2191k\n\u22a2 n \u2223 m\n[PROOFSTEP]\nexact \u27e8k, by exact_mod_cast hk\u27e9\n[GOAL]\na b : Cardinal.{u}\nn m k : \u2115\nhk : \u2191m = \u2191n * \u2191k\n\u22a2 m = n * k\n[PROOFSTEP]\nexact_mod_cast hk\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 Prime \u2191n \u2194 Nat.Prime n\n[PROOFSTEP]\nsimp only [Prime, Nat.prime_iff]\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 (\u2191n \u2260 0 \u2227 \u00acIsUnit \u2191n \u2227 \u2200 (a b : Cardinal.{u_1}), \u2191n \u2223 a * b \u2192 \u2191n \u2223 a \u2228 \u2191n \u2223 b) \u2194\n    n \u2260 0 \u2227 \u00acIsUnit n \u2227 \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\n[PROOFSTEP]\nrefine' and_congr (by simp) (and_congr _ \u27e8fun h b c hbc => _, fun h b c hbc => _\u27e9)\n[GOAL]\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 \u2191n \u2260 0 \u2194 n \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 \u00acIsUnit \u2191n \u2194 \u00acIsUnit n\n[PROOFSTEP]\nsimp only [isUnit_iff, Nat.isUnit_iff]\n[GOAL]\ncase refine'_1\na b : Cardinal.{u}\nn m : \u2115\n\u22a2 \u00ac\u2191n = 1 \u2194 \u00acn = 1\n[PROOFSTEP]\nexact_mod_cast Iff.rfl\n[GOAL]\ncase refine'_2\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : Cardinal.{u_1}), \u2191n \u2223 a * b \u2192 \u2191n \u2223 a \u2228 \u2191n \u2223 b\nb c : \u2115\nhbc : n \u2223 b * c\n\u22a2 n \u2223 b \u2228 n \u2223 c\n[PROOFSTEP]\nexact_mod_cast h b c (by exact_mod_cast hbc)\n[GOAL]\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : Cardinal.{u_1}), \u2191n \u2223 a * b \u2192 \u2191n \u2223 a \u2228 \u2191n \u2223 b\nb c : \u2115\nhbc : n \u2223 b * c\n\u22a2 \u2191n \u2223 \u2191b * \u2191c\n[PROOFSTEP]\nexact_mod_cast hbc\n[GOAL]\ncase refine'_3\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\ncases' lt_or_le (b * c) \u2135\u2080 with h' h'\n[GOAL]\ncase refine'_3.inl\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : b * c < \u2135\u2080\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nrcases mul_lt_aleph0_iff.mp h' with (rfl | rfl | \u27e8hb, hc\u27e9)\n[GOAL]\ncase refine'_3.inl.inl\na b : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nc : Cardinal.{u_1}\nhbc : \u2191n \u2223 0 * c\nh' : 0 * c < \u2135\u2080\n\u22a2 \u2191n \u2223 0 \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3.inl.inr.inl\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * 0\nh' : b * 0 < \u2135\u2080\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3.inl.inr.inr.intro\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : b * c < \u2135\u2080\nhb : b < \u2135\u2080\nhc : c < \u2135\u2080\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nlift b to \u2115 using hb\n[GOAL]\ncase refine'_3.inl.inr.inr.intro.intro\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nc : Cardinal.{u_1}\nhc : c < \u2135\u2080\nb : \u2115\nhbc : \u2191n \u2223 \u2191b * c\nh' : \u2191b * c < \u2135\u2080\n\u22a2 \u2191n \u2223 \u2191b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nlift c to \u2115 using hc\n[GOAL]\ncase refine'_3.inl.inr.inr.intro.intro.intro\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : \u2115\nhbc : \u2191n \u2223 \u2191b * \u2191c\nh' : \u2191b * \u2191c < \u2135\u2080\n\u22a2 \u2191n \u2223 \u2191b \u2228 \u2191n \u2223 \u2191c\n[PROOFSTEP]\nexact_mod_cast h b c (by exact_mod_cast hbc)\n[GOAL]\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : \u2115\nhbc : \u2191n \u2223 \u2191b * \u2191c\nh' : \u2191b * \u2191c < \u2135\u2080\n\u22a2 n \u2223 b * c\n[PROOFSTEP]\nexact_mod_cast hbc\n[GOAL]\ncase refine'_3.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nrcases aleph0_le_mul_iff.mp h' with \u27e8hb, hc, h\u2135\u2080\u27e9\n[GOAL]\ncase refine'_3.inr.intro.intro\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nhave hn : (n : Cardinal) \u2260 0 := by\n  intro h\n  rw [h, zero_dvd_iff, mul_eq_zero] at hbc \n  cases hbc <;> contradiction\n[GOAL]\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\na b\u271d : Cardinal.{u}\nn m : \u2115\nh\u271d : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nh : \u2191n = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h, zero_dvd_iff, mul_eq_zero] at hbc \n[GOAL]\na b\u271d : Cardinal.{u}\nn m : \u2115\nh\u271d : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : b = 0 \u2228 c = 0\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nh : \u2191n = 0\n\u22a2 False\n[PROOFSTEP]\ncases hbc\n[GOAL]\ncase inl\na b\u271d : Cardinal.{u}\nn m : \u2115\nh\u271d\u00b9 : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nh : \u2191n = 0\nh\u271d : b = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nh\u271d\u00b9 : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nh : \u2191n = 0\nh\u271d : c = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase refine'_3.inr.intro.intro\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nwlog h\u2135\u2080b : \u2135\u2080 \u2264 b\n[GOAL]\ncase refine'_3.inr.intro.intro.inr\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nrefine' (this h c b _ _ hc hb h\u2135\u2080.symm hn (h\u2135\u2080.resolve_left h\u2135\u2080b)).symm\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_1\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 Cardinal.{u}\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_1\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 Cardinal.{u}\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_2\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 Cardinal.{u}\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_2\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 Cardinal.{u}\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_3\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2115\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_3\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2115\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_4\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 c * b\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_4\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 c * b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_5\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2135\u2080 \u2264 c * b\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_5\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2135\u2080 \u2264 c * b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_4\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 c * b\ncase refine'_3.inr.intro.intro.inr.refine'_5\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2135\u2080 \u2264 c * b\nn\u271d : \u2115\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nh\u2135\u2080b : \u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nrwa [mul_comm] at hbc \n[GOAL]\ncase refine'_3.inr.intro.intro.inr.refine'_5\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nthis :\n  \u2200 {a b : Cardinal.{u}} {n : \u2115} {m : \u2115},\n    (\u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b) \u2192\n      \u2200 (b c : Cardinal.{u_1}),\n        \u2191n \u2223 b * c \u2192 \u2135\u2080 \u2264 b * c \u2192 b \u2260 0 \u2192 c \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c \u2192 \u2191n \u2260 0 \u2192 \u2135\u2080 \u2264 b \u2192 \u2191n \u2223 b \u2228 \u2191n \u2223 c\nh\u2135\u2080b : \u00ac\u2135\u2080 \u2264 b\n\u22a2 \u2135\u2080 \u2264 c * b\nn\u271d : \u2115\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nh\u2135\u2080b : \u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nrwa [mul_comm] at h' \n[GOAL]\nn\u271d : \u2115\na b\u271d : Cardinal.{u}\nn m : \u2115\nh : \u2200 (a b : \u2115), n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\nb c : Cardinal.{u_1}\nhbc : \u2191n \u2223 b * c\nh' : \u2135\u2080 \u2264 b * c\nhb : b \u2260 0\nhc : c \u2260 0\nh\u2135\u2080 : \u2135\u2080 \u2264 b \u2228 \u2135\u2080 \u2264 c\nhn : \u2191n \u2260 0\nh\u2135\u2080b : \u2135\u2080 \u2264 b\n\u22a2 \u2191n \u2223 b \u2228 \u2191n \u2223 c\n[PROOFSTEP]\nexact Or.inl (dvd_of_le_of_aleph0_le hn ((nat_lt_aleph0 n).le.trans h\u2135\u2080b) h\u2135\u2080b)\n[GOAL]\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\n\u22a2 Prime a \u2194 \u2135\u2080 \u2264 a \u2228 \u2203 p, a = \u2191p \u2227 Nat.Prime p\n[PROOFSTEP]\ncases' le_or_lt \u2135\u2080 a with h h\n[GOAL]\ncase inl\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\nh : \u2135\u2080 \u2264 a\n\u22a2 Prime a \u2194 \u2135\u2080 \u2264 a \u2228 \u2203 p, a = \u2191p \u2227 Nat.Prime p\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\nh : a < \u2135\u2080\n\u22a2 Prime a \u2194 \u2135\u2080 \u2264 a \u2228 \u2203 p, a = \u2191p \u2227 Nat.Prime p\n[PROOFSTEP]\nlift a to \u2115 using id h\n[GOAL]\ncase inr.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u2191a < \u2135\u2080\n\u22a2 Prime \u2191a \u2194 \u2135\u2080 \u2264 \u2191a \u2228 \u2203 p, \u2191a = \u2191p \u2227 Nat.Prime p\n[PROOFSTEP]\nsimp [not_le.mpr h]\n[GOAL]\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\n\u22a2 IsPrimePow a \u2194 \u2135\u2080 \u2264 a \u2228 \u2203 n, a = \u2191n \u2227 IsPrimePow n\n[PROOFSTEP]\nby_cases h : \u2135\u2080 \u2264 a\n[GOAL]\ncase pos\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\nh : \u2135\u2080 \u2264 a\n\u22a2 IsPrimePow a \u2194 \u2135\u2080 \u2264 a \u2228 \u2203 n, a = \u2191n \u2227 IsPrimePow n\n[PROOFSTEP]\nsimp [h, (prime_of_aleph0_le h).isPrimePow]\n[GOAL]\ncase neg\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\nh : \u00ac\u2135\u2080 \u2264 a\n\u22a2 IsPrimePow a \u2194 \u2135\u2080 \u2264 a \u2228 \u2203 n, a = \u2191n \u2227 IsPrimePow n\n[PROOFSTEP]\nsimp only [h, Nat.cast_inj, exists_eq_left', false_or_iff, isPrimePow_nat_iff]\n[GOAL]\ncase neg\na\u271d b : Cardinal.{u}\nn m : \u2115\na : Cardinal.{u_1}\nh : \u00ac\u2135\u2080 \u2264 a\n\u22a2 IsPrimePow a \u2194 \u2203 n, a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nlift a to \u2115 using not_le.mp h\n[GOAL]\ncase neg.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\n\u22a2 IsPrimePow \u2191a \u2194 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrw [isPrimePow_def]\n[GOAL]\ncase neg.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\n\u22a2 (\u2203 p k, Prime p \u2227 0 < k \u2227 p ^ k = \u2191a) \u2194 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrefine' \u27e8_, fun \u27e8n, han, p, k, hp, hk, h\u27e9 => \u27e8p, k, nat_is_prime_iff.2 hp, hk, by rw [han]; exact_mod_cast h\u27e9\u27e9\n[GOAL]\na\u271d b : Cardinal.{u}\nn\u271d m a : \u2115\nh\u271d : \u00ac\u2135\u2080 \u2264 \u2191a\nx\u271d : \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\nn : \u2115\nhan : \u2191a = \u2191n\np k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\nh : p ^ k = n\n\u22a2 \u2191p ^ k = \u2191a\n[PROOFSTEP]\nrw [han]\n[GOAL]\na\u271d b : Cardinal.{u}\nn\u271d m a : \u2115\nh\u271d : \u00ac\u2135\u2080 \u2264 \u2191a\nx\u271d : \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\nn : \u2115\nhan : \u2191a = \u2191n\np k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\nh : p ^ k = n\n\u22a2 \u2191p ^ k = \u2191n\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\ncase neg.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\n\u22a2 (\u2203 p k, Prime p \u2227 0 < k \u2227 p ^ k = \u2191a) \u2192 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrintro \u27e8p, k, hp, hk, hpk\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\np : Cardinal.{u_1}\nk : \u2115\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = \u2191a\n\u22a2 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nhave key : p ^ 1 \u2264 \u2191a := by rw [\u2190 hpk]; apply power_le_power_left hp.ne_zero; exact_mod_cast hk\n[GOAL]\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\np : Cardinal.{u_1}\nk : \u2115\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = \u2191a\n\u22a2 p ^ 1 \u2264 \u2191a\n[PROOFSTEP]\nrw [\u2190 hpk]\n[GOAL]\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\np : Cardinal.{u_1}\nk : \u2115\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = \u2191a\n\u22a2 p ^ 1 \u2264 p ^ k\n[PROOFSTEP]\napply power_le_power_left hp.ne_zero\n[GOAL]\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\np : Cardinal.{u_1}\nk : \u2115\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = \u2191a\n\u22a2 1 \u2264 \u2191k\n[PROOFSTEP]\nexact_mod_cast hk\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\np : Cardinal.{u_1}\nk : \u2115\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = \u2191a\nkey : p ^ 1 \u2264 \u2191a\n\u22a2 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrw [power_one] at key \n[GOAL]\ncase neg.intro.intro.intro.intro.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\np : Cardinal.{u_1}\nk : \u2115\nhp : Prime p\nhk : 0 < k\nhpk : p ^ k = \u2191a\nkey : p \u2264 \u2191a\n\u22a2 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nlift p to \u2115 using key.trans_lt (nat_lt_aleph0 a)\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\nk : \u2115\nhk : 0 < k\np : \u2115\nhp : Prime \u2191p\nhpk : \u2191p ^ k = \u2191a\nkey : \u2191p \u2264 \u2191a\n\u22a2 \u2203 n, \u2191a = \u2191n \u2227 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nexact \u27e8a, rfl, p, k, nat_is_prime_iff.mp hp, hk, by exact_mod_cast hpk\u27e9\n[GOAL]\na\u271d b : Cardinal.{u}\nn m a : \u2115\nh : \u00ac\u2135\u2080 \u2264 \u2191a\nk : \u2115\nhk : 0 < k\np : \u2115\nhp : Prime \u2191p\nhpk : \u2191p ^ k = \u2191a\nkey : \u2191p \u2264 \u2191a\n\u22a2 p ^ k = a\n[PROOFSTEP]\nexact_mod_cast hpk\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Cardinal.Divisibility", "llama_tokens": 20826, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950947024555, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.5556352933339149}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (leftDerived F n).map f =\n    (leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) \u226b\n        (leftDerivedObjIso F n Q).inv\n[PROOFSTEP]\ndsimp only [Functor.leftDerived, Functor.leftDerivedObjIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (projectiveResolutions C \u22d9\n          mapHomotopyCategory F (ComplexShape.down \u2115) \u22d9 HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      f =\n    ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).mapIso\n            (HomotopyCategory.isoOfHomotopyEquiv\n              (mapHomotopyEquiv F (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P))) \u226a\u226b\n          (HomotopyCategory.homologyFactors D (ComplexShape.down \u2115) n).app\n            ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) \u226b\n        ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).mapIso\n              (HomotopyCategory.isoOfHomotopyEquiv\n                (mapHomotopyEquiv F (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q))) \u226a\u226b\n            (HomotopyCategory.homologyFactors D (ComplexShape.down \u2115) n).app\n              ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj Q.complex)).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((mapHomotopyCategory F (ComplexShape.down \u2115)).map ((projectiveResolutions C).map f)) =\n    ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n        \ud835\udfd9\n          ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj\n              ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj Q.complex) n \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj Q.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n).right) \u226b\n        \ud835\udfd9 (HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj Q.complex) n) \u226b\n          (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n              ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n                (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv))\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((mapHomotopyCategory F (ComplexShape.down \u2115)).map ((projectiveResolutions C).map f)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj Q.complex) n \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down \u2115)).obj Q.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g) n).right) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv))\n[PROOFSTEP]\nrw [\u2190 homologyFunctor_map, HomotopyCategory.homologyFunctor_map_factors]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((mapHomotopyCategory F (ComplexShape.down \u2115)).map ((projectiveResolutions C).map f)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex F (ComplexShape.down \u2115)).map g)) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv))\n[PROOFSTEP]\nsimp only [\u2190 Functor.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((mapHomotopyCategory F (ComplexShape.down \u2115)).map ((projectiveResolutions C).map f)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n          ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n            g \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 (mapHomotopyCategory F (ComplexShape.down \u2115)).map ((projectiveResolutions C).map f) =\n    (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n        ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          g \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv))\n[PROOFSTEP]\napply HomotopyCategory.eq_of_homotopy\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 Homotopy ((mapHomologicalComplex F (ComplexShape.down \u2115)).map (projectiveResolution.lift f))\n    ((mapHomologicalComplex F (ComplexShape.down \u2115)).map\n      ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        g \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv))\n[PROOFSTEP]\napply Functor.mapHomotopy\n[GOAL]\ncase e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 Homotopy (projectiveResolution.lift f)\n    ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n      g \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv)\n[PROOFSTEP]\napply ProjectiveResolution.liftHomotopy f\n[GOAL]\ncase e_a.h.h.g_comm\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 projectiveResolution.lift f \u226b (projectiveResolution Y).\u03c0 = (projectiveResolution X).\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.h.h.h_comm\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Additive F\nn : \u2115\nX Y : C\nf : X \u27f6 Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\ng : P.complex \u27f6 Q.complex\nw : g \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n\u22a2 ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        g \u226b (ProjectiveResolution.homotopyEquiv (projectiveResolution Y) Q).inv) \u226b\n      (projectiveResolution Y).\u03c0 =\n    (projectiveResolution X).\u03c0 \u226b (ChainComplex.single\u2080 C).map f\n[PROOFSTEP]\nsimp [w]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nn : \u2115\n\u22a2 leftDerived (\ud835\udfd9 F) n = \ud835\udfd9 (Functor.leftDerived F n)\n[PROOFSTEP]\nsimp [NatTrans.leftDerived]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b9 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u2070 : Preadditive C\ninst\u271d\u2079 : HasZeroObject C\ninst\u271d\u2078 : HasEqualizers C\ninst\u271d\u2077 : HasImages C\ninst\u271d\u2076 : HasProjectiveResolutions C\ninst\u271d\u2075 : Preadditive D\ninst\u271d\u2074 : HasEqualizers D\ninst\u271d\u00b3 : HasCokernels D\ninst\u271d\u00b2 : HasImages D\ninst\u271d\u00b9 : HasImageMaps D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nn : \u2115\n\u22a2 \ud835\udfd9\n      (projectiveResolutions C \u22d9\n        Functor.mapHomotopyCategory F (ComplexShape.down \u2115) \u22d9\n          HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n) =\n    \ud835\udfd9 (Functor.leftDerived F n)\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u2074 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b3 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b2 : Preadditive C\ninst\u271d\u00b9\u00b9 : HasZeroObject C\ninst\u271d\u00b9\u2070 : HasEqualizers C\ninst\u271d\u2079 : HasImages C\ninst\u271d\u2078 : HasProjectiveResolutions C\ninst\u271d\u2077 : Preadditive D\ninst\u271d\u2076 : HasEqualizers D\ninst\u271d\u2075 : HasCokernels D\ninst\u271d\u2074 : HasImages D\ninst\u271d\u00b3 : HasImageMaps D\nF G H : C \u2964 D\ninst\u271d\u00b2 : Functor.Additive F\ninst\u271d\u00b9 : Functor.Additive G\ninst\u271d : Functor.Additive H\n\u03b1 : F \u27f6 G\n\u03b2 : G \u27f6 H\nn : \u2115\n\u22a2 leftDerived (\u03b1 \u226b \u03b2) n = leftDerived \u03b1 n \u226b leftDerived \u03b2 n\n[PROOFSTEP]\nsimp [NatTrans.leftDerived]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 app (leftDerived \u03b1 n) X =\n    (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv\n[PROOFSTEP]\nsymm\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv =\n    app (leftDerived \u03b1 n) X\n[PROOFSTEP]\ndsimp [NatTrans.leftDerived, Functor.leftDerivedObjIso]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n        \ud835\udfd9\n          ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n)\n          (HomologicalComplex.Hom.sqFrom (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n).right) \u226b\n        \ud835\udfd9 (HomologicalComplex.homology ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n) \u226b\n          (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n              ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n                (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as))\n[PROOFSTEP]\nsimp only [Category.comp_id, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n)\n          (HomologicalComplex.Hom.sqFrom (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex) n).right) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as))\n[PROOFSTEP]\nrw [\u2190 homologyFunctor_map, HomotopyCategory.homologyFunctor_map_factors]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex)) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as))\n[PROOFSTEP]\nsimp only [\u2190 Functor.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex \u226b\n            (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex \u226b\n          (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) =\n    (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as)\n[PROOFSTEP]\napply HomotopyCategory.eq_of_homotopy\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n      app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) P.complex \u226b\n        (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)\n    (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as)\n[PROOFSTEP]\nsimp only [NatTrans.mapHomologicalComplex_naturality_assoc, \u2190 Functor.map_comp]\n[GOAL]\ncase e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as \u226b\n      (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n        ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    (app (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)) ((projectiveResolutions C).obj X).as)\n[PROOFSTEP]\napply Homotopy.compLeftId\n[GOAL]\ncase e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n      ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))\n[PROOFSTEP]\nrefine' (Functor.mapHomotopy _ (HomotopyEquiv.homotopyHomInvId _)).trans _\n[GOAL]\ncase e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))\n[PROOFSTEP]\napply Homotopy.ofEq\n[GOAL]\ncase e_a.h.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as) =\n    \ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as)\n[PROOFSTEP]\nsimp only [Functor.map_id]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Functor.LeftDerived", "llama_tokens": 12226, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950907764118, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.5556352907489391}}
{"text": "[GOAL]\nn : \u2115\nhn1 : n \u2260 1\nhn : Odd n\n\u22a2 (n + 1)# = n#\n[PROOFSTEP]\nrefine prod_congr ?_ fun _ _ \u21a6 rfl\n[GOAL]\nn : \u2115\nhn1 : n \u2260 1\nhn : Odd n\n\u22a2 filter Nat.Prime (range (n + 1 + 1)) = filter Nat.Prime (range (n + 1))\n[PROOFSTEP]\nrw [range_succ, filter_insert, if_neg fun h \u21a6 odd_iff_not_even.mp hn _]\n[GOAL]\nn : \u2115\nhn1 : n \u2260 1\nhn : Odd n\n\u22a2 Nat.Prime (n + 1) \u2192 Even n\n[PROOFSTEP]\nexact fun h \u21a6 h.even_sub_one <| mt succ.inj hn1\n[GOAL]\nm n : \u2115\n\u22a2 (m + n)# = m# * \u220f p in filter Nat.Prime (Ico (m + 1) (m + n + 1)), p\n[PROOFSTEP]\nrw [primorial, primorial, \u2190 Ico_zero_eq_range, \u2190 prod_union, \u2190 filter_union, Ico_union_Ico_eq_Ico]\n[GOAL]\ncase hab\nm n : \u2115\n\u22a2 0 \u2264 m + 1\ncase hbc\nm n : \u2115\n\u22a2 m + 1 \u2264 m + n + 1\nm n : \u2115 \u22a2 Disjoint (filter Nat.Prime (Ico 0 (m + 1))) (filter Nat.Prime (Ico (m + 1) (m + n + 1)))\n[PROOFSTEP]\nexacts [Nat.zero_le _, add_le_add_right (Nat.le_add_right _ _) _,\n  disjoint_filter_filter <| Ico_disjoint_Ico_consecutive _ _ _]\n[GOAL]\nm n : \u2115\nh : n \u2264 m\np : \u2115\nhp : p \u2208 filter Nat.Prime (Ico (m + 1) (m + n + 1))\n\u22a2 p \u2223 Nat.choose (m + n) m\n[PROOFSTEP]\nrw [mem_filter, mem_Ico] at hp \n[GOAL]\nm n : \u2115\nh : n \u2264 m\np : \u2115\nhp : (m + 1 \u2264 p \u2227 p < m + n + 1) \u2227 Nat.Prime p\n\u22a2 p \u2223 Nat.choose (m + n) m\n[PROOFSTEP]\nexact hp.2.dvd_choose_add hp.1.1 (h.trans_lt (m.lt_succ_self.trans_le hp.1.1)) (Nat.lt_succ_iff.1 hp.1.2)\n[GOAL]\nn : \u2115\n\u22a2 n# \u2264 4 ^ n\n[PROOFSTEP]\ninduction' n using Nat.strong_induction_on with n ihn\n[GOAL]\ncase h\nn : \u2115\nihn : \u2200 (m : \u2115), m < n \u2192 m# \u2264 4 ^ m\n\u22a2 n# \u2264 4 ^ n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase h.zero\nihn : \u2200 (m : \u2115), m < zero \u2192 m# \u2264 4 ^ m\n\u22a2 zero# \u2264 4 ^ zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.succ\nn : \u2115\nihn : \u2200 (m : \u2115), m < succ n \u2192 m# \u2264 4 ^ m\n\u22a2 succ n# \u2264 4 ^ succ n\n[PROOFSTEP]\nrcases n.even_or_odd with (\u27e8m, rfl\u27e9 | ho)\n[GOAL]\ncase h.succ.inl.intro\nm : \u2115\nihn : \u2200 (m_1 : \u2115), m_1 < succ (m + m) \u2192 m_1# \u2264 4 ^ m_1\n\u22a2 succ (m + m)# \u2264 4 ^ succ (m + m)\n[PROOFSTEP]\nrcases m.eq_zero_or_pos with (rfl | hm)\n[GOAL]\ncase h.succ.inl.intro.inl\nihn : \u2200 (m : \u2115), m < succ (0 + 0) \u2192 m# \u2264 4 ^ m\n\u22a2 succ (0 + 0)# \u2264 4 ^ succ (0 + 0)\n[PROOFSTEP]\ndecide\n[GOAL]\ncase h.succ.inl.intro.inr\nm : \u2115\nihn : \u2200 (m_1 : \u2115), m_1 < succ (m + m) \u2192 m_1# \u2264 4 ^ m_1\nhm : m > 0\n\u22a2 succ (m + m)# \u2264 4 ^ succ (m + m)\n[PROOFSTEP]\ncalc\n  (m + m + 1)# = (m + 1 + m)# := by rw [add_right_comm]\n  _ \u2264 (m + 1)# * choose (m + 1 + m) (m + 1) := (primorial_add_le m.le_succ)\n  _ = (m + 1)# * choose (2 * m + 1) m := by rw [choose_symm_add, two_mul, add_right_comm]\n  _ \u2264 4 ^ (m + 1) * 4 ^ m :=\n    (mul_le_mul' (ihn _ <| succ_lt_succ <| (lt_add_iff_pos_left _).2 hm) (choose_middle_le_pow _))\n  _ \u2264 4 ^ (m + m + 1) := by rw [\u2190 pow_add, add_right_comm]\n[GOAL]\nm : \u2115\nihn : \u2200 (m_1 : \u2115), m_1 < succ (m + m) \u2192 m_1# \u2264 4 ^ m_1\nhm : m > 0\n\u22a2 (m + m + 1)# = (m + 1 + m)#\n[PROOFSTEP]\nrw [add_right_comm]\n[GOAL]\nm : \u2115\nihn : \u2200 (m_1 : \u2115), m_1 < succ (m + m) \u2192 m_1# \u2264 4 ^ m_1\nhm : m > 0\n\u22a2 (m + 1)# * Nat.choose (m + 1 + m) (m + 1) = (m + 1)# * Nat.choose (2 * m + 1) m\n[PROOFSTEP]\nrw [choose_symm_add, two_mul, add_right_comm]\n[GOAL]\nm : \u2115\nihn : \u2200 (m_1 : \u2115), m_1 < succ (m + m) \u2192 m_1# \u2264 4 ^ m_1\nhm : m > 0\n\u22a2 4 ^ (m + 1) * 4 ^ m \u2264 4 ^ (m + m + 1)\n[PROOFSTEP]\nrw [\u2190 pow_add, add_right_comm]\n[GOAL]\ncase h.succ.inr\nn : \u2115\nihn : \u2200 (m : \u2115), m < succ n \u2192 m# \u2264 4 ^ m\nho : Odd n\n\u22a2 succ n# \u2264 4 ^ succ n\n[PROOFSTEP]\nrcases Decidable.eq_or_ne n 1 with (rfl | hn)\n[GOAL]\ncase h.succ.inr.inl\nihn : \u2200 (m : \u2115), m < succ 1 \u2192 m# \u2264 4 ^ m\nho : Odd 1\n\u22a2 succ 1# \u2264 4 ^ succ 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase h.succ.inr.inr\nn : \u2115\nihn : \u2200 (m : \u2115), m < succ n \u2192 m# \u2264 4 ^ m\nho : Odd n\nhn : n \u2260 1\n\u22a2 succ n# \u2264 4 ^ succ n\n[PROOFSTEP]\ncalc\n  (n + 1)# = n# := primorial_succ hn ho\n  _ \u2264 4 ^ n := (ihn n n.lt_succ_self)\n  _ \u2264 4 ^ (n + 1) := pow_le_pow_of_le_right four_pos n.le_succ\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Primorial", "llama_tokens": 2119, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.7122321903471563, "lm_q1q2_score": 0.5555360518032135}}
{"text": "[GOAL]\n\u22a2 StableUnderComposition @FiniteType\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 : FiniteType f\nhg : FiniteType g\n\u22a2 FiniteType (comp g f)\n[PROOFSTEP]\nexact hg.comp hf\n[GOAL]\n\u22a2 HoldsForLocalizationAway @FiniteType\n[PROOFSTEP]\nintrov R _\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\n\u22a2 FiniteType (algebraMap R S)\n[PROOFSTEP]\nsuffices Algebra.FiniteType R S by\n  rw [RingHom.FiniteType]\n  convert this; ext; rw [Algebra.smul_def]; rfl\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 FiniteType (algebraMap R S)\n[PROOFSTEP]\nrw [RingHom.FiniteType]\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 Algebra.FiniteType R S\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_5\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 toAlgebra (algebraMap R S) = inst\u271d\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\nr\u271d : R\nx\u271d : S\n\u22a2 (let_fun I := toAlgebra (algebraMap R S);\n    r\u271d \u2022 x\u271d) =\n    r\u271d \u2022 x\u271d\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\nr\u271d : R\nx\u271d : S\n\u22a2 (let_fun I := toAlgebra (algebraMap R S);\n    r\u271d \u2022 x\u271d) =\n    \u2191(algebraMap R S) r\u271d * x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\n\u22a2 Algebra.FiniteType R S\n[PROOFSTEP]\nexact IsLocalization.finiteType_of_monoid_fg (Submonoid.powers r) S\n[GOAL]\n\u22a2 OfLocalizationSpanTarget @FiniteType\n[PROOFSTEP]\nrw [ofLocalizationSpanTarget_iff_finite]\n[GOAL]\n\u22a2 OfLocalizationFiniteSpanTarget @FiniteType\n[PROOFSTEP]\nintrov R hs H\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), FiniteType (comp (algebraMap S (Localization.Away \u2191r)) f)\n\u22a2 FiniteType f\n[PROOFSTEP]\nclassical\nletI := f.toAlgebra\nreplace H : \u2200 r : s, Algebra.FiniteType R (Localization.Away (r : S))\n\u00b7 intro r; simp_rw [RingHom.FiniteType] at H ; convert H r; ext; simp_rw [Algebra.smul_def]; rfl\nreplace H := fun r => (H r).1\nconstructor\n  -- Suppose `s : Finset S` spans `S`, and each `S\u1d63` is finitely generated as an `R`-algebra.\n    -- Say `t r : Finset S\u1d63` generates `S\u1d63`. By assumption, we may find `l\u1d62` such that\n    -- `\u2211 l\u1d62 * s\u1d62 = 1`. I claim that all `s` and `l` and the numerators of `t` and generates `S`.\nchoose t ht using H\nobtain \u27e8l, hl\u27e9 :=\n  (Finsupp.mem_span_iff_total S (s : Set S) 1).mp (show (1 : S) \u2208 Ideal.span (s : Set S) by rw [hs]; trivial)\nlet sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (x : S)) (t x)\nuse s.attach.biUnion sf \u222a s \u222a l.support.image l\nrw [eq_top_iff]\n  -- We need to show that every `x` falls in the subalgebra generated by those elements.\n    -- Since all `s` and `l` are in the subalgebra, it suffices to check that `s\u1d62 ^ n\u1d62 \u2022 x` falls in\n    -- the algebra for each `s\u1d62` and some `n\u1d62`.\nrintro x -\napply Subalgebra.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set S) l hl _ _ x _\n\u00b7 intro x hx\n  apply Algebra.subset_adjoin\n  rw [Finset.coe_union, Finset.coe_union]\n  exact Or.inl (Or.inr hx)\n\u00b7 intro i\n  by_cases h : l i = 0; \u00b7 rw [h]; exact zero_mem _\n  apply Algebra.subset_adjoin\n  rw [Finset.coe_union, Finset.coe_image]\n  exact Or.inr (Set.mem_image_of_mem _ (Finsupp.mem_support_iff.mpr h))\n\u00b7 intro r\n  rw [Finset.coe_union, Finset.coe_union, Finset.coe_biUnion]\n    -- Since all `s\u1d62` and numerators of `t r` are in the algebra, it suffices to show that the\n        -- image of `x` in `S\u1d63` falls in the `R`-adjoin of `t r`, which is of course true.\n        -- Porting note: The following `obtain` fails because Lean wants to know right away what the\n        -- placeholders are, so we need to provide a little more guidance\n        -- obtain \u27e8\u27e8_, n\u2082, rfl\u27e9, hn\u2082\u27e9 := IsLocalization.exists_smul_mem_of_mem_adjoin\n        --   (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) _ _ _\n  rw [show\n      \u2200 A : Set S,\n        (\u2203 n, (r : S) ^ n \u2022 x \u2208 Algebra.adjoin R A) \u2194\n          (\u2203 m : (Submonoid.powers (r : S)), (m : S) \u2022 x \u2208 Algebra.adjoin R A)\n      by {exact fun _ => by simp [Submonoid.mem_powers_iff]\n    }]\n  refine IsLocalization.exists_smul_mem_of_mem_adjoin (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) ?_ ?_ ?_\n  \u00b7 intro x hx\n    apply Algebra.subset_adjoin\n    exact Or.inl (Or.inl \u27e8_, \u27e8r, rfl\u27e9, _, \u27e8s.mem_attach r, rfl\u27e9, hx\u27e9)\n  \u00b7 rw [Submonoid.powers_eq_closure, Submonoid.closure_le, Set.singleton_subset_iff]\n    apply Algebra.subset_adjoin\n    exact Or.inl (Or.inr r.2)\n  \u00b7 rw [ht]; trivial\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), FiniteType (comp (algebraMap S (Localization.Away \u2191r)) f)\n\u22a2 FiniteType f\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), FiniteType (comp (algebraMap S (Localization.Away \u2191r)) f)\nthis : Algebra R S := toAlgebra f\n\u22a2 FiniteType f\n[PROOFSTEP]\nreplace H : \u2200 r : s, Algebra.FiniteType R (Localization.Away (r : S))\n[GOAL]\ncase H\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), FiniteType (comp (algebraMap S (Localization.Away \u2191r)) f)\nthis : Algebra R S := toAlgebra f\n\u22a2 \u2200 (r : { x // x \u2208 s }), Algebra.FiniteType R (Localization.Away \u2191r)\n[PROOFSTEP]\nintro r\n[GOAL]\ncase H\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), FiniteType (comp (algebraMap S (Localization.Away \u2191r)) f)\nthis : Algebra R S := toAlgebra f\nr : { x // x \u2208 s }\n\u22a2 Algebra.FiniteType R (Localization.Away \u2191r)\n[PROOFSTEP]\nsimp_rw [RingHom.FiniteType] at H \n[GOAL]\ncase H\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), Algebra.FiniteType R (Localization.Away \u2191r)\nthis : Algebra R S := toAlgebra f\nr : { x // x \u2208 s }\n\u22a2 Algebra.FiniteType R (Localization.Away \u2191r)\n[PROOFSTEP]\nconvert H r\n[GOAL]\ncase h.e'_5\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), Algebra.FiniteType R (Localization.Away \u2191r)\nthis : Algebra R S := toAlgebra f\nr : { x // x \u2208 s }\n\u22a2 Localization.algebra = toAlgebra (comp (algebraMap S (Localization.Away \u2191r)) f)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), Algebra.FiniteType R (Localization.Away \u2191r)\nthis : Algebra R S := toAlgebra f\nr : { x // x \u2208 s }\nr\u271d : R\nx\u271d : Localization (Submonoid.powers \u2191r)\n\u22a2 (let_fun I := Localization.algebra;\n    r\u271d \u2022 x\u271d) =\n    r\u271d \u2022 x\u271d\n[PROOFSTEP]\nsimp_rw [Algebra.smul_def]\n[GOAL]\ncase h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nH : \u2200 (r : { x // x \u2208 s }), Algebra.FiniteType R (Localization.Away \u2191r)\nthis : Algebra R S := toAlgebra f\nr : { x // x \u2208 s }\nr\u271d : R\nx\u271d : Localization (Submonoid.powers \u2191r)\n\u22a2 \u2191(algebraMap R (Localization (Submonoid.powers \u2191r))) r\u271d * x\u271d = r\u271d \u2022 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nH : \u2200 (r : { x // x \u2208 s }), Algebra.FiniteType R (Localization.Away \u2191r)\n\u22a2 FiniteType f\n[PROOFSTEP]\nreplace H := fun r => (H r).1\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nH : \u2200 (r : { x // x \u2208 s }), Subalgebra.FG \u22a4\n\u22a2 FiniteType f\n[PROOFSTEP]\nconstructor\n  -- Suppose `s : Finset S` spans `S`, and each `S\u1d63` is finitely generated as an `R`-algebra.\n    -- Say `t r : Finset S\u1d63` generates `S\u1d63`. By assumption, we may find `l\u1d62` such that\n    -- `\u2211 l\u1d62 * s\u1d62 = 1`. I claim that all `s` and `l` and the numerators of `t` and generates `S`.\n[GOAL]\ncase out\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nH : \u2200 (r : { x // x \u2208 s }), Subalgebra.FG \u22a4\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nchoose t ht using H\n[GOAL]\ncase out\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nobtain \u27e8l, hl\u27e9 :=\n  (Finsupp.mem_span_iff_total S (s : Set S) 1).mp (show (1 : S) \u2208 Ideal.span (s : Set S) by rw [hs]; trivial)\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\n\u22a2 1 \u2208 Ideal.span \u2191s\n[PROOFSTEP]\nrw [hs]\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\n\u22a2 1 \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase out.intro\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nlet sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (x : S)) (t x)\n[GOAL]\ncase out.intro\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\n\u22a2 Subalgebra.FG \u22a4\n[PROOFSTEP]\nuse s.attach.biUnion sf \u222a s \u222a l.support.image l\n[GOAL]\ncase h\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\n\u22a2 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n  -- We need to show that every `x` falls in the subalgebra generated by those elements.\n    -- Since all `s` and `l` are in the subalgebra, it suffices to check that `s\u1d62 ^ n\u1d62 \u2022 x` falls in\n    -- the algebra for each `s\u1d62` and some `n\u1d62`.\n[GOAL]\ncase h\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\n\u22a2 \u22a4 \u2264 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nrintro x -\n[GOAL]\ncase h\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\n\u22a2 x \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\napply Subalgebra.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set S) l hl _ _ x _\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\n\u22a2 \u2191s \u2286 \u2191(Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support))\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx\u271d x : S\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u2191(Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support))\n[PROOFSTEP]\napply Algebra.subset_adjoin\n[GOAL]\ncase a\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx\u271d x : S\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nrw [Finset.coe_union, Finset.coe_union]\n[GOAL]\ncase a\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx\u271d x : S\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u2191(Finset.biUnion (Finset.attach s) sf) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nexact Or.inl (Or.inr hx)\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\n\u22a2 \u2200 (i : \u2191\u2191s), \u2191l i \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nintro i\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\ni : \u2191\u2191s\n\u22a2 \u2191l i \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nby_cases h : l i = 0\n[GOAL]\ncase pos\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\ni : \u2191\u2191s\nh : \u2191l i = 0\n\u22a2 \u2191l i \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\ni : \u2191\u2191s\nh : \u2191l i = 0\n\u22a2 0 \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nexact zero_mem _\n[GOAL]\ncase neg\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\ni : \u2191\u2191s\nh : \u00ac\u2191l i = 0\n\u22a2 \u2191l i \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\napply Algebra.subset_adjoin\n[GOAL]\ncase neg.a\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\ni : \u2191\u2191s\nh : \u00ac\u2191l i = 0\n\u22a2 \u2191l i \u2208 \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nrw [Finset.coe_union, Finset.coe_image]\n[GOAL]\ncase neg.a\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\ni : \u2191\u2191s\nh : \u00ac\u2191l i = 0\n\u22a2 \u2191l i \u2208 \u2191(Finset.biUnion (Finset.attach s) sf \u222a s) \u222a \u2191l '' \u2191l.support\n[PROOFSTEP]\nexact Or.inr (Set.mem_image_of_mem _ (Finsupp.mem_support_iff.mpr h))\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\n\u22a2 \u2200 (r : \u2191\u2191s),\n    \u2203 n, \u2191r ^ n \u2022 x \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nintro r\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 Algebra.adjoin R \u2191(Finset.biUnion (Finset.attach s) sf \u222a s \u222a Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nrw [Finset.coe_union, Finset.coe_union, Finset.coe_biUnion]\n  -- Since all `s\u1d62` and numerators of `t r` are in the algebra, it suffices to show that the\n      -- image of `x` in `S\u1d63` falls in the `R`-adjoin of `t r`, which is of course true.\n      -- Porting note: The following `obtain` fails because Lean wants to know right away what the\n      -- placeholders are, so we need to provide a little more guidance\n      -- obtain \u27e8\u27e8_, n\u2082, rfl\u27e9, hn\u2082\u27e9 := IsLocalization.exists_smul_mem_of_mem_adjoin\n      --   (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) _ _ _\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2203 n,\n    \u2191r ^ n \u2022 x \u2208\n      Algebra.adjoin R\n        ((\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support))\n[PROOFSTEP]\nrw [show\n    \u2200 A : Set S,\n      (\u2203 n, (r : S) ^ n \u2022 x \u2208 Algebra.adjoin R A) \u2194 (\u2203 m : (Submonoid.powers (r : S)), (m : S) \u2022 x \u2208 Algebra.adjoin R A)\n    by {exact fun _ => by simp [Submonoid.mem_powers_iff]\n  }]\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2200 (A : Set S), (\u2203 n, \u2191r ^ n \u2022 x \u2208 Algebra.adjoin R A) \u2194 \u2203 m, \u2191m \u2022 x \u2208 Algebra.adjoin R A\n[PROOFSTEP]\n{exact fun _ => by simp [Submonoid.mem_powers_iff]\n}\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2200 (A : Set S), (\u2203 n, \u2191r ^ n \u2022 x \u2208 Algebra.adjoin R A) \u2194 \u2203 m, \u2191m \u2022 x \u2208 Algebra.adjoin R A\n[PROOFSTEP]\nexact fun _ => by simp [Submonoid.mem_powers_iff]\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\nx\u271d : Set S\n\u22a2 (\u2203 n, \u2191r ^ n \u2022 x \u2208 Algebra.adjoin R x\u271d) \u2194 \u2203 m, \u2191m \u2022 x \u2208 Algebra.adjoin R x\u271d\n[PROOFSTEP]\nsimp [Submonoid.mem_powers_iff]\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2203 m,\n    \u2191m \u2022 x \u2208\n      Algebra.adjoin R\n        ((\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support))\n[PROOFSTEP]\nrefine IsLocalization.exists_smul_mem_of_mem_adjoin (Submonoid.powers (r : S)) x (t r) (Algebra.adjoin R _) ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191r) (t r)) \u2286\n    \u2191(Algebra.adjoin R\n        ((\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support)))\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine_1\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx\u271d : S\nr : \u2191\u2191s\nx : S\nhx : x \u2208 \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191r) (t r))\n\u22a2 x \u2208\n    \u2191(Algebra.adjoin R\n        ((\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support)))\n[PROOFSTEP]\napply Algebra.subset_adjoin\n[GOAL]\ncase refine_1.a\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx\u271d : S\nr : \u2191\u2191s\nx : S\nhx : x \u2208 \u2191(IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191r) (t r))\n\u22a2 x \u2208 (\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nexact Or.inl (Or.inl \u27e8_, \u27e8r, rfl\u27e9, _, \u27e8s.mem_attach r, rfl\u27e9, hx\u27e9)\n[GOAL]\ncase refine_2\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 Submonoid.powers \u2191r \u2264\n    (Algebra.adjoin R\n          ((\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a\n            \u2191(Finset.image (\u2191l) l.support))).toSubsemiring.toSubmonoid\n[PROOFSTEP]\nrw [Submonoid.powers_eq_closure, Submonoid.closure_le, Set.singleton_subset_iff]\n[GOAL]\ncase refine_2\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191r \u2208\n    \u2191(Algebra.adjoin R\n            ((\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a\n              \u2191(Finset.image (\u2191l) l.support))).toSubsemiring.toSubmonoid\n[PROOFSTEP]\napply Algebra.subset_adjoin\n[GOAL]\ncase refine_2.a\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191r \u2208 (\u22c3 (x : { x // x \u2208 s }) (_ : x \u2208 \u2191(Finset.attach s)), \u2191(sf x)) \u222a \u2191s \u222a \u2191(Finset.image (\u2191l) l.support)\n[PROOFSTEP]\nexact Or.inl (Or.inr r.2)\n[GOAL]\ncase refine_3\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(algebraMap S (Localization.Away \u2191r)) x \u2208 Algebra.adjoin R \u2191(t r)\n[PROOFSTEP]\nrw [ht]\n[GOAL]\ncase refine_3\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\ns : Finset S\nhs : Ideal.span \u2191s = \u22a4\nthis : Algebra R S := toAlgebra f\nt : (r : { x // x \u2208 s }) \u2192 Finset (Localization.Away \u2191r)\nht : \u2200 (r : { x // x \u2208 s }), Algebra.adjoin R \u2191(t r) = \u22a4\nl : \u2191\u2191s \u2192\u2080 S\nhl : \u2191(Finsupp.total (\u2191\u2191s) S S Subtype.val) l = 1\nsf : { x // x \u2208 s } \u2192 Finset S := fun x => IsLocalization.finsetIntegerMultiple (Submonoid.powers \u2191x) (t x)\nx : S\nr : \u2191\u2191s\n\u22a2 \u2191(algebraMap S (Localization.Away \u2191r)) x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.RingHom.FiniteType", "llama_tokens": 14432, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936484231889, "lm_q2_score": 0.7248702761768248, "lm_q1q2_score": 0.5554634885650636}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1 \u00d7 \u03b1\n\u22a2 Rel \u03b1 x y \u2192 Rel \u03b1 y x\n[PROOFSTEP]\naesop (rule_sets [Sym2])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z : \u03b1 \u00d7 \u03b1\na : Rel \u03b1 x y\nb : Rel \u03b1 y z\n\u22a2 Rel \u03b1 x z\n[PROOFSTEP]\naesop (rule_sets [Sym2])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z w : \u03b1\n\u22a2 Rel \u03b1 (x, y) (z, w) \u2194 x = z \u2227 y = w \u2228 x = w \u2227 y = z\n[PROOFSTEP]\naesop (rule_sets [Sym2])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : Sym2 \u03b1 \u2192 Sym2 \u03b2 \u2192 Prop\ni : Sym2 \u03b1\nj : Sym2 \u03b2\nhf : \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b1) (b : \u03b2 \u00d7 \u03b2), f (Quotient.mk (Rel.setoid \u03b1) a) (Quotient.mk (Rel.setoid \u03b2) b)\n[PROOFSTEP]\nintro \u27e8a\u2081, a\u2082\u27e9 \u27e8b\u2081, b\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : Sym2 \u03b1 \u2192 Sym2 \u03b2 \u2192 Prop\ni : Sym2 \u03b1\nj : Sym2 \u03b2\nhf : \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 f (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))\n[PROOFSTEP]\nexact hf _ _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (b, a)\n[PROOFSTEP]\nrw [Quotient.eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 (a, b) \u2248 (b, a)\n[PROOFSTEP]\napply Rel.swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u00d7 \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (Prod.swap p) = Quotient.mk (Rel.setoid \u03b1) p\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nfst\u271d snd\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (Prod.swap (fst\u271d, snd\u271d)) = Quotient.mk (Rel.setoid \u03b1) (fst\u271d, snd\u271d)\n[PROOFSTEP]\nexact eq_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (a, c) \u2194 b = c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (a, c) \u2192 b = c\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 b = c \u2192 Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (a, c)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (a, c)\n\u22a2 b = c\n[PROOFSTEP]\nrw [Quotient.eq] at h \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\nh : (a, b) \u2248 (a, c)\n\u22a2 b = c\n[PROOFSTEP]\ncases h\n[GOAL]\ncase mp.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 b = b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\n\u22a2 a = a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\nh : b = c\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (a, c)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (b, a) = Quotient.mk (Rel.setoid \u03b1) (c, a) \u2194 b = c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (b, a) = Quotient.mk (Rel.setoid \u03b1) (c, a) \u2192 b = c\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 b = c \u2192 Quotient.mk (Rel.setoid \u03b1) (b, a) = Quotient.mk (Rel.setoid \u03b1) (c, a)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (b, a) = Quotient.mk (Rel.setoid \u03b1) (c, a)\n\u22a2 b = c\n[PROOFSTEP]\nrw [Quotient.eq] at h \n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\nh : (b, a) \u2248 (c, a)\n\u22a2 b = c\n[PROOFSTEP]\ncases h\n[GOAL]\ncase mp.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 b = b\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mp.swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\n\u22a2 a = a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\nh : b = c\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (b, a) = Quotient.mk (Rel.setoid \u03b1) (c, a)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y z w : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (z, w) \u2194 x = z \u2227 y = w \u2228 x = w \u2227 y = z\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np q : \u03b1 \u00d7 \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) p = Quotient.mk (Rel.setoid \u03b1) q \u2194 p = q \u2228 p = Prod.swap q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nq : \u03b1 \u00d7 \u03b1\nfst\u271d snd\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (fst\u271d, snd\u271d) = Quotient.mk (Rel.setoid \u03b1) q \u2194 (fst\u271d, snd\u271d) = q \u2228 (fst\u271d, snd\u271d) = Prod.swap q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nfst\u271d\u00b9 snd\u271d\u00b9 fst\u271d snd\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (fst\u271d\u00b9, snd\u271d\u00b9) = Quotient.mk (Rel.setoid \u03b1) (fst\u271d, snd\u271d) \u2194\n    (fst\u271d\u00b9, snd\u271d\u00b9) = (fst\u271d, snd\u271d) \u2228 (fst\u271d\u00b9, snd\u271d\u00b9) = Prod.swap (fst\u271d, snd\u271d)\n[PROOFSTEP]\nsimp only [eq_iff, Prod.mk.inj_iff, Prod.swap_prod_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1), f a\u2081 a\u2082 = f a\u2082 a\u2081 }\n\u22a2 \u2200 (a b : \u03b1 \u00d7 \u03b1), a \u2248 b \u2192 uncurry (\u2191f) a = uncurry (\u2191f) b\n[PROOFSTEP]\nrintro _ _ \u27e8\u27e9\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1), f a\u2081 a\u2082 = f a\u2082 a\u2081 }\nx\u271d y\u271d : \u03b1\n\u22a2 uncurry \u2191f (x\u271d, y\u271d) = uncurry \u2191f (x\u271d, y\u271d)\ncase swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1), f a\u2081 a\u2082 = f a\u2082 a\u2081 }\nx\u271d y\u271d : \u03b1\n\u22a2 uncurry \u2191f (x\u271d, y\u271d) = uncurry \u2191f (y\u271d, x\u271d)\n[PROOFSTEP]\nexacts [rfl, f.prop _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2082 a\u2081 b\u2081 b\u2082 \u2227 f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2081 a\u2082 b\u2082 b\u2081 }\n\u22a2 \u2200 (a\u2081 : \u03b1 \u00d7 \u03b1) (b\u2081 : \u03b2 \u00d7 \u03b2) (a\u2082 : \u03b1 \u00d7 \u03b1) (b\u2082 : \u03b2 \u00d7 \u03b2),\n    a\u2081 \u2248 a\u2082 \u2192 b\u2081 \u2248 b\u2082 \u2192 (fun a b => \u2191f a.fst a.snd b.fst b.snd) a\u2081 b\u2081 = (fun a b => \u2191f a.fst a.snd b.fst b.snd) a\u2082 b\u2082\n[PROOFSTEP]\nrintro _ _ _ _ \u27e8\u27e9 \u27e8\u27e9\n[GOAL]\ncase refl.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2082 a\u2081 b\u2081 b\u2082 \u2227 f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2081 a\u2082 b\u2082 b\u2081 }\nx\u271d\u00b9 y\u271d\u00b9 : \u03b1\nx\u271d y\u271d : \u03b2\n\u22a2 (fun a b => \u2191f a.fst a.snd b.fst b.snd) (x\u271d\u00b9, y\u271d\u00b9) (x\u271d, y\u271d) =\n    (fun a b => \u2191f a.fst a.snd b.fst b.snd) (x\u271d\u00b9, y\u271d\u00b9) (x\u271d, y\u271d)\ncase refl.swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2082 a\u2081 b\u2081 b\u2082 \u2227 f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2081 a\u2082 b\u2082 b\u2081 }\nx\u271d\u00b9 y\u271d\u00b9 : \u03b1\nx\u271d y\u271d : \u03b2\n\u22a2 (fun a b => \u2191f a.fst a.snd b.fst b.snd) (x\u271d\u00b9, y\u271d\u00b9) (x\u271d, y\u271d) =\n    (fun a b => \u2191f a.fst a.snd b.fst b.snd) (x\u271d\u00b9, y\u271d\u00b9) (y\u271d, x\u271d)\ncase swap.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2082 a\u2081 b\u2081 b\u2082 \u2227 f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2081 a\u2082 b\u2082 b\u2081 }\nx\u271d\u00b9 y\u271d\u00b9 : \u03b1\nx\u271d y\u271d : \u03b2\n\u22a2 (fun a b => \u2191f a.fst a.snd b.fst b.snd) (x\u271d\u00b9, y\u271d\u00b9) (x\u271d, y\u271d) =\n    (fun a b => \u2191f a.fst a.snd b.fst b.snd) (y\u271d\u00b9, x\u271d\u00b9) (x\u271d, y\u271d)\ncase swap.swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : { f // \u2200 (a\u2081 a\u2082 : \u03b1) (b\u2081 b\u2082 : \u03b2), f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2082 a\u2081 b\u2081 b\u2082 \u2227 f a\u2081 a\u2082 b\u2081 b\u2082 = f a\u2081 a\u2082 b\u2082 b\u2081 }\nx\u271d\u00b9 y\u271d\u00b9 : \u03b1\nx\u271d y\u271d : \u03b2\n\u22a2 (fun a b => \u2191f a.fst a.snd b.fst b.snd) (x\u271d\u00b9, y\u271d\u00b9) (x\u271d, y\u271d) =\n    (fun a b => \u2191f a.fst a.snd b.fst b.snd) (y\u271d\u00b9, x\u271d\u00b9) (y\u271d, x\u271d)\n[PROOFSTEP]\nexacts [rfl, (f.2 _ _ _ _).2, (f.2 _ _ _ _).1, (f.2 _ _ _ _).1.trans (f.2 _ _ _ _).2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nF : Sym2 \u03b1 \u2192 Sym2 \u03b2 \u2192 \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2081 a\u2082 b\u2081 b\u2082 =\n      (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2082 a\u2081 b\u2081 b\u2082 \u2227\n    (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2081 a\u2082 b\u2081 b\u2082 =\n      (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2081 a\u2082 b\u2082 b\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nF : Sym2 \u03b1 \u2192 Sym2 \u03b2 \u2192 \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2081 a\u2082 b\u2081 b\u2082 =\n    (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2082 a\u2081 b\u2081 b\u2082\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nF : Sym2 \u03b1 \u2192 Sym2 \u03b2 \u2192 \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2081 a\u2082 b\u2081 b\u2082 =\n    (fun a\u2081 a\u2082 b\u2081 b\u2082 => F (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b2) (b\u2081, b\u2082))) a\u2081 a\u2082 b\u2082 b\u2081\n[PROOFSTEP]\nexacts [congr_arg\u2082 F eq_swap rfl, congr_arg\u2082 F rfl eq_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\n\u22a2 ((fun x x_1 => x \u2248 x_1) \u21d2 fun x x_1 => x \u2248 x_1) (Prod.map f f) (Prod.map f f)\n[PROOFSTEP]\nintro _ _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\na\u271d b\u271d : \u03b1 \u00d7 \u03b1\nh : a\u271d \u2248 b\u271d\n\u22a2 Prod.map f f a\u271d \u2248 Prod.map f f b\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\n\u22a2 Prod.map f f (x\u271d, y\u271d) \u2248 Prod.map f f (x\u271d, y\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\n\u22a2 Prod.map f f (x\u271d, y\u271d) \u2248 Prod.map f f (y\u271d, x\u271d)\n[PROOFSTEP]\napply Rel.swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 map id = id\n[PROOFSTEP]\next \u27e8\u27e8x, y\u27e9\u27e9\n[GOAL]\ncase h.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym2 \u03b1\nx y : \u03b1\n\u22a2 map id (Quot.mk Setoid.r (x, y)) = id (Quot.mk Setoid.r (x, y))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 map (g \u2218 f) = map g \u2218 map f\n[PROOFSTEP]\next \u27e8\u27e8x, y\u27e9\u27e9\n[GOAL]\ncase h.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nx\u271d : Sym2 \u03b1\nx y : \u03b1\n\u22a2 map (g \u2218 f) (Quot.mk Setoid.r (x, y)) = (map g \u2218 map f) (Quot.mk Setoid.r (x, y))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nx : Sym2 \u03b1\n\u22a2 map g (map f x) = map (g \u2218 f) x\n[PROOFSTEP]\nrevert x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : Sym2 \u03b1), map g (map f x) = map (g \u2218 f) x\n[PROOFSTEP]\napply Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x y : \u03b1), map g (map f (Quotient.mk (Rel.setoid \u03b1) (x, y))) = map (g \u2218 f) (Quotient.mk (Rel.setoid \u03b1) (x, y))\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\n\u22a2 Injective (map f)\n[PROOFSTEP]\nintro z z'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\n\u22a2 map f z = map f z' \u2192 z = z'\n[PROOFSTEP]\nrefine' Quotient.ind\u2082 (fun z z' => _) z z'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz\u271d z'\u271d : Sym2 \u03b1\nz z' : \u03b1 \u00d7 \u03b1\n\u22a2 map f (Quotient.mk (Rel.setoid \u03b1) z) = map f (Quotient.mk (Rel.setoid \u03b1) z') \u2192\n    Quotient.mk (Rel.setoid \u03b1) z = Quotient.mk (Rel.setoid \u03b1) z'\n[PROOFSTEP]\ncases' z with x y\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z'\u271d : Sym2 \u03b1\nz' : \u03b1 \u00d7 \u03b1\nx y : \u03b1\n\u22a2 map f (Quotient.mk (Rel.setoid \u03b1) (x, y)) = map f (Quotient.mk (Rel.setoid \u03b1) z') \u2192\n    Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) z'\n[PROOFSTEP]\ncases' z' with x' y'\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\n\u22a2 map f (Quotient.mk (Rel.setoid \u03b1) (x, y)) = map f (Quotient.mk (Rel.setoid \u03b1) (x', y')) \u2192\n    Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nrepeat' rw [map_pair_eq, eq_iff]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\n\u22a2 map f (Quotient.mk (Rel.setoid \u03b1) (x, y)) = map f (Quotient.mk (Rel.setoid \u03b1) (x', y')) \u2192\n    Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nrw [map_pair_eq, eq_iff]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b2) (f x, f y) = map f (Quotient.mk (Rel.setoid \u03b1) (x', y')) \u2192\n    x = x' \u2227 y = y' \u2228 x = y' \u2227 y = x'\n[PROOFSTEP]\nrw [map_pair_eq, eq_iff]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\n\u22a2 f x = f x' \u2227 f y = f y' \u2228 f x = f y' \u2227 f y = f x' \u2192 x = x' \u2227 y = y' \u2228 x = y' \u2227 y = x'\n[PROOFSTEP]\nrw [map_pair_eq, eq_iff]\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\n\u22a2 f x = f x' \u2227 f y = f y' \u2228 f x = f y' \u2227 f y = f x' \u2192 x = x' \u2227 y = y' \u2228 x = y' \u2227 y = x'\n[PROOFSTEP]\nrintro (h | h)\n[GOAL]\ncase mk.mk.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\nh : f x = f x' \u2227 f y = f y'\n\u22a2 x = x' \u2227 y = y' \u2228 x = y' \u2227 y = x'\n[PROOFSTEP]\nsimp [hinj h.1, hinj h.2]\n[GOAL]\ncase mk.mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nz z' : Sym2 \u03b1\nx y x' y' : \u03b1\nh : f x = f y' \u2227 f y = f x'\n\u22a2 x = x' \u2227 y = y' \u2228 x = y' \u2227 y = x'\n[PROOFSTEP]\nsimp [hinj h.1, hinj h.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 Sym2.Mem a (Quotient.mk (Rel.setoid \u03b1) (b, c)) \u2192 a = b \u2228 a = c\n[PROOFSTEP]\nrintro \u27e8_, h\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c w\u271d : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (b, c) = Quotient.mk (Rel.setoid \u03b1) (a, w\u271d)\n\u22a2 a = b \u2228 a = c\n[PROOFSTEP]\nrw [eq_iff] at h \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c w\u271d : \u03b1\nh : b = a \u2227 c = w\u271d \u2228 b = w\u271d \u2227 c = a\n\u22a2 a = b \u2228 a = c\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c : \u03b1\n\u22a2 a = b \u2228 a = c \u2192 Sym2.Mem a (Quotient.mk (Rel.setoid \u03b1) (b, c))\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na c : \u03b1\n\u22a2 Sym2.Mem a (Quotient.mk (Rel.setoid \u03b1) (a, c))\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 Sym2.Mem a (Quotient.mk (Rel.setoid \u03b1) (b, a))\n[PROOFSTEP]\nrw [eq_swap]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b : \u03b1\n\u22a2 Sym2.Mem a (Quotient.mk (Rel.setoid \u03b1) (a, b))\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh : (fun z => {x | Sym2.Mem x z}) z = (fun z => {x | Sym2.Mem x z}) z'\n\u22a2 z = z'\n[PROOFSTEP]\nsimp only [Set.ext_iff, Set.mem_setOf_eq] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\n\u22a2 z = z'\n[PROOFSTEP]\ninduction' z using Sym2.ind with x y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = z'\n[PROOFSTEP]\ninduction' z' using Sym2.ind with x' y'\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d\u00b2 : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh\u271d\u00b9 : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\nx' y' : \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x', y'))\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nhave hx := h x\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d\u00b2 : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh\u271d\u00b9 : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\nx' y' : \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx : Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nhave hy := h y\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d\u00b2 : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh\u271d\u00b9 : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\nx' y' : \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx : Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhy : Sym2.Mem y (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem y (Quotient.mk (Rel.setoid \u03b1) (x', y'))\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nhave hx' := h x'\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d\u00b2 : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh\u271d\u00b9 : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\nx' y' : \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx : Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhy : Sym2.Mem y (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem y (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx' : Sym2.Mem x' (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x' (Quotient.mk (Rel.setoid \u03b1) (x', y'))\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nhave hy' := h y'\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d\u00b2 : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh\u271d\u00b9 : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\nx' y' : \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx : Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhy : Sym2.Mem y (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem y (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx' : Sym2.Mem x' (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x' (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhy' : Sym2.Mem y' (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem y' (Quotient.mk (Rel.setoid \u03b1) (x', y'))\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\nsimp only [mem_iff', eq_self_iff_true, or_true_iff, iff_true_iff, true_or_iff, true_iff_iff] at hx hy hx' hy' \n[GOAL]\ncase h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz z' : Sym2 \u03b1\nh\u271d\u00b2 : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x z'\nx y : \u03b1\nh\u271d\u00b9 : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 z'\nx' y' : \u03b1\nh\u271d : \u2200 (x : \u03b1), Sym2.Mem x z \u2194 Sym2.Mem x (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nh : \u2200 (x_1 : \u03b1), Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 Sym2.Mem x_1 (Quotient.mk (Rel.setoid \u03b1) (x', y'))\nhx : x = x' \u2228 x = y'\nhy : y = x' \u2228 y = y'\nhx' : x' = x \u2228 x' = y\nhy' : y' = x \u2228 y' = y\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, y) = Quotient.mk (Rel.setoid \u03b1) (x', y')\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Sym2 \u03b1\n\u22a2 e = Quotient.mk (Rel.setoid \u03b1) ((Quotient.out e).fst, (Quotient.out e).snd)\n[PROOFSTEP]\nrw [Prod.mk.eta, e.out_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ne : Sym2 \u03b1\n\u22a2 e = Quotient.mk (Rel.setoid \u03b1) ((Quotient.out e).snd, (Quotient.out e).fst)\n[PROOFSTEP]\nrw [eq_swap, Prod.mk.eta, e.out_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 (\u2200 (c : \u03b1), c \u2208 Quotient.mk (Rel.setoid \u03b1) (a, b) \u2192 p c) \u2194 p a \u2227 p b\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h _ <| mem_mk''_left _ _, h _ <| mem_mk''_right _ _\u27e9, fun h c hc => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\na b : \u03b1\nh : p a \u2227 p b\nc : \u03b1\nhc : c \u2208 Quotient.mk (Rel.setoid \u03b1) (a, b)\n\u22a2 p c\n[PROOFSTEP]\nobtain rfl | rfl := Sym2.mem_iff.1 hc\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\nb c : \u03b1\nh : p c \u2227 p b\nhc : c \u2208 Quotient.mk (Rel.setoid \u03b1) (c, b)\n\u22a2 p c\n[PROOFSTEP]\nexact h.1\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u03b1 \u2192 Prop\na c : \u03b1\nh : p a \u2227 p c\nhc : c \u2208 Quotient.mk (Rel.setoid \u03b1) (a, c)\n\u22a2 p c\n[PROOFSTEP]\nexact h.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, Mem.other h) = z\n[PROOFSTEP]\nerw [\u2190 Classical.choose_spec h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Mem.other h \u2208 z\n[PROOFSTEP]\nconvert mem_mk''_right a <| Mem.other h\n[GOAL]\ncase h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 z = Quotient.mk (Rel.setoid \u03b1) (a, Mem.other h)\n[PROOFSTEP]\nrw [other_spec h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nz : Sym2 \u03b1\nhne : x \u2260 y\n\u22a2 x \u2208 z \u2227 y \u2208 z \u2194 z = Quotient.mk (Rel.setoid \u03b1) (x, y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nz : Sym2 \u03b1\nhne : x \u2260 y\n\u22a2 x \u2208 z \u2227 y \u2208 z \u2192 z = Quotient.mk (Rel.setoid \u03b1) (x, y)\n[PROOFSTEP]\ninduction' z using Sym2.ind with x' y'\n[GOAL]\ncase mp.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nhne : x \u2260 y\nx' y' : \u03b1\n\u22a2 x \u2208 Quotient.mk (Rel.setoid \u03b1) (x', y') \u2227 y \u2208 Quotient.mk (Rel.setoid \u03b1) (x', y') \u2192\n    Quotient.mk (Rel.setoid \u03b1) (x', y') = Quotient.mk (Rel.setoid \u03b1) (x, y)\n[PROOFSTEP]\nrw [mem_iff, mem_iff]\n[GOAL]\ncase mp.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nhne : x \u2260 y\nx' y' : \u03b1\n\u22a2 (x = x' \u2228 x = y') \u2227 (y = x' \u2228 y = y') \u2192 Quotient.mk (Rel.setoid \u03b1) (x', y') = Quotient.mk (Rel.setoid \u03b1) (x, y)\n[PROOFSTEP]\naesop\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nz : Sym2 \u03b1\nhne : x \u2260 y\n\u22a2 z = Quotient.mk (Rel.setoid \u03b1) (x, y) \u2192 x \u2208 z \u2227 y \u2208 z\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nhne : x \u2260 y\n\u22a2 x \u2208 Quotient.mk (Rel.setoid \u03b1) (x, y) \u2227 y \u2208 Quotient.mk (Rel.setoid \u03b1) (x, y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nz : Sym2 \u03b1\n\u22a2 b \u2208 map f z \u2194 \u2203 a, a \u2208 z \u2227 f a = b\n[PROOFSTEP]\ninduction' z using Sym2.ind with x y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nx y : \u03b1\n\u22a2 b \u2208 map f (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2194 \u2203 a, a \u2208 Quotient.mk (Rel.setoid \u03b1) (x, y) \u2227 f a = b\n[PROOFSTEP]\nsimp only [map, Quotient.map_mk, Prod.map_mk, mem_iff]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nx y : \u03b1\n\u22a2 b = f x \u2228 b = f y \u2194 \u2203 a, (a = x \u2228 a = y) \u2227 f a = b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nx y : \u03b1\n\u22a2 b = f x \u2228 b = f y \u2192 \u2203 a, (a = x \u2228 a = y) \u2227 f a = b\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase h.mp.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 \u2203 a, (a = x \u2228 a = y) \u2227 f a = f x\n[PROOFSTEP]\nexact \u27e8x, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 (x = x \u2228 x = y) \u2227 f x = f x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mp.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 \u2203 a, (a = x \u2228 a = y) \u2227 f a = f y\n[PROOFSTEP]\nexact \u27e8y, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 (y = x \u2228 y = y) \u2227 f y = f y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\nx y : \u03b1\n\u22a2 (\u2203 a, (a = x \u2228 a = y) \u2227 f a = b) \u2192 b = f x \u2228 b = f y\n[PROOFSTEP]\nrintro \u27e8w, rfl | rfl, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ny w : \u03b1\n\u22a2 f w = f w \u2228 f w = f y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mpr.intro.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nx w : \u03b1\n\u22a2 f w = f x \u2228 f w = f w\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\n\u22a2 map f s = map g s\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\ny : \u03b2\n\u22a2 y \u2208 map f s \u2194 y \u2208 map g s\n[PROOFSTEP]\nsimp only [mem_map]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\ny : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 f a = y) \u2194 \u2203 a, a \u2208 s \u2227 g a = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\ny : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 f a = y) \u2192 \u2203 a, a \u2208 s \u2227 g a = y\n[PROOFSTEP]\nrintro \u27e8w, hw, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\nw : \u03b1\nhw : w \u2208 s\n\u22a2 \u2203 a, a \u2208 s \u2227 g a = f w\n[PROOFSTEP]\nexact \u27e8w, hw, by simp [hw, h]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\nw : \u03b1\nhw : w \u2208 s\n\u22a2 g w = f w\n[PROOFSTEP]\nsimp [hw, h]\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\ny : \u03b2\n\u22a2 (\u2203 a, a \u2208 s \u2227 g a = y) \u2192 \u2203 a, a \u2208 s \u2227 f a = y\n[PROOFSTEP]\nrintro \u27e8w, hw, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\nw : \u03b1\nhw : w \u2208 s\n\u22a2 \u2203 a, a \u2208 s \u2227 f a = g w\n[PROOFSTEP]\nexact \u27e8w, hw, by simp [hw, h]\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\ns : Sym2 \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x\nw : \u03b1\nhw : w \u2208 s\n\u22a2 f w = g w\n[PROOFSTEP]\nsimp [hw, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\nh : diag x = diag y\n\u22a2 x = y\n[PROOFSTEP]\ncases Quotient.exact h\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : \u03b1\nh : diag x = diag x\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : \u03b1\nh : diag x = diag x\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nz : Sym2 \u03b1\n\u22a2 IsDiag z \u2192 z \u2208 Set.range diag\n[PROOFSTEP]\ninduction' z using Sym2.ind with x y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx y : \u03b1\n\u22a2 IsDiag (Quotient.mk (Rel.setoid \u03b1) (x, y)) \u2192 Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 Set.range diag\n[PROOFSTEP]\nrintro (rfl : x = y)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x, x) \u2208 Set.range diag\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\n\u22a2 DecidablePred IsDiag\n[PROOFSTEP]\nrefine' fun z => Quotient.recOnSubsingleton z fun a => _\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nz : Sym2 \u03b1\na : \u03b1 \u00d7 \u03b1\n\u22a2 Decidable (IsDiag (Quotient.mk (Rel.setoid \u03b1) a))\n[PROOFSTEP]\nerw [isDiag_iff_proj_eq]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nz : Sym2 \u03b1\na : \u03b1 \u00d7 \u03b1\n\u22a2 Decidable (a.fst = a.snd)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nhd : \u00acIsDiag z\nh : a \u2208 z\n\u22a2 Mem.other h \u2260 a\n[PROOFSTEP]\ncontrapose! hd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\nhd : Mem.other h = a\n\u22a2 IsDiag z\n[PROOFSTEP]\nhave h' := Sym2.other_spec h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\nhd : Mem.other h = a\nh' : Quotient.mk (Rel.setoid \u03b1) (a, Mem.other h) = z\n\u22a2 IsDiag z\n[PROOFSTEP]\nrw [hd] at h' \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\nhd : Mem.other h = a\nh' : Quotient.mk (Rel.setoid \u03b1) (a, a) = z\n\u22a2 IsDiag z\n[PROOFSTEP]\nrw [\u2190 h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\nhd : Mem.other h = a\nh' : Quotient.mk (Rel.setoid \u03b1) (a, a) = z\n\u22a2 IsDiag (Quotient.mk (Rel.setoid \u03b1) (a, a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y) = \u2205\n[PROOFSTEP]\napply Set.eq_empty_of_forall_not_mem fun e => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (e : Sym2 \u03b1), \u00ace \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y)\n[PROOFSTEP]\napply Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (x y : \u03b1), \u00acQuotient.mk (Rel.setoid \u03b1) (x, y) \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y)\n[PROOFSTEP]\nsimp [-Set.bot_eq_empty, Prop.bot_eq_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 fromRel (_ : \u2200 (x y : \u03b1), \u22a4 x y \u2192 \u22a4 x y) = Set.univ\n[PROOFSTEP]\napply Set.eq_univ_of_forall fun e => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (e : Sym2 \u03b1), e \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a4 x y \u2192 \u22a4 x y)\n[PROOFSTEP]\napply Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (x y : \u03b1), Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a4 x y \u2192 \u22a4 x y)\n[PROOFSTEP]\nsimp [-Set.top_eq_univ, Prop.top_eq_true]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nsym : Symmetric r\n\u22a2 Irreflexive r \u2192 \u2200 {z : Sym2 \u03b1}, z \u2208 fromRel sym \u2192 \u00acIsDiag z\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nsym : Symmetric r\nh : Irreflexive r\n\u22a2 \u2200 {z : Sym2 \u03b1}, z \u2208 fromRel sym \u2192 \u00acIsDiag z\n[PROOFSTEP]\napply Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nsym : Symmetric r\nh : Irreflexive r\n\u22a2 \u2200 (x y : \u03b1), Quotient.mk (Rel.setoid \u03b1) (x, y) \u2208 fromRel sym \u2192 \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x, y))\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set (Sym2 \u03b1)\nx y : \u03b1\n\u22a2 ToRel s x y \u2192 ToRel s y x\n[PROOFSTEP]\nsimp [eq_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\n\u22a2 [a\u2081, b\u2081] ~ [a\u2082, b\u2082] \u2192 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 a\u2081 = b\u2082 \u2227 b\u2081 = a\u2082\n[PROOFSTEP]\nsimp [\u2190 Multiset.coe_eq_coe, \u2190 Multiset.cons_coe, Multiset.cons_eq_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\n\u22a2 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 \u00aca\u2081 = a\u2082 \u2227 b\u2081 = a\u2082 \u2227 b\u2082 = a\u2081 \u2192 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082 \u2228 a\u2081 = b\u2082 \u2227 b\u2081 = a\u2082\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh\u2081 : a\u2081 = a\u2082\nh\u2082 : b\u2081 = b\u2082\n\u22a2 [a\u2081, b\u2081] ~ [a\u2082, b\u2082]\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\n\n[PROOFSTEP]\nfirst\n| done\n| apply List.Perm.swap'; rfl\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh\u2081 : a\u2081 = b\u2082\nh\u2082 : b\u2081 = a\u2082\n\u22a2 [a\u2081, b\u2081] ~ [a\u2082, b\u2082]\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh\u2081 : a\u2081 = b\u2082\nh\u2082 : b\u2081 = a\u2082\n\u22a2 [b\u2082, a\u2082] ~ [a\u2082, b\u2082]\n[PROOFSTEP]\nfirst\n| done\n| apply List.Perm.swap'; rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh\u2081 : a\u2081 = b\u2082\nh\u2082 : b\u2081 = a\u2082\n\u22a2 [b\u2082, a\u2082] ~ [a\u2082, b\u2082]\n[PROOFSTEP]\ndone\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh\u2081 : a\u2081 = b\u2082\nh\u2082 : b\u2081 = a\u2082\n\u22a2 [b\u2082, a\u2082] ~ [a\u2082, b\u2082]\n[PROOFSTEP]\napply List.Perm.swap'\n[GOAL]\ncase p\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh\u2081 : a\u2081 = b\u2082\nh\u2082 : b\u2081 = a\u2082\n\u22a2 [] ~ []\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 ((fun x x_1 => x \u2248 x_1) \u21d2 fun x x_1 => x \u2248 x_1)\n    (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n    fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) }\n[PROOFSTEP]\nrintro _ _ \u27e8_\u27e9\n[GOAL]\ncase refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d y\u271d : \u03b1\n\u22a2 (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (x\u271d, y\u271d) \u2248\n    (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (x\u271d, y\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase refl.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d y\u271d : \u03b1\n\u22a2 [(x\u271d, y\u271d).snd] ~ [(x\u271d, y\u271d).snd]\n[PROOFSTEP]\napply List.Perm.refl\n[GOAL]\ncase swap\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d y\u271d : \u03b1\n\u22a2 (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (x\u271d, y\u271d) \u2248\n    (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (y\u271d, x\u271d)\n[PROOFSTEP]\napply List.Perm.swap'\n[GOAL]\ncase swap.p\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d y\u271d : \u03b1\n\u22a2 [] ~ []\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 ((fun x x_1 => x \u2248 x_1) \u21d2 fun x x_1 => x \u2248 x_1) Sym2.fromVector Sym2.fromVector\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9 \u27e8y, hy\u27e9 h\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : List \u03b1\nhx : List.length x = 2\ny : List \u03b1\nhy : List.length y = 2\nh : { val := x, property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := x, property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase mk.mk.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhx : List.length [] = 2\nh : { val := [], property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := [], property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase mk.mk.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d :: x) = 2\nh : { val := head\u271d :: x, property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := head\u271d :: x, property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase mk.mk.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d : \u03b1\nhx : List.length [head\u271d] = 2\nh : { val := [head\u271d], property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d], property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase mk.mk.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b9 :: head\u271d :: x) = 2\nh : { val := head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase mk.mk.cons.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d\u00b9 head\u271d : \u03b1\nhx : List.length [head\u271d\u00b9, head\u271d] = 2\nh : { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\ncase mk.mk.cons.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x) = 2\nh : { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248\n    Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\nswap\n[GOAL]\ncase mk.mk.cons.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x) = 2\nh : { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248\n    Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase mk.mk.cons.cons.cons.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x) = 2\nh : { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx } \u2248 { val := y, property := hy }\n\u22a2 False\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase mk.mk.cons.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ny : List \u03b1\nhy : List.length y = 2\nhead\u271d\u00b9 head\u271d : \u03b1\nhx : List.length [head\u271d\u00b9, head\u271d] = 2\nh : { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 { val := y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 Sym2.fromVector { val := y, property := hy }\n[PROOFSTEP]\ncases' y with _ y\n[GOAL]\ncase mk.mk.cons.cons.nil.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b9 head\u271d : \u03b1\nhx : List.length [head\u271d\u00b9, head\u271d] = 2\nhy : List.length [] = 2\nh : { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 { val := [], property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 Sym2.fromVector { val := [], property := hy }\n[PROOFSTEP]\nsimp at hy \n[GOAL]\ncase mk.mk.cons.cons.nil.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b2 head\u271d\u00b9 : \u03b1\nhx : List.length [head\u271d\u00b2, head\u271d\u00b9] = 2\nhead\u271d : \u03b1\ny : List \u03b1\nhy : List.length (head\u271d :: y) = 2\nh : { val := [head\u271d\u00b2, head\u271d\u00b9], property := hx } \u2248 { val := head\u271d :: y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b2, head\u271d\u00b9], property := hx } \u2248 Sym2.fromVector { val := head\u271d :: y, property := hy }\n[PROOFSTEP]\ncases' y with _ y\n[GOAL]\ncase mk.mk.cons.cons.nil.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b2 head\u271d\u00b9 : \u03b1\nhx : List.length [head\u271d\u00b2, head\u271d\u00b9] = 2\nhead\u271d : \u03b1\nhy : List.length [head\u271d] = 2\nh : { val := [head\u271d\u00b2, head\u271d\u00b9], property := hx } \u2248 { val := [head\u271d], property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b2, head\u271d\u00b9], property := hx } \u2248 Sym2.fromVector { val := [head\u271d], property := hy }\n[PROOFSTEP]\nsimp at hy \n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b3 head\u271d\u00b2 : \u03b1\nhx : List.length [head\u271d\u00b3, head\u271d\u00b2] = 2\nhead\u271d\u00b9 head\u271d : \u03b1\ny : List \u03b1\nhy : List.length (head\u271d\u00b9 :: head\u271d :: y) = 2\nh : { val := [head\u271d\u00b3, head\u271d\u00b2], property := hx } \u2248 { val := head\u271d\u00b9 :: head\u271d :: y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b3, head\u271d\u00b2], property := hx } \u2248\n    Sym2.fromVector { val := head\u271d\u00b9 :: head\u271d :: y, property := hy }\n[PROOFSTEP]\ncases' y with _ y\n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b3 head\u271d\u00b2 : \u03b1\nhx : List.length [head\u271d\u00b3, head\u271d\u00b2] = 2\nhead\u271d\u00b9 head\u271d : \u03b1\nhy : List.length [head\u271d\u00b9, head\u271d] = 2\nh : { val := [head\u271d\u00b3, head\u271d\u00b2], property := hx } \u2248 { val := [head\u271d\u00b9, head\u271d], property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b3, head\u271d\u00b2], property := hx } \u2248\n    Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hy }\ncase mk.mk.cons.cons.nil.cons.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u2074 head\u271d\u00b3 : \u03b1\nhx : List.length [head\u271d\u2074, head\u271d\u00b3] = 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\ny : List \u03b1\nhy : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y) = 2\nh : { val := [head\u271d\u2074, head\u271d\u00b3], property := hx } \u2248 { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u2074, head\u271d\u00b3], property := hx } \u2248\n    Sym2.fromVector { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y, property := hy }\n[PROOFSTEP]\nswap\n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u2074 head\u271d\u00b3 : \u03b1\nhx : List.length [head\u271d\u2074, head\u271d\u00b3] = 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\ny : List \u03b1\nhy : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y) = 2\nh : { val := [head\u271d\u2074, head\u271d\u00b3], property := hx } \u2248 { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y, property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u2074, head\u271d\u00b3], property := hx } \u2248\n    Sym2.fromVector { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y, property := hy }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons.cons.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u2074 head\u271d\u00b3 : \u03b1\nhx : List.length [head\u271d\u2074, head\u271d\u00b3] = 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\ny : List \u03b1\nhy : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y) = 2\nh : { val := [head\u271d\u2074, head\u271d\u00b3], property := hx } \u2248 { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: y, property := hy }\n\u22a2 False\n[PROOFSTEP]\nsimp at hy \n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b3 head\u271d\u00b2 : \u03b1\nhx : List.length [head\u271d\u00b3, head\u271d\u00b2] = 2\nhead\u271d\u00b9 head\u271d : \u03b1\nhy : List.length [head\u271d\u00b9, head\u271d] = 2\nh : { val := [head\u271d\u00b3, head\u271d\u00b2], property := hx } \u2248 { val := [head\u271d\u00b9, head\u271d], property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b3, head\u271d\u00b2], property := hx } \u2248\n    Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hy }\n[PROOFSTEP]\nrcases perm_card_two_iff.mp h with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons.nil.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b9 head\u271d : \u03b1\nhx hy : List.length [head\u271d\u00b9, head\u271d] = 2\nh : { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 { val := [head\u271d\u00b9, head\u271d], property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248\n    Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hy }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.mk.cons.cons.nil.cons.cons.nil.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nhead\u271d\u00b9 head\u271d : \u03b1\nhx : List.length [head\u271d\u00b9, head\u271d] = 2\nhy : List.length [head\u271d, head\u271d\u00b9] = 2\nh : { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248 { val := [head\u271d, head\u271d\u00b9], property := hy }\n\u22a2 Sym2.fromVector { val := [head\u271d\u00b9, head\u271d], property := hx } \u2248\n    Sym2.fromVector { val := [head\u271d, head\u271d\u00b9], property := hy }\n[PROOFSTEP]\napply Sym2.Rel.swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 LeftInverse\n    (Quotient.map Sym2.fromVector\n      (_ :\n        \u2200 \u2983a b : Vector \u03b1 2\u2984,\n          (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b)))\n    (Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b))\n[PROOFSTEP]\napply Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 \u2200 (x y : \u03b1),\n    Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.map\n          (fun x =>\n            { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n          (_ :\n            \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n              a \u2248 b \u2192\n                (fun x =>\n                      { val := [x.fst, x.snd],\n                        property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                    a \u2248\n                  (fun x =>\n                      { val := [x.fst, x.snd],\n                        property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                    b)\n          (Quotient.mk (Rel.setoid \u03b1) (x, y))) =\n      Quotient.mk (Rel.setoid \u03b1) (x, y)\n[PROOFSTEP]\naesop (add norm unfold [Sym2.fromVector])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : Sym' \u03b1 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        x) =\n    x\n[PROOFSTEP]\nrefine' Quotient.recOnSubsingleton x fun x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nx : Vector \u03b1 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) x)) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) x\n[PROOFSTEP]\ncases' x with x hx\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nx : List \u03b1\nhx : List.length x = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := x, property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := x, property := hx }\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase mk.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : Sym' \u03b1 2\nhx : List.length [] = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [], property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [], property := hx }\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase mk.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nhead\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d :: x) = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d :: x, property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d :: x, property := hx }\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase mk.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : Sym' \u03b1 2\nhead\u271d : \u03b1\nhx : List.length [head\u271d] = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [head\u271d], property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [head\u271d], property := hx }\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase mk.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nhead\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b9 :: head\u271d :: x) = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d\u00b9 :: head\u271d :: x, property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d\u00b9 :: head\u271d :: x, property := hx }\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase mk.cons.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : Sym' \u03b1 2\nhead\u271d\u00b9 head\u271d : \u03b1\nhx : List.length [head\u271d\u00b9, head\u271d] = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [head\u271d\u00b9, head\u271d], property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [head\u271d\u00b9, head\u271d], property := hx }\ncase mk.cons.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x) = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx }\n[PROOFSTEP]\nswap\n[GOAL]\ncase mk.cons.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x) = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x, property := hx }\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase mk.cons.cons.cons.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx\u271d : Sym' \u03b1 2\nhead\u271d\u00b2 head\u271d\u00b9 head\u271d : \u03b1\nx : List \u03b1\nhx : List.length (head\u271d\u00b2 :: head\u271d\u00b9 :: head\u271d :: x) = 2\n\u22a2 False\n[PROOFSTEP]\nsimp at hx \n[GOAL]\ncase mk.cons.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nx : Sym' \u03b1 2\nhead\u271d\u00b9 head\u271d : \u03b1\nhx : List.length [head\u271d\u00b9, head\u271d] = 2\n\u22a2 Quotient.map\n      (fun x => { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n      (_ :\n        \u2200 \u2983a b : \u03b1 \u00d7 \u03b1\u2984,\n          a \u2248 b \u2192\n            (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                a \u2248\n              (fun x =>\n                  { val := [x.fst, x.snd], property := (_ : List.length [x.fst, x.snd] = List.length [x.fst, x.snd]) })\n                b)\n      (Quotient.map Sym2.fromVector\n        (_ :\n          \u2200 \u2983a b : Vector \u03b1 2\u2984,\n            (fun x x_1 => x \u2248 x_1) a b \u2192 (fun x x_1 => x \u2248 x_1) (Sym2.fromVector a) (Sym2.fromVector b))\n        (Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [head\u271d\u00b9, head\u271d], property := hx })) =\n    Quotient.mk (Vector.Perm.isSetoid \u03b1 2) { val := [head\u271d\u00b9, head\u271d], property := hx }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1 \u00d7 \u03b1\n\u22a2 relBool x y = true \u2194 Rel \u03b1 x y\n[PROOFSTEP]\ncases' x with x\u2081 x\u2082\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ny : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\n\u22a2 relBool (x\u2081, x\u2082) y = true \u2194 Rel \u03b1 (x\u2081, x\u2082) y\n[PROOFSTEP]\ncases' y with y\u2081 y\u2082\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nx\u2081 x\u2082 y\u2081 y\u2082 : \u03b1\n\u22a2 relBool (x\u2081, x\u2082) (y\u2081, y\u2082) = true \u2194 Rel \u03b1 (x\u2081, x\u2082) (y\u2081, y\u2082)\n[PROOFSTEP]\naesop (rule_sets [Sym2]) (add norm unfold [relBool])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\n\u22a2 \u2200 (a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1),\n    (fun x\u2081 x\u2082 y\u2081 y\u2082 => relBool (x\u2081, x\u2082) (y\u2081, y\u2082)) a\u2081 a\u2082 b\u2081 b\u2082 =\n        (fun x\u2081 x\u2082 y\u2081 y\u2082 => relBool (x\u2081, x\u2082) (y\u2081, y\u2082)) a\u2082 a\u2081 b\u2081 b\u2082 \u2227\n      (fun x\u2081 x\u2082 y\u2081 y\u2082 => relBool (x\u2081, x\u2082) (y\u2081, y\u2082)) a\u2081 a\u2082 b\u2081 b\u2082 =\n        (fun x\u2081 x\u2082 y\u2081 y\u2082 => relBool (x\u2081, x\u2082) (y\u2081, y\u2082)) a\u2081 a\u2082 b\u2082 b\u2081\n[PROOFSTEP]\naesop (add norm unfold [relBool])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na b : Sym2 \u03b1\n\u22a2 \u2200 (a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1),\n    eqBool (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b1) (b\u2081, b\u2082)) = true \u2194\n      Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082) = Quotient.mk (Rel.setoid \u03b1) (b\u2081, b\u2082)\n[PROOFSTEP]\naesop (rule_sets [Sym2])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 \u2200 (a_1 b : \u03b1 \u00d7 \u03b1) (p : a_1 \u2248 b),\n    Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x => Sym2.pairOther a a_1)\n        (_ : Quotient.mk (Rel.setoid \u03b1) a_1 = Quotient.mk (Rel.setoid \u03b1) b) =\n      fun x => Sym2.pairOther a b\n[PROOFSTEP]\nclear h z\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 \u2200 (a_1 b : \u03b1 \u00d7 \u03b1) (p : a_1 \u2248 b),\n    Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x => Sym2.pairOther a a_1)\n        (_ : Quotient.mk (Rel.setoid \u03b1) a_1 = Quotient.mk (Rel.setoid \u03b1) b) =\n      fun x => Sym2.pairOther a b\n[PROOFSTEP]\nintro x y h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\n\u22a2 Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x)\n      (_ : Quotient.mk (Rel.setoid \u03b1) x = Quotient.mk (Rel.setoid \u03b1) y) =\n    fun x => Sym2.pairOther a y\n[PROOFSTEP]\next hy\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\n\u22a2 Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x)\n      (_ : Quotient.mk (Rel.setoid \u03b1) x = Quotient.mk (Rel.setoid \u03b1) y) hy =\n    Sym2.pairOther a y\n[PROOFSTEP]\nconvert_to Sym2.pairOther a x = _\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\n\u22a2 Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x)\n      (_ : Quotient.mk (Rel.setoid \u03b1) x = Quotient.mk (Rel.setoid \u03b1) y) hy =\n    Sym2.pairOther a x\n[PROOFSTEP]\nhave :\n  \u2200 {c e h},\n    @Eq.ndrec (Quotient (Rel.setoid \u03b1)) (Quotient.mk (Rel.setoid \u03b1) x) (fun x => a \u2208 x \u2192 \u03b1)\n        (fun _ => Sym2.pairOther a x) c e h =\n      Sym2.pairOther a x :=\n  by intro _ e _; subst e; rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\n\u22a2 \u2200 {c : Quotient (Rel.setoid \u03b1)} {e : Quotient.mk (Rel.setoid \u03b1) x = c} {h : a \u2208 c},\n    Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x) e h = Sym2.pairOther a x\n[PROOFSTEP]\nintro _ e _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\nc\u271d : Quotient (Rel.setoid \u03b1)\ne : Quotient.mk (Rel.setoid \u03b1) x = c\u271d\nh\u271d : a \u2208 c\u271d\n\u22a2 Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x) e h\u271d = Sym2.pairOther a x\n[PROOFSTEP]\nsubst e\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\nh\u271d : a \u2208 Quotient.mk (Rel.setoid \u03b1) x\n\u22a2 Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x)\n      (_ : Quotient.mk (Rel.setoid \u03b1) x = Quotient.mk (Rel.setoid \u03b1) x) h\u271d =\n    Sym2.pairOther a x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\nthis :\n  \u2200 {c : Quotient (Rel.setoid \u03b1)} {e : Quotient.mk (Rel.setoid \u03b1) x = c} {h : a \u2208 c},\n    Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x) e h = Sym2.pairOther a x\n\u22a2 Eq.ndrec (motive := fun x => a \u2208 x \u2192 \u03b1) (fun x_1 => Sym2.pairOther a x)\n      (_ : Quotient.mk (Rel.setoid \u03b1) x = Quotient.mk (Rel.setoid \u03b1) y) hy =\n    Sym2.pairOther a x\n[PROOFSTEP]\napply this\n[GOAL]\ncase h.convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a \u2208 Quotient.mk (Rel.setoid \u03b1) y\n\u22a2 Sym2.pairOther a x = Sym2.pairOther a y\n[PROOFSTEP]\nrw [mem_iff] at hy \n[GOAL]\ncase h.convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a = y.fst \u2228 a = y.snd\n\u22a2 Sym2.pairOther a x = Sym2.pairOther a y\n[PROOFSTEP]\nhave : relBool x y := (relBool_spec x y).mpr h\n[GOAL]\ncase h.convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx y : \u03b1 \u00d7 \u03b1\nh : x \u2248 y\nhy : a = y.fst \u2228 a = y.snd\nthis : relBool x y = true\n\u22a2 Sym2.pairOther a x = Sym2.pairOther a y\n[PROOFSTEP]\naesop (add norm unfold [pairOther, relBool])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, Mem.other' h) = z\n[PROOFSTEP]\ninduction z using Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na x\u271d y\u271d : \u03b1\nh : a \u2208 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, Mem.other' h) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nhave h' := mem_iff.mp h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na x\u271d y\u271d : \u03b1\nh : a \u2208 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nh' : a = x\u271d \u2228 a = y\u271d\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, Mem.other' h) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\naesop (add norm unfold [Quotient.rec, Quot.rec]) (rule_sets [Sym2])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Mem.other h = Mem.other' h\n[PROOFSTEP]\nrw [\u2190 congr_right, other_spec' h, other_spec]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Mem.other' h \u2208 z\n[PROOFSTEP]\nrw [\u2190 other_eq_other']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Mem.other h \u2208 z\n[PROOFSTEP]\nexact other_mem h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nz : Sym2 \u03b1\nha : a \u2208 z\nhb : Mem.other' ha \u2208 z\n\u22a2 Mem.other' hb = a\n[PROOFSTEP]\ninduction z using Sym2.ind\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\na x\u271d y\u271d : \u03b1\nha : a \u2208 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nhb : Mem.other' ha \u2208 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 Mem.other' hb = a\n[PROOFSTEP]\naesop (rule_sets [Sym2]) (add norm unfold [Quotient.rec, Quot.rec])\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nha : a \u2208 z\nhb : Mem.other ha \u2208 z\n\u22a2 Mem.other hb = a\n[PROOFSTEP]\nclassical\nrw [other_eq_other'] at hb \u22a2\nconvert other_invol' ha hb using 2\napply other_eq_other'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nha : a \u2208 z\nhb : Mem.other ha \u2208 z\n\u22a2 Mem.other hb = a\n[PROOFSTEP]\nrw [other_eq_other'] at hb \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nha : a \u2208 z\nhb\u271d : Mem.other ha \u2208 z\nhb : Mem.other' ha \u2208 z\n\u22a2 Mem.other' hb\u271d = a\n[PROOFSTEP]\nconvert other_invol' ha hb using 2\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nha : a \u2208 z\nhb\u271d : Mem.other ha \u2208 z\nhb : Mem.other' ha \u2208 z\n\u22a2 Mem.other ha = Mem.other' ha\n[PROOFSTEP]\napply other_eq_other'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) = image Quotient.mk'' (Finset.diag s)\n[PROOFSTEP]\next z\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nz : Sym2 \u03b1\n\u22a2 z \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194 z \u2208 image Quotient.mk'' (Finset.diag s)\n[PROOFSTEP]\ninduction' z using Sym2.inductionOn\n[GOAL]\ncase a.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194\n    Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 image Quotient.mk'' (Finset.diag s)\n[PROOFSTEP]\nsimp only [mem_image, mem_diag, exists_prop, mem_filter, Prod.exists, mem_product]\n[GOAL]\ncase a.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2194\n    \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.hf.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nrintro \u27e8\u27e8a, b, \u27e8ha, hb\u27e9, (h : Quotient.mk _ _ = _)\u27e9, hab\u27e9\n[GOAL]\ncase a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\nhab : IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\na b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nrw [\u2190 h, Sym2.mk''_isDiag_iff] at hab \n[GOAL]\ncase a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a b : \u03b1\nhab : a = b\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nexact \u27e8a, b, \u27e8ha, hab\u27e9, h\u27e9\n[GOAL]\ncase a.hf.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\n[PROOFSTEP]\nrintro \u27e8a, b, \u27e8ha, rfl\u27e9, h\u27e9\n[GOAL]\ncase a.hf.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a : \u03b1\nha : a \u2208 s\nh : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n    IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase a.hf.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a : \u03b1\nha : a \u2208 s\nh : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 (\u2203 a_1 b, (a_1 \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a_1, b) = Quotient.mk'' (a, a)) \u2227 IsDiag (Quotient.mk'' (a, a))\n[PROOFSTEP]\nexact \u27e8\u27e8a, a, \u27e8ha, ha\u27e9, rfl\u27e9, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 filter (fun a => \u00acIsDiag a) (image Quotient.mk'' (s \u00d7\u02e2 s)) = image Quotient.mk'' (offDiag s)\n[PROOFSTEP]\next z\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nz : Sym2 \u03b1\n\u22a2 z \u2208 filter (fun a => \u00acIsDiag a) (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194 z \u2208 image Quotient.mk'' (offDiag s)\n[PROOFSTEP]\ninduction z using Sym2.inductionOn\n[GOAL]\ncase a.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 filter (fun a => \u00acIsDiag a) (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194\n    Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 image Quotient.mk'' (offDiag s)\n[PROOFSTEP]\nsimp only [mem_image, mem_offDiag, mem_filter, Prod.exists, mem_product]\n[GOAL]\ncase a.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2194\n    \u2203 a b, (a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.hf.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    \u2203 a b, (a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nrintro \u27e8\u27e8a, b, \u27e8ha, hb\u27e9, (h : Quotient.mk _ _ = _)\u27e9, hab\u27e9\n[GOAL]\ncase a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\nhab : \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\na b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nrw [\u2190 h, Sym2.mk''_isDiag_iff] at hab \n[GOAL]\ncase a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a b : \u03b1\nhab : \u00aca = b\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n[PROOFSTEP]\nexact \u27e8a, b, \u27e8ha, hb, hab\u27e9, h\u27e9\n[GOAL]\ncase a.hf.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s \u2227 a \u2260 b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\n[PROOFSTEP]\nrintro \u27e8a, b, \u27e8ha, hb, hab\u27e9, (h : Quotient.mk _ _ = _)\u27e9\n[GOAL]\ncase a.hf.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\nhab : a \u2260 b\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n    \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\n[PROOFSTEP]\nrw [Ne.def, \u2190 Sym2.mk''_isDiag_iff, h] at hab \n[GOAL]\ncase a.hf.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\nhab : \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n    \u00acIsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\n[PROOFSTEP]\nexact \u27e8\u27e8a, b, \u27e8ha, hb\u27e9, h\u27e9, hab\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Sym.Sym2", "llama_tokens": 35764, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812553, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.5553267826439315}}
{"text": "[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\n\u22a2 \u2016x\u2016 = infDist 0 {m | \u2191m = x}\n[PROOFSTEP]\nsimp only [AddSubgroup.quotient_norm_eq, infDist_eq_iInf, sInf_image', dist_zero_left]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M\n\u22a2 \u2016\u2191x\u2016 = infDist x \u2191S\n[PROOFSTEP]\nrw [norm_eq_infDist, \u2190 infDist_image (IsometryEquiv.subLeft x).isometry, IsometryEquiv.subLeft_apply, sub_zero, \u2190\n  IsometryEquiv.preimage_symm]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M\n\u22a2 infDist x (\u2191(IsometryEquiv.symm (IsometryEquiv.subLeft x)) \u207b\u00b9' {m | \u2191m = \u2191x}) = infDist x \u2191S\n[PROOFSTEP]\ncongr 1 with y\n[GOAL]\ncase e_s.h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n\u22a2 y \u2208 \u2191(IsometryEquiv.symm (IsometryEquiv.subLeft x)) \u207b\u00b9' {m | \u2191m = \u2191x} \u2194 y \u2208 \u2191S\n[PROOFSTEP]\nsimp only [mem_preimage, IsometryEquiv.subLeft_symm_apply, mem_setOf_eq, QuotientAddGroup.eq, neg_add, neg_neg,\n  neg_add_cancel_right, SetLike.mem_coe]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\n\u22a2 \u2016-x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimp only [AddSubgroup.quotient_norm_eq]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\n\u22a2 sInf (norm '' {m | \u2191m = -x}) = sInf (norm '' {m | \u2191m = x})\n[PROOFSTEP]\ncongr 1 with r\n[GOAL]\ncase e_a.h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 r \u2208 norm '' {m | \u2191m = -x} \u2194 r \u2208 norm '' {m | \u2191m = x}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase e_a.h.mp\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 r \u2208 norm '' {m | \u2191m = -x} \u2192 r \u2208 norm '' {m | \u2191m = x}\n[PROOFSTEP]\n{rintro \u27e8m, hm, rfl\u27e9; use-m; simpa [neg_eq_iff_eq_neg] using hm\n}\n[GOAL]\ncase e_a.h.mp\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 r \u2208 norm '' {m | \u2191m = -x} \u2192 r \u2208 norm '' {m | \u2191m = x}\n[PROOFSTEP]\nrintro \u27e8m, hm, rfl\u27e9\n[GOAL]\ncase e_a.h.mp.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nm : M\nhm : m \u2208 {m | \u2191m = -x}\n\u22a2 \u2016m\u2016 \u2208 norm '' {m | \u2191m = x}\n[PROOFSTEP]\nuse-m\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nm : M\nhm : m \u2208 {m | \u2191m = -x}\n\u22a2 -m \u2208 {m | \u2191m = x} \u2227 \u2016-m\u2016 = \u2016m\u2016\n[PROOFSTEP]\nsimpa [neg_eq_iff_eq_neg] using hm\n[GOAL]\ncase e_a.h.mpr\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 r \u2208 norm '' {m | \u2191m = x} \u2192 r \u2208 norm '' {m | \u2191m = -x}\n[PROOFSTEP]\n{rintro \u27e8m, hm, rfl\u27e9; use-m; simpa [neg_eq_iff_eq_neg] using hm\n}\n[GOAL]\ncase e_a.h.mpr\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 r \u2208 norm '' {m | \u2191m = x} \u2192 r \u2208 norm '' {m | \u2191m = -x}\n[PROOFSTEP]\nrintro \u27e8m, hm, rfl\u27e9\n[GOAL]\ncase e_a.h.mpr.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nm : M\nhm : m \u2208 {m | \u2191m = x}\n\u22a2 \u2016m\u2016 \u2208 norm '' {m | \u2191m = -x}\n[PROOFSTEP]\nuse-m\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nm : M\nhm : m \u2208 {m | \u2191m = x}\n\u22a2 -m \u2208 {m | \u2191m = -x} \u2227 \u2016-m\u2016 = \u2016m\u2016\n[PROOFSTEP]\nsimpa [neg_eq_iff_eq_neg] using hm\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M \u29f8 S\n\u22a2 \u2016x - y\u2016 = \u2016y - x\u2016\n[PROOFSTEP]\nrw [\u2190 neg_sub, quotient_norm_neg]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u22a2 \u2016\u2191(mk' S) m\u2016 = sInf ((fun x => \u2016m + x\u2016) '' \u2191S)\n[PROOFSTEP]\nrw [mk'_apply, norm_mk, sInf_image', \u2190 infDist_image isometry_neg, image_neg, neg_coe_set (H := S), infDist_eq_iInf]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u22a2 \u2a05 (y : \u2191\u2191S), dist (-m) \u2191y = \u2a05 (a : \u2191\u2191S), \u2016m + \u2191a\u2016\n[PROOFSTEP]\nsimp only [dist_eq_norm', sub_neg_eq_add, add_comm]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u22a2 \u2016\u2191(mk' S) m\u2016 = 0 \u2194 m \u2208 closure \u2191S\n[PROOFSTEP]\nrw [mk'_apply, norm_mk, \u2190 mem_closure_iff_infDist_zero]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u22a2 Set.Nonempty \u2191S\n[PROOFSTEP]\nexact \u27e80, S.zero_mem\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 \u2016x\u2016 < r \u2194 \u2203 m, \u2191m = x \u2227 \u2016m\u2016 < r\n[PROOFSTEP]\nrw [isGLB_lt_iff (isGLB_quotient_norm _), bex_image_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\nr : \u211d\n\u22a2 (\u2203 x_1, x_1 \u2208 {m | \u2191(mk' S) m = x} \u2227 \u2016x_1\u2016 < r) \u2194 \u2203 m, \u2191m = x \u2227 \u2016m\u2016 < r\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 s, s \u2208 S \u2227 \u2016m + s\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n[PROOFSTEP]\nobtain \u27e8n : M, hn : mk' S n = mk' S m, hn' : \u2016n\u2016 < \u2016mk' S m\u2016 + \u03b5\u27e9 := norm_mk_lt (QuotientAddGroup.mk' S m) h\u03b5\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nn : M\nhn : \u2191(mk' S) n = \u2191(mk' S) m\nhn' : \u2016n\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n\u22a2 \u2203 s, s \u2208 S \u2227 \u2016m + s\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n[PROOFSTEP]\nerw [eq_comm, QuotientAddGroup.eq] at hn \n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nn : M\nhn : -m + n \u2208 S\nhn' : \u2016n\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n\u22a2 \u2203 s, s \u2208 S \u2227 \u2016m + s\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n[PROOFSTEP]\nuse-m + n, hn\n[GOAL]\ncase right\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nn : M\nhn : -m + n \u2208 S\nhn' : \u2016n\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n\u22a2 \u2016m + (-m + n)\u2016 < \u2016\u2191(mk' S) m\u2016 + \u03b5\n[PROOFSTEP]\nrwa [add_neg_cancel_left]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M \u29f8 S\n\u22a2 \u2016x + y\u2016 \u2264 \u2016x\u2016 + \u2016y\u2016\n[PROOFSTEP]\nrcases And.intro (mk_surjective x) (mk_surjective y) with \u27e8\u27e8x, rfl\u27e9, \u27e8y, rfl\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n\u22a2 \u2016\u2191x + \u2191y\u2016 \u2264 \u2016\u2191x\u2016 + \u2016\u2191y\u2016\n[PROOFSTEP]\nsimp only [\u2190 mk'_apply, \u2190 map_add, quotient_norm_mk_eq, sInf_image']\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n\u22a2 \u2a05 (a : \u2191\u2191S), \u2016x + y + \u2191a\u2016 \u2264 (\u2a05 (a : \u2191\u2191S), \u2016x + \u2191a\u2016) + \u2a05 (a : \u2191\u2191S), \u2016y + \u2191a\u2016\n[PROOFSTEP]\nrefine le_ciInf_add_ciInf fun a b \u21a6 ?_\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\na b : \u2191\u2191S\n\u22a2 \u2a05 (a : \u2191\u2191S), \u2016x + y + \u2191a\u2016 \u2264 \u2016x + \u2191a\u2016 + \u2016y + \u2191b\u2016\n[PROOFSTEP]\nrefine ciInf_le_of_le \u27e80, forall_range_iff.2 fun _ \u21a6 norm_nonneg _\u27e9 (a + b) ?_\n[GOAL]\ncase intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\na b : \u2191\u2191S\n\u22a2 \u2016x + y + \u2191(a + b)\u2016 \u2264 \u2016x + \u2191a\u2016 + \u2016y + \u2191b\u2016\n[PROOFSTEP]\nexact (congr_arg norm (add_add_add_comm _ _ _ _)).trans_le (norm_add_le _ _)\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u22a2 \u20160\u2016 = 0\n[PROOFSTEP]\nerw [quotient_norm_eq_zero_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u22a2 0 \u2208 closure \u2191S\n[PROOFSTEP]\nexact subset_closure S.zero_mem\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nhS : IsClosed \u2191S\nm : M\nh : \u2016\u2191(mk' S) m\u2016 = 0\n\u22a2 m \u2208 S\n[PROOFSTEP]\nrwa [quotient_norm_eq_zero_iff, hS.closure_eq] at h \n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u22a2 Filter.HasBasis (\ud835\udcdd 0) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {x | \u2016x\u2016 < \u03b5}\n[PROOFSTEP]\nhave : \u2200 \u03b5 : \u211d, mk '' ball (0 : M) \u03b5 = {x : M \u29f8 S | \u2016x\u2016 < \u03b5}\n[GOAL]\ncase this\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u22a2 \u2200 (\u03b5 : \u211d), mk '' ball 0 \u03b5 = {x | \u2016x\u2016 < \u03b5}\n[PROOFSTEP]\nrefine fun \u03b5 \u21a6 Set.ext <| forall_mk.2 fun x \u21a6 ?_\n[GOAL]\ncase this\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u03b5 : \u211d\nx : M\n\u22a2 \u2191x \u2208 mk '' ball 0 \u03b5 \u2194 \u2191x \u2208 {x | \u2016x\u2016 < \u03b5}\n[PROOFSTEP]\nrw [ball_zero_eq, mem_setOf_eq, norm_lt_iff, mem_image]\n[GOAL]\ncase this\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u03b5 : \u211d\nx : M\n\u22a2 (\u2203 x_1, x_1 \u2208 {x | \u2016x\u2016 < \u03b5} \u2227 \u2191x_1 = \u2191x) \u2194 \u2203 m, \u2191m = \u2191x \u2227 \u2016m\u2016 < \u03b5\n[PROOFSTEP]\nexact exists_congr fun _ \u21a6 and_comm\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nthis : \u2200 (\u03b5 : \u211d), mk '' ball 0 \u03b5 = {x | \u2016x\u2016 < \u03b5}\n\u22a2 Filter.HasBasis (\ud835\udcdd 0) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {x | \u2016x\u2016 < \u03b5}\n[PROOFSTEP]\nrw [\u2190 mk_zero, nhds_eq, \u2190 funext this]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nthis : \u2200 (\u03b5 : \u211d), mk '' ball 0 \u03b5 = {x | \u2016x\u2016 < \u03b5}\n\u22a2 Filter.HasBasis (Filter.map mk (\ud835\udcdd 0)) (fun \u03b5 => 0 < \u03b5) fun x => mk '' ball 0 x\n[PROOFSTEP]\nexact .map _ Metric.nhds_basis_ball\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M \u29f8 S\n\u22a2 dist x x = 0\n[PROOFSTEP]\nsimp only [norm_mk_zero, sub_self]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y z : M \u29f8 S\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\nrefine le_trans ?_ (quotient_norm_add_le _ _ _)\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y z : M \u29f8 S\n\u22a2 dist x z \u2264 \u2016x - y + (y - z)\u2016\n[PROOFSTEP]\nexact (congr_arg norm (sub_add_sub_cancel _ _ _).symm).le\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M \u29f8 S\n\u22a2 (fun x y => \u2191{ val := \u2016x - y\u2016, property := (_ : 0 \u2264 \u2016x - y\u2016) }) x y = ENNReal.ofReal (dist x y)\n[PROOFSTEP]\nexact ENNReal.coe_nnreal_eq _\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u22a2 uniformity (M \u29f8 S) = \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), Filter.principal {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nrw [uniformity_eq_comap_nhds_zero', ((quotient_nhd_basis S).comap _).eq_biInf]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\n\u22a2 \u2a05 (i : \u211d) (_ : 0 < i), Filter.principal ((fun p => p.snd - p.fst) \u207b\u00b9' {x | \u2016x\u2016 < i}) =\n    \u2a05 (\u03b5 : \u211d) (_ : \u03b5 > 0), Filter.principal {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp only [dist, quotient_norm_sub_rev (Prod.fst _), preimage_setOf_eq]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nsrc\u271d : M \u2192+ M \u29f8 S := mk' S\nm : M\n\u22a2 \u2016ZeroHom.toFun (\u2191src\u271d) m\u2016 \u2264 1 * \u2016m\u2016\n[PROOFSTEP]\nsimpa [one_mul] using quotient_norm_mk_le _ m\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u22a2 \u2016\u2191(normedMk S) m\u2016 \u2264 1 * \u2016m\u2016\n[PROOFSTEP]\nsimp [quotient_norm_mk_le']\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\ncases (norm_nonneg f).eq_or_gt with\n| inl h =>\n  rcases mk_surjective x with \u27e8x, rfl\u27e9\n  simpa [h] using le_opNorm f x\n| inr h =>\n  rw [\u2190 not_lt, \u2190 _root_.lt_div_iff' h, norm_lt_iff]\n  rintro \u27e8x, rfl, hx\u27e9\n  exact ((lt_div_iff' h).1 hx).not_le (le_opNorm f x)\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\nx\u271d : \u2016f\u2016 = 0 \u2228 0 < \u2016f\u2016\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\ncases (norm_nonneg f).eq_or_gt with\n| inl h =>\n  rcases mk_surjective x with \u27e8x, rfl\u27e9\n  simpa [h] using le_opNorm f x\n| inr h =>\n  rw [\u2190 not_lt, \u2190 _root_.lt_div_iff' h, norm_lt_iff]\n  rintro \u27e8x, rfl, hx\u27e9\n  exact ((lt_div_iff' h).1 hx).not_le (le_opNorm f x)\n[GOAL]\ncase inl\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\nh : \u2016f\u2016 = 0\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\n\n| inl h =>\n  rcases mk_surjective x with \u27e8x, rfl\u27e9\n  simpa [h] using le_opNorm f x\n[GOAL]\ncase inl\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\nh : \u2016f\u2016 = 0\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrcases mk_surjective x with \u27e8x, rfl\u27e9\n[GOAL]\ncase inl.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nh : \u2016f\u2016 = 0\nx : M\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) \u2191x\u2016 \u2264 \u2016f\u2016 * \u2016\u2191x\u2016\n[PROOFSTEP]\nsimpa [h] using le_opNorm f x\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\nh : 0 < \u2016f\u2016\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\n\n| inr h =>\n  rw [\u2190 not_lt, \u2190 _root_.lt_div_iff' h, norm_lt_iff]\n  rintro \u27e8x, rfl, hx\u27e9\n  exact ((lt_div_iff' h).1 hx).not_le (le_opNorm f x)\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\nh : 0 < \u2016f\u2016\n\u22a2 \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [\u2190 not_lt, \u2190 _root_.lt_div_iff' h, norm_lt_iff]\n[GOAL]\ncase inr\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nx : M \u29f8 S\nh : 0 < \u2016f\u2016\n\u22a2 \u00ac\u2203 m, \u2191m = x \u2227 \u2016m\u2016 < \u2016\u2191(lift S (toAddMonoidHom f) hf) x\u2016 / \u2016f\u2016\n[PROOFSTEP]\nrintro \u27e8x, rfl, hx\u27e9\n[GOAL]\ncase inr.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (x : M), x \u2208 S \u2192 \u2191f x = 0\nh : 0 < \u2016f\u2016\nx : M\nhx : \u2016x\u2016 < \u2016\u2191(lift S (toAddMonoidHom f) hf) \u2191x\u2016 / \u2016f\u2016\n\u22a2 False\n[PROOFSTEP]\nexact ((lt_div_iff' h).1 hx).not_le (le_opNorm f x)\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) \u2260 univ\n\u22a2 \u2016normedMk S\u2016 = 1\n[PROOFSTEP]\nrefine le_antisymm (norm_normedMk_le S) ?_\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) \u2260 univ\n\u22a2 1 \u2264 \u2016normedMk S\u2016\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x : M, 0 < \u2016(x : M \u29f8 S)\u2016\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) \u2260 univ\n\u22a2 \u2203 x, 0 < \u2016\u2191x\u2016\n[PROOFSTEP]\nrefine (Set.nonempty_compl.2 h).imp fun x hx \u21a6 ?_\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) \u2260 univ\nx : M\nhx : x \u2208 (\u2191(topologicalClosure S))\u1d9c\n\u22a2 0 < \u2016\u2191x\u2016\n[PROOFSTEP]\nexact (norm_nonneg _).lt_of_ne' <| mt (quotient_norm_eq_zero_iff S x).1 hx\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) \u2260 univ\nx : M\nhx : 0 < \u2016\u2191x\u2016\n\u22a2 1 \u2264 \u2016normedMk S\u2016\n[PROOFSTEP]\nrefine (le_mul_iff_one_le_left hx).1 ?_\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) \u2260 univ\nx : M\nhx : 0 < \u2016\u2191x\u2016\n\u22a2 \u2016\u2191x\u2016 \u2264 \u2016normedMk S\u2016 * \u2016\u2191x\u2016\n[PROOFSTEP]\nexact norm_lift_apply_le S.normedMk (fun x \u21a6 (eq_zero_iff x).2) x\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) = univ\n\u22a2 \u2016normedMk S\u2016 = 0\n[PROOFSTEP]\nrefine' le_antisymm (opNorm_le_bound _ le_rfl fun x => _) (norm_nonneg _)\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) = univ\nx : M\n\u22a2 \u2016\u2191(normedMk S) x\u2016 \u2264 0 * \u2016x\u2016\n[PROOFSTEP]\nhave hker : x \u2208 S.normedMk.ker.topologicalClosure :=\n  by\n  rw [S.ker_normedMk, \u2190 SetLike.mem_coe, h]\n  trivial\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) = univ\nx : M\n\u22a2 x \u2208 topologicalClosure (ker (normedMk S))\n[PROOFSTEP]\nrw [S.ker_normedMk, \u2190 SetLike.mem_coe, h]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) = univ\nx : M\n\u22a2 x \u2208 univ\n[PROOFSTEP]\ntrivial\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) = univ\nx : M\nhker : x \u2208 topologicalClosure (ker (normedMk S))\n\u22a2 \u2016\u2191(normedMk S) x\u2016 \u2264 0 * \u2016x\u2016\n[PROOFSTEP]\nrw [ker_normedMk] at hker \n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nh : \u2191(topologicalClosure S) = univ\nx : M\nhker : x \u2208 topologicalClosure S\n\u22a2 \u2016\u2191(normedMk S) x\u2016 \u2264 0 * \u2016x\u2016\n[PROOFSTEP]\nsimp only [(quotient_norm_eq_zero_iff S x).mpr hker, normedMk.apply, zero_mul, le_rfl]\n[GOAL]\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\ng : NormedAddGroupHom (M \u29f8 S) N\nh : NormedAddGroupHom.comp g (AddSubgroup.normedMk S) = f\n\u22a2 g = lift S f hf\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\ng : NormedAddGroupHom (M \u29f8 S) N\nh : NormedAddGroupHom.comp g (AddSubgroup.normedMk S) = f\nx : M \u29f8 S\n\u22a2 \u2191g x = \u2191(lift S f hf) x\n[PROOFSTEP]\nrcases AddSubgroup.surjective_normedMk _ x with \u27e8x, rfl\u27e9\n[GOAL]\ncase H.intro\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\ng : NormedAddGroupHom (M \u29f8 S) N\nh : NormedAddGroupHom.comp g (AddSubgroup.normedMk S) = f\nx : M\n\u22a2 \u2191g (\u2191(AddSubgroup.normedMk S) x) = \u2191(lift S f hf) (\u2191(AddSubgroup.normedMk S) x)\n[PROOFSTEP]\nchange g.comp S.normedMk x = _\n[GOAL]\ncase H.intro\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\ng : NormedAddGroupHom (M \u29f8 S) N\nh : NormedAddGroupHom.comp g (AddSubgroup.normedMk S) = f\nx : M\n\u22a2 \u2191(NormedAddGroupHom.comp g (AddSubgroup.normedMk S)) x = \u2191(lift S f hf) (\u2191(AddSubgroup.normedMk S) x)\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase H.intro\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\ng : NormedAddGroupHom (M \u29f8 S) N\nh : NormedAddGroupHom.comp g (AddSubgroup.normedMk S) = f\nx : M\n\u22a2 \u2191f x = \u2191(lift S f hf) (\u2191(AddSubgroup.normedMk S) x)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nm : M\n\u22a2 \u2016\u2191(AddSubgroup.normedMk S) m\u2016 = sInf ((fun m_1 => \u2016m + m_1\u2016) '' \u2191(ker (AddSubgroup.normedMk S)))\n[PROOFSTEP]\nsimpa [S.ker_normedMk] using quotient_norm_mk_eq _ m\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nn : N\n\u22a2 \u2203 m, \u2191f m = n \u2227 \u2016m\u2016 < \u2016n\u2016 + \u03b5\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := hquot.surjective n\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\n\u22a2 \u2203 m_1, \u2191f m_1 = \u2191f m \u2227 \u2016m_1\u2016 < \u2016\u2191f m\u2016 + \u03b5\n[PROOFSTEP]\nhave nonemp : ((fun m' => \u2016m + m'\u2016) '' f.ker).Nonempty :=\n  by\n  rw [Set.nonempty_image_iff]\n  exact \u27e80, f.ker.zero_mem\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\n\u22a2 Set.Nonempty ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f))\n[PROOFSTEP]\nrw [Set.nonempty_image_iff]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\n\u22a2 Set.Nonempty \u2191(ker f)\n[PROOFSTEP]\nexact \u27e80, f.ker.zero_mem\u27e9\n[GOAL]\ncase intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\nnonemp : Set.Nonempty ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f))\n\u22a2 \u2203 m_1, \u2191f m_1 = \u2191f m \u2227 \u2016m_1\u2016 < \u2016\u2191f m\u2016 + \u03b5\n[PROOFSTEP]\nrcases Real.lt_sInf_add_pos nonemp h\u03b5 with\n  \u27e8_, \u27e8\u27e8x, hx, rfl\u27e9, H : \u2016m + x\u2016 < sInf ((fun m' : M => \u2016m + m'\u2016) '' f.ker) + \u03b5\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\nnonemp : Set.Nonempty ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f))\nx : M\nhx : x \u2208 \u2191(ker f)\nH : \u2016m + x\u2016 < sInf ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f)) + \u03b5\n\u22a2 \u2203 m_1, \u2191f m_1 = \u2191f m \u2227 \u2016m_1\u2016 < \u2016\u2191f m\u2016 + \u03b5\n[PROOFSTEP]\nexact \u27e8m + x, by rw [map_add, (NormedAddGroupHom.mem_ker f x).mp hx, add_zero], by rwa [hquot.norm]\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\nnonemp : Set.Nonempty ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f))\nx : M\nhx : x \u2208 \u2191(ker f)\nH : \u2016m + x\u2016 < sInf ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f)) + \u03b5\n\u22a2 \u2191f (m + x) = \u2191f m\n[PROOFSTEP]\nrw [map_add, (NormedAddGroupHom.mem_ker f x).mp hx, add_zero]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nm : M\nnonemp : Set.Nonempty ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f))\nx : M\nhx : x \u2208 \u2191(ker f)\nH : \u2016m + x\u2016 < sInf ((fun m' => \u2016m + m'\u2016) '' \u2191(ker f)) + \u03b5\n\u22a2 \u2016m + x\u2016 < \u2016\u2191f m\u2016 + \u03b5\n[PROOFSTEP]\nrwa [hquot.norm]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n\u22a2 \u2016\u2191f m\u2016 \u2264 \u2016m\u2016\n[PROOFSTEP]\nrw [hquot.norm]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n\u22a2 sInf ((fun m_1 => \u2016m + m_1\u2016) '' \u2191(ker f)) \u2264 \u2016m\u2016\n[PROOFSTEP]\napply csInf_le\n[GOAL]\ncase h\u2081\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n\u22a2 BddBelow ((fun m_1 => \u2016m + m_1\u2016) '' \u2191(ker f))\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n\u22a2 0 \u2208 lowerBounds ((fun m_1 => \u2016m + m_1\u2016) '' \u2191(ker f))\n[PROOFSTEP]\nrintro _ \u27e8m', -, rfl\u27e9\n[GOAL]\ncase h.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm m' : M\n\u22a2 0 \u2264 (fun m_1 => \u2016m + m_1\u2016) m'\n[PROOFSTEP]\napply norm_nonneg\n[GOAL]\ncase h\u2082\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n\u22a2 \u2016m\u2016 \u2208 (fun m_1 => \u2016m + m_1\u2016) '' \u2191(ker f)\n[PROOFSTEP]\nexact \u27e80, f.ker.zero_mem, by simp\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup M\ninst\u271d : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n\u22a2 (fun m_1 => \u2016m + m_1\u2016) 0 = \u2016m\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\nfb : NormNoninc f\nx : M \u29f8 S\n\u22a2 \u2016\u2191(lift S f hf) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nhave fb' : \u2016f\u2016 \u2264 (1 : \u211d\u22650) := NormNoninc.normNoninc_iff_norm_le_one.mp fb\n[GOAL]\nM : Type u_1\nN\u271d : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\u271d\nN : Type u_3\ninst\u271d : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : \u2200 (s : M), s \u2208 S \u2192 \u2191f s = 0\nfb : NormNoninc f\nx : M \u29f8 S\nfb' : \u2016f\u2016 \u2264 \u21911\n\u22a2 \u2016\u2191(lift S f hf) x\u2016 \u2264 \u2016x\u2016\n[PROOFSTEP]\nsimpa using le_of_opNorm_le _ (f.lift_norm_le _ _ fb') _\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\nx : M \u29f8 S\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2016k \u2022 x\u2016 \u2264 \u2016k\u2016 * \u2016x\u2016 + \u03b5\n[PROOFSTEP]\nhave :=\n  (nhds_basis_ball.tendsto_iff nhds_basis_ball).mp ((@Real.uniformContinuous_const_mul \u2016k\u2016).continuous.tendsto \u2016x\u2016) \u03b5 h\u03b5\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\nx : M \u29f8 S\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis :\n  \u2203 ia,\n    0 < ia \u2227 \u2200 (x_1 : \u211d), x_1 \u2208 ball \u2016x\u2016 ia \u2192 (fun x x_2 => x * x_2) \u2016k\u2016 x_1 \u2208 ball ((fun x x_2 => x * x_2) \u2016k\u2016 \u2016x\u2016) \u03b5\n\u22a2 \u2016k \u2022 x\u2016 \u2264 \u2016k\u2016 * \u2016x\u2016 + \u03b5\n[PROOFSTEP]\nsimp only [mem_ball, exists_prop, dist, abs_sub_lt_iff] at this \n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\nx : M \u29f8 S\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis :\n  \u2203 ia, 0 < ia \u2227 \u2200 (x_1 : \u211d), x_1 - \u2016x\u2016 < ia \u2227 \u2016x\u2016 - x_1 < ia \u2192 \u2016k\u2016 * x_1 - \u2016k\u2016 * \u2016x\u2016 < \u03b5 \u2227 \u2016k\u2016 * \u2016x\u2016 - \u2016k\u2016 * x_1 < \u03b5\n\u22a2 \u2016k \u2022 x\u2016 \u2264 \u2016k\u2016 * \u2016x\u2016 + \u03b5\n[PROOFSTEP]\nrcases this with \u27e8\u03b4, h\u03b4, h\u27e9\n[GOAL]\ncase intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\nx : M \u29f8 S\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 (x_1 : \u211d), x_1 - \u2016x\u2016 < \u03b4 \u2227 \u2016x\u2016 - x_1 < \u03b4 \u2192 \u2016k\u2016 * x_1 - \u2016k\u2016 * \u2016x\u2016 < \u03b5 \u2227 \u2016k\u2016 * \u2016x\u2016 - \u2016k\u2016 * x_1 < \u03b5\n\u22a2 \u2016k \u2022 x\u2016 \u2264 \u2016k\u2016 * \u2016x\u2016 + \u03b5\n[PROOFSTEP]\nobtain \u27e8a, rfl, ha\u27e9 := Submodule.Quotient.norm_mk_lt x h\u03b4\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\na : M\nh : \u2200 (x : \u211d), x - \u2016mk a\u2016 < \u03b4 \u2227 \u2016mk a\u2016 - x < \u03b4 \u2192 \u2016k\u2016 * x - \u2016k\u2016 * \u2016mk a\u2016 < \u03b5 \u2227 \u2016k\u2016 * \u2016mk a\u2016 - \u2016k\u2016 * x < \u03b5\nha : \u2016a\u2016 < \u2016mk a\u2016 + \u03b4\n\u22a2 \u2016k \u2022 mk a\u2016 \u2264 \u2016k\u2016 * \u2016mk a\u2016 + \u03b5\n[PROOFSTEP]\nspecialize h \u2016a\u2016 \u27e8by linarith, by linarith [Submodule.Quotient.norm_mk_le S a]\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\na : M\nh : \u2200 (x : \u211d), x - \u2016mk a\u2016 < \u03b4 \u2227 \u2016mk a\u2016 - x < \u03b4 \u2192 \u2016k\u2016 * x - \u2016k\u2016 * \u2016mk a\u2016 < \u03b5 \u2227 \u2016k\u2016 * \u2016mk a\u2016 - \u2016k\u2016 * x < \u03b5\nha : \u2016a\u2016 < \u2016mk a\u2016 + \u03b4\n\u22a2 \u2016a\u2016 - \u2016mk a\u2016 < \u03b4\n[PROOFSTEP]\nlinarith\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\na : M\nh : \u2200 (x : \u211d), x - \u2016mk a\u2016 < \u03b4 \u2227 \u2016mk a\u2016 - x < \u03b4 \u2192 \u2016k\u2016 * x - \u2016k\u2016 * \u2016mk a\u2016 < \u03b5 \u2227 \u2016k\u2016 * \u2016mk a\u2016 - \u2016k\u2016 * x < \u03b5\nha : \u2016a\u2016 < \u2016mk a\u2016 + \u03b4\n\u22a2 \u2016mk a\u2016 - \u2016a\u2016 < \u03b4\n[PROOFSTEP]\nlinarith [Submodule.Quotient.norm_mk_le S a]\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u2077 : SeminormedAddCommGroup M\ninst\u271d\u2076 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d\u2075 : Ring R\ninst\u271d\u2074 : Module R M\nS : Submodule R M\n\ud835\udd5c : Type u_4\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c M\ninst\u271d\u00b9 : SMul \ud835\udd5c R\ninst\u271d : IsScalarTower \ud835\udd5c R M\nsrc\u271d : Module \ud835\udd5c (M \u29f8 S) := module' S\nk : \ud835\udd5c\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\na : M\nha : \u2016a\u2016 < \u2016mk a\u2016 + \u03b4\nh : \u2016k\u2016 * \u2016a\u2016 - \u2016k\u2016 * \u2016mk a\u2016 < \u03b5 \u2227 \u2016k\u2016 * \u2016mk a\u2016 - \u2016k\u2016 * \u2016a\u2016 < \u03b5\n\u22a2 \u2016k \u2022 mk a\u2016 \u2264 \u2016k\u2016 * \u2016mk a\u2016 + \u03b5\n[PROOFSTEP]\ncalc\n  _ \u2264 \u2016k\u2016 * \u2016a\u2016 := (quotient_norm_mk_le S.toAddSubgroup (k \u2022 a)).trans_eq (norm_smul k a)\n  _ \u2264 _ := (sub_lt_iff_lt_add'.mp h.1).le\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\nx y : R \u29f8 I\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2016x * y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016 + \u03b5\n[PROOFSTEP]\nhave :=\n  ((nhds_basis_ball.prod_nhds nhds_basis_ball).tendsto_iff nhds_basis_ball).mp (continuous_mul.tendsto (\u2016x\u2016, \u2016y\u2016)) \u03b5 h\u03b5\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\nx y : R \u29f8 I\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis :\n  \u2203 ia,\n    (0 < ia.fst \u2227 0 < ia.snd) \u2227\n      \u2200 (x_1 : \u211d \u00d7 \u211d),\n        x_1 \u2208 ball \u2016x\u2016 ia.fst \u00d7\u02e2 ball \u2016y\u2016 ia.snd \u2192 x_1.fst * x_1.snd \u2208 ball ((\u2016x\u2016, \u2016y\u2016).fst * (\u2016x\u2016, \u2016y\u2016).snd) \u03b5\n\u22a2 \u2016x * y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016 + \u03b5\n[PROOFSTEP]\nsimp only [Set.mem_prod, mem_ball, and_imp, Prod.forall, exists_prop, Prod.exists] at this \n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\nx y : R \u29f8 I\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nthis : \u2203 a b, (0 < a \u2227 0 < b) \u2227 \u2200 (a_1 b_1 : \u211d), dist a_1 \u2016x\u2016 < a \u2192 dist b_1 \u2016y\u2016 < b \u2192 dist (a_1 * b_1) (\u2016x\u2016 * \u2016y\u2016) < \u03b5\n\u22a2 \u2016x * y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016 + \u03b5\n[PROOFSTEP]\nrcases this with \u27e8\u03b5\u2081, \u03b5\u2082, \u27e8h\u2081, h\u2082\u27e9, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\nx y : R \u29f8 I\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh : \u2200 (a b : \u211d), dist a \u2016x\u2016 < \u03b5\u2081 \u2192 dist b \u2016y\u2016 < \u03b5\u2082 \u2192 dist (a * b) (\u2016x\u2016 * \u2016y\u2016) < \u03b5\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\n\u22a2 \u2016x * y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016 + \u03b5\n[PROOFSTEP]\nobtain \u27e8\u27e8a, rfl, ha\u27e9, \u27e8b, rfl, hb\u27e9\u27e9 := Ideal.Quotient.norm_mk_lt x h\u2081, Ideal.Quotient.norm_mk_lt y h\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nh :\n  \u2200 (a_1 b_1 : \u211d),\n    dist a_1 \u2016\u2191(mk I) a\u2016 < \u03b5\u2081 \u2192 dist b_1 \u2016\u2191(mk I) b\u2016 < \u03b5\u2082 \u2192 dist (a_1 * b_1) (\u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016) < \u03b5\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\n\u22a2 \u2016\u2191(mk I) a * \u2191(mk I) b\u2016 \u2264 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 + \u03b5\n[PROOFSTEP]\nsimp only [dist, abs_sub_lt_iff] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\nh :\n  \u2200 (a_1 b_1 : \u211d),\n    a_1 - \u2016\u2191(mk I) a\u2016 < \u03b5\u2081 \u2227 \u2016\u2191(mk I) a\u2016 - a_1 < \u03b5\u2081 \u2192\n      b_1 - \u2016\u2191(mk I) b\u2016 < \u03b5\u2082 \u2227 \u2016\u2191(mk I) b\u2016 - b_1 < \u03b5\u2082 \u2192\n        a_1 * b_1 - \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 < \u03b5 \u2227 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 - a_1 * b_1 < \u03b5\n\u22a2 \u2016\u2191(mk I) a * \u2191(mk I) b\u2016 \u2264 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 + \u03b5\n[PROOFSTEP]\nspecialize\n  h \u2016a\u2016 \u2016b\u2016 \u27e8by linarith, by linarith [Ideal.Quotient.norm_mk_le I a]\u27e9\n    \u27e8by linarith, by linarith [Ideal.Quotient.norm_mk_le I b]\u27e9\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\nh :\n  \u2200 (a_1 b_1 : \u211d),\n    a_1 - \u2016\u2191(mk I) a\u2016 < \u03b5\u2081 \u2227 \u2016\u2191(mk I) a\u2016 - a_1 < \u03b5\u2081 \u2192\n      b_1 - \u2016\u2191(mk I) b\u2016 < \u03b5\u2082 \u2227 \u2016\u2191(mk I) b\u2016 - b_1 < \u03b5\u2082 \u2192\n        a_1 * b_1 - \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 < \u03b5 \u2227 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 - a_1 * b_1 < \u03b5\n\u22a2 \u2016a\u2016 - \u2016\u2191(mk I) a\u2016 < \u03b5\u2081\n[PROOFSTEP]\nlinarith\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\nh :\n  \u2200 (a_1 b_1 : \u211d),\n    a_1 - \u2016\u2191(mk I) a\u2016 < \u03b5\u2081 \u2227 \u2016\u2191(mk I) a\u2016 - a_1 < \u03b5\u2081 \u2192\n      b_1 - \u2016\u2191(mk I) b\u2016 < \u03b5\u2082 \u2227 \u2016\u2191(mk I) b\u2016 - b_1 < \u03b5\u2082 \u2192\n        a_1 * b_1 - \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 < \u03b5 \u2227 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 - a_1 * b_1 < \u03b5\n\u22a2 \u2016\u2191(mk I) a\u2016 - \u2016a\u2016 < \u03b5\u2081\n[PROOFSTEP]\nlinarith [Ideal.Quotient.norm_mk_le I a]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\nh :\n  \u2200 (a_1 b_1 : \u211d),\n    a_1 - \u2016\u2191(mk I) a\u2016 < \u03b5\u2081 \u2227 \u2016\u2191(mk I) a\u2016 - a_1 < \u03b5\u2081 \u2192\n      b_1 - \u2016\u2191(mk I) b\u2016 < \u03b5\u2082 \u2227 \u2016\u2191(mk I) b\u2016 - b_1 < \u03b5\u2082 \u2192\n        a_1 * b_1 - \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 < \u03b5 \u2227 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 - a_1 * b_1 < \u03b5\n\u22a2 \u2016b\u2016 - \u2016\u2191(mk I) b\u2016 < \u03b5\u2082\n[PROOFSTEP]\nlinarith\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\nh :\n  \u2200 (a_1 b_1 : \u211d),\n    a_1 - \u2016\u2191(mk I) a\u2016 < \u03b5\u2081 \u2227 \u2016\u2191(mk I) a\u2016 - a_1 < \u03b5\u2081 \u2192\n      b_1 - \u2016\u2191(mk I) b\u2016 < \u03b5\u2082 \u2227 \u2016\u2191(mk I) b\u2016 - b_1 < \u03b5\u2082 \u2192\n        a_1 * b_1 - \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 < \u03b5 \u2227 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 - a_1 * b_1 < \u03b5\n\u22a2 \u2016\u2191(mk I) b\u2016 - \u2016b\u2016 < \u03b5\u2082\n[PROOFSTEP]\nlinarith [Ideal.Quotient.norm_mk_le I b]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup M\ninst\u271d\u00b9 : SeminormedAddCommGroup N\nR : Type u_3\ninst\u271d : SeminormedCommRing R\nI : Ideal R\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5\u2081 \u03b5\u2082 : \u211d\nh\u2081 : 0 < \u03b5\u2081\nh\u2082 : 0 < \u03b5\u2082\na : R\nha : \u2016a\u2016 < \u2016\u2191(mk I) a\u2016 + \u03b5\u2081\nb : R\nhb : \u2016b\u2016 < \u2016\u2191(mk I) b\u2016 + \u03b5\u2082\nh : \u2016a\u2016 * \u2016b\u2016 - \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 < \u03b5 \u2227 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 - \u2016a\u2016 * \u2016b\u2016 < \u03b5\n\u22a2 \u2016\u2191(mk I) a * \u2191(mk I) b\u2016 \u2264 \u2016\u2191(mk I) a\u2016 * \u2016\u2191(mk I) b\u2016 + \u03b5\n[PROOFSTEP]\ncalc\n  _ \u2264 \u2016a\u2016 * \u2016b\u2016 := (Ideal.Quotient.norm_mk_le I (a * b)).trans (norm_mul_le a b)\n  _ \u2264 _ := (sub_lt_iff_lt_add'.mp h.1).le\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.Quotient", "llama_tokens": 20079, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.7461389930307512, "lm_q1q2_score": 0.5545066295987853}}
{"text": "[GOAL]\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 Real.toNNReal r = { val := r, property := hr }\n[PROOFSTEP]\nsimp_rw [Real.toNNReal, max_eq_left hr]\n[GOAL]\n\u22a2 Zero \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 One \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Add \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Sub \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Mul \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Inv \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Div \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 LE \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Bot \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Inhabited \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 Nontrivial \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nr\u2081 r\u2082 : \u211d\u22650\nh : r\u2082 \u2264 r\u2081\n\u22a2 \u2191r\u2082 \u2264 \u2191r\u2081 - 0\n[PROOFSTEP]\nsimp [show (r\u2082 : \u211d) \u2264 r\u2081 from h]\n[GOAL]\nr : \u211d\u22650\n\u22a2 \u2191r = 0 \u2194 r = 0\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_zero, NNReal.coe_eq]\n[GOAL]\nr : \u211d\u22650\n\u22a2 \u2191r = 1 \u2194 r = 1\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_one, NNReal.coe_eq]\n[GOAL]\n\u22a2 CommSemiring \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra \u211d A\nr : \u211d\u22650\nx : (fun x => A) r\n\u22a2 \u2191(RingHom.comp (algebraMap \u211d A) toRealHom) r * x = x * \u2191(RingHom.comp (algebraMap \u211d A) toRealHom) r\n[PROOFSTEP]\nsimp [Algebra.commutes]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra \u211d A\nr : \u211d\u22650\nx : (fun x => A) r\n\u22a2 r \u2022 x = \u2191(RingHom.comp (algebraMap \u211d A) toRealHom) r * x\n[PROOFSTEP]\nsimp [\u2190 Algebra.smul_def (r : \u211d) x, smul_def]\n[GOAL]\n\u22a2 Algebra \u211d\u22650 \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 DistribMulAction \u211d\u22650\u02e3 \u211d\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 MonoidWithZero \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 CommMonoidWithZero \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 CommGroupWithZero \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a\n\u22a2 Real.toNNReal (\u2211 a in s, f a) = \u2211 a in s, Real.toNNReal (f a)\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, NNReal.coe_sum, Real.coe_toNNReal _ (Finset.sum_nonneg hf)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a\n\u22a2 \u2211 i in s, f i = \u2211 a in s, \u2191(Real.toNNReal (f a))\n[PROOFSTEP]\nexact Finset.sum_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a\nx : \u03b1\nhxs : x \u2208 s\n\u22a2 f x = \u2191(Real.toNNReal (f x))\n[PROOFSTEP]\nrw [Real.coe_toNNReal _ (hf x hxs)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a\n\u22a2 Real.toNNReal (\u220f a in s, f a) = \u220f a in s, Real.toNNReal (f a)\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a\n\u22a2 \u220f i in s, f i = \u220f a in s, \u2191(Real.toNNReal (f a))\n[PROOFSTEP]\nexact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]\n[GOAL]\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a\nx : \u03b1\nhxs : x \u2208 s\n\u22a2 f x = \u2191(Real.toNNReal (f x))\n[PROOFSTEP]\nrw [Real.coe_toNNReal _ (hf x hxs)]\n[GOAL]\n\u22a2 LinearOrder \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nn : \u2115\n\u22a2 \u2191(Real.toNNReal \u2191n) = \u2191\u2191n\n[PROOFSTEP]\nsimp [Real.coe_toNNReal]\n[GOAL]\n\u22a2 OrderBot \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 PartialOrder \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 CanonicallyLinearOrderedAddMonoid \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 LinearOrderedAddCommMonoid \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 DistribLattice \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 SemilatticeInf \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 SemilatticeSup \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 LinearOrderedSemiring \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 OrderedCommSemiring \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 LinearOrderedCommMonoid \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 LinearOrderedCommMonoidWithZero \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 LinearOrderedCommGroupWithZero \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 CanonicallyOrderedCommSemiring \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 DenselyOrdered \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u22a2 NoMaxOrder \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 \u211d\u22650\n\u22a2 \u2191(\u2a06 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191(s i)\n[PROOFSTEP]\nrw [iSup, iSup, coe_sSup, \u2190 Set.range_comp]\n[GOAL]\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 \u211d\u22650\n\u22a2 sSup (Set.range (toReal \u2218 fun i => s i)) = sSup (Set.range fun i => \u2191(s i))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 sInf \u2205 = 0\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq_zero, coe_sInf, Set.image_empty, Real.sInf_empty]\n[GOAL]\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 \u211d\u22650\n\u22a2 \u2191(\u2a05 (i : \u03b9), s i) = \u2a05 (i : \u03b9), \u2191(s i)\n[PROOFSTEP]\nrw [iInf, iInf, coe_sInf, \u2190 Set.range_comp]\n[GOAL]\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 \u211d\u22650\n\u22a2 sInf (Set.range (toReal \u2218 fun i => s i)) = sInf (Set.range fun i => \u2191(s i))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Sort u_1\n\u03b9' : Sort u_2\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Nonempty \u03b9'\nf : \u03b9 \u2192 \u211d\u22650\ng : \u03b9' \u2192 \u211d\u22650\na : \u211d\u22650\nh : \u2200 (i : \u03b9) (j : \u03b9'), a \u2264 f i + g j\n\u22a2 a \u2264 (\u2a05 (i : \u03b9), f i) + \u2a05 (j : \u03b9'), g j\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_le_coe, NNReal.coe_add, coe_iInf, coe_iInf]\n[GOAL]\n\u03b9 : Sort u_1\n\u03b9' : Sort u_2\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Nonempty \u03b9'\nf : \u03b9 \u2192 \u211d\u22650\ng : \u03b9' \u2192 \u211d\u22650\na : \u211d\u22650\nh : \u2200 (i : \u03b9) (j : \u03b9'), a \u2264 f i + g j\n\u22a2 \u2191a \u2264 (\u2a05 (i : \u03b9), \u2191(f i)) + \u2a05 (i : \u03b9'), \u2191(g i)\n[PROOFSTEP]\nexact le_ciInf_add_ciInf h\n[GOAL]\n\u22a2 Archimedean \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n  -- porting note: TODO: remove?\n[GOAL]\na b : \u211d\u22650\nh : \u2191a < \u2191b\nq : \u211a\nhaq : \u2191a < \u2191q\nhqb : \u2191q < \u2191b\nthis : 0 \u2264 \u2191q\n\u22a2 a < Real.toNNReal \u2191q \u2227 Real.toNNReal \u2191q < b\n[PROOFSTEP]\nsimp [Real.coe_toNNReal _ this, NNReal.coe_lt_coe.symm, haq, hqb]\n[GOAL]\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\u22650\ns : Finset \u03b1\nr : \u211d\u22650\n\u22a2 Finset.sup s f / r = Finset.sup s fun a => f a / r\n[PROOFSTEP]\nsimp only [div_eq_inv_mul, mul_finset_sup]\n[GOAL]\nr : \u211d\n\u22a2 0 < toNNReal r \u2194 0 < r\n[PROOFSTEP]\nsimp [\u2190 NNReal.coe_lt_coe, lt_irrefl]\n[GOAL]\nr : \u211d\n\u22a2 toNNReal r = 0 \u2194 r \u2264 0\n[PROOFSTEP]\nsimpa [-toNNReal_pos] using not_iff_not.2 (@toNNReal_pos r)\n[GOAL]\nr p : \u211d\nhp : 0 \u2264 p\n\u22a2 toNNReal r \u2264 toNNReal p \u2194 r \u2264 p\n[PROOFSTEP]\nsimp [\u2190 NNReal.coe_le_coe, hp]\n[GOAL]\nr p : \u211d\nhr : 0 \u2264 r\nhp : 0 \u2264 p\n\u22a2 toNNReal r = toNNReal p \u2194 r = p\n[PROOFSTEP]\nsimp [\u2190 NNReal.coe_eq, coe_toNNReal, hr, hp]\n[GOAL]\nr p : \u211d\nhr : 0 \u2264 r\nhp : 0 \u2264 p\n\u22a2 \u2191(toNNReal (r + p)) = \u2191(toNNReal r + toNNReal p)\n[PROOFSTEP]\nsimp [hr, hp, add_nonneg]\n[GOAL]\nr : \u211d\u22650\np : \u211d\nhp : 0 \u2264 p\n\u22a2 r \u2264 toNNReal p \u2194 \u2191r \u2264 p\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_le_coe, Real.coe_toNNReal p hp]\n[GOAL]\nr : \u211d\u22650\np : \u211d\nhr : 0 < r\nhp : p < 0\n\u22a2 r \u2264 toNNReal p \u2194 \u2191r \u2264 p\n[PROOFSTEP]\nsimp only [(hp.trans_le r.coe_nonneg).not_le, toNNReal_eq_zero.2 hp.le, hr.not_le]\n[GOAL]\nr : \u211d\np : \u211d\u22650\nha : 0 \u2264 r\n\u22a2 toNNReal r < p \u2194 r < \u2191p\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_lt_coe, Real.coe_toNNReal r ha]\n[GOAL]\nx : \u211d\nhx : 0 \u2264 x\nn : \u2115\n\u22a2 toNNReal (x ^ n) = toNNReal x ^ n\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, NNReal.coe_pow, Real.coe_toNNReal _ (pow_nonneg hx _), Real.coe_toNNReal x hx]\n[GOAL]\np q : \u211d\nhp : 0 \u2264 p\n\u22a2 \u2191(toNNReal (p * q)) = \u2191(toNNReal p * toNNReal q)\n[PROOFSTEP]\nsimp [mul_max_of_nonneg, hp]\n[GOAL]\na b c : \u211d\u22650\nh : a \u2260 0\n\u22a2 a * b = a * c \u2194 b = c\n[PROOFSTEP]\nrw [mul_eq_mul_left_iff, or_iff_left h]\n[GOAL]\na b : \u211d\u22650\nha : 0 < a\nhb : b < 1\n\u22a2 \u2203 n, b ^ n < a\n[PROOFSTEP]\nsimpa only [\u2190 coe_pow, NNReal.coe_lt_coe] using exists_pow_lt_of_lt_one (NNReal.coe_pos.2 ha) (NNReal.coe_lt_coe.2 hb)\n[GOAL]\n\u22a2 OrderedSub \u211d\u22650\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nr p : \u211d\u22650\nh : r \u2260 0\n\u22a2 r\u207b\u00b9 \u2264 p \u2194 1 \u2264 r * p\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h]\n[GOAL]\nr p : \u211d\u22650\nh : 1 \u2264 r * p\n\u22a2 r\u207b\u00b9 \u2264 p\n[PROOFSTEP]\nby_cases r = 0\n[GOAL]\nr p : \u211d\u22650\nh : 1 \u2264 r * p\n\u22a2 r\u207b\u00b9 \u2264 p\n[PROOFSTEP]\nby_cases r = 0\n[GOAL]\ncase pos\nr p : \u211d\u22650\nh\u271d : 1 \u2264 r * p\nh : r = 0\n\u22a2 r\u207b\u00b9 \u2264 p\n[PROOFSTEP]\nsimp [*, inv_le]\n[GOAL]\ncase neg\nr p : \u211d\u22650\nh\u271d : 1 \u2264 r * p\nh : \u00acr = 0\n\u22a2 r\u207b\u00b9 \u2264 p\n[PROOFSTEP]\nsimp [*, inv_le]\n[GOAL]\nr p : \u211d\u22650\nh : p \u2260 0\n\u22a2 r \u2264 p\u207b\u00b9 \u2194 r * p \u2264 1\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]\n[GOAL]\nr p : \u211d\u22650\nh : p \u2260 0\n\u22a2 r < p\u207b\u00b9 \u2194 r * p < 1\n[PROOFSTEP]\nrw [\u2190 mul_lt_mul_left (pos_iff_ne_zero.2 h), mul_inv_cancel h, mul_comm]\n[GOAL]\na b r : \u211d\u22650\nhr : r \u2260 0\n\u22a2 r * a \u2264 b \u2194 a \u2264 r\u207b\u00b9 * b\n[PROOFSTEP]\nhave : 0 < r := lt_of_le_of_ne (zero_le r) hr.symm\n[GOAL]\na b r : \u211d\u22650\nhr : r \u2260 0\nthis : 0 < r\n\u22a2 r * a \u2264 b \u2194 a \u2264 r\u207b\u00b9 * b\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_left (inv_pos.mpr this), \u2190 mul_assoc, inv_mul_cancel hr, one_mul]\n[GOAL]\na b c : \u211d\u22650\nh : a \u2264 b * c\nh0 : c = 0\n\u22a2 a / c \u2264 b\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\na b r : \u211d\u22650\nh : a < b / r\nhr : r = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [hr] at h \n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\n\u22a2 a \u2264 y\n[PROOFSTEP]\nhave hx : x \u2260 0 := pos_iff_ne_zero.1 (lt_of_le_of_lt (zero_le _) ha)\n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\nhx : x \u2260 0\n\u22a2 a \u2264 y\n[PROOFSTEP]\nhave hx' : x\u207b\u00b9 \u2260 0 := by rwa [Ne.def, inv_eq_zero]\n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\nhx : x \u2260 0\n\u22a2 x\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, inv_eq_zero]\n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\nhx : x \u2260 0\nhx' : x\u207b\u00b9 \u2260 0\n\u22a2 a \u2264 y\n[PROOFSTEP]\nhave : a * x\u207b\u00b9 < 1 := by rwa [\u2190 lt_inv_iff_mul_lt hx', inv_inv]\n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\nhx : x \u2260 0\nhx' : x\u207b\u00b9 \u2260 0\n\u22a2 a * x\u207b\u00b9 < 1\n[PROOFSTEP]\nrwa [\u2190 lt_inv_iff_mul_lt hx', inv_inv]\n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\nhx : x \u2260 0\nhx' : x\u207b\u00b9 \u2260 0\nthis : a * x\u207b\u00b9 < 1\n\u22a2 a \u2264 y\n[PROOFSTEP]\nhave : a * x\u207b\u00b9 * x \u2264 y := h _ this\n[GOAL]\nx y : \u211d\u22650\nh : \u2200 (a : \u211d\u22650), a < 1 \u2192 a * x \u2264 y\na : \u211d\u22650\nha : a < x\nhx : x \u2260 0\nhx' : x\u207b\u00b9 \u2260 0\nthis\u271d : a * x\u207b\u00b9 < 1\nthis : a * x\u207b\u00b9 * x \u2264 y\n\u22a2 a \u2264 y\n[PROOFSTEP]\nrwa [mul_assoc, inv_mul_cancel hx, mul_one] at this \n[GOAL]\na b : \u211d\u22650\nh : a < b\n\u22a2 a / b < 1\n[PROOFSTEP]\nrwa [div_lt_iff, one_mul]\n[GOAL]\na b : \u211d\u22650\nh : a < b\n\u22a2 b \u2260 0\n[PROOFSTEP]\nexact ne_of_gt (lt_of_le_of_lt (zero_le _) h)\n[GOAL]\nx : \u211d\n\u22a2 toNNReal x\u207b\u00b9 = (toNNReal x)\u207b\u00b9\n[PROOFSTEP]\ncases' le_total 0 x with hx hx\n[GOAL]\ncase inl\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 toNNReal x\u207b\u00b9 = (toNNReal x)\u207b\u00b9\n[PROOFSTEP]\nnth_rw 1 [\u2190 Real.coe_toNNReal x hx]\n[GOAL]\ncase inl\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 toNNReal (\u2191(toNNReal x))\u207b\u00b9 = (toNNReal x)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_inv, Real.toNNReal_coe]\n[GOAL]\ncase inr\nx : \u211d\nhx : x \u2264 0\n\u22a2 toNNReal x\u207b\u00b9 = (toNNReal x)\u207b\u00b9\n[PROOFSTEP]\nrw [toNNReal_eq_zero.mpr hx, inv_zero, toNNReal_eq_zero.mpr (inv_nonpos.mpr hx)]\n[GOAL]\nx y : \u211d\nhx : 0 \u2264 x\n\u22a2 toNNReal (x / y) = toNNReal x / toNNReal y\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, \u2190 Real.toNNReal_inv, \u2190 Real.toNNReal_mul hx]\n[GOAL]\nx y : \u211d\nhy : 0 \u2264 y\n\u22a2 toNNReal (x / y) = toNNReal x / toNNReal y\n[PROOFSTEP]\nrw [div_eq_inv_mul, div_eq_inv_mul, Real.toNNReal_mul (inv_nonneg.2 hy), Real.toNNReal_inv]\n[GOAL]\nx : \u211d\u22650\nhx : x \u2260 0\n\u22a2 x\u207b\u00b9 < 1 \u2194 1 < x\n[PROOFSTEP]\nrw [\u2190 one_div, div_lt_iff hx, one_mul]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ns : Set \u211d\u22650\nhs : \u00acBddAbove s\n\u22a2 sSup s = 0\n[PROOFSTEP]\nrw [\u2190 bddAbove_coe] at hs \n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ns : Set \u211d\u22650\nhs : \u00acBddAbove (toReal '' s)\n\u22a2 sSup s = 0\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, coe_sSup, NNReal.coe_zero]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ns : Set \u211d\u22650\nhs : \u00acBddAbove (toReal '' s)\n\u22a2 sSup (toReal '' s) = 0\n[PROOFSTEP]\nexact sSup_of_not_bddAbove hs\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d : \u03b9 \u2192 \u211d\u22650\ninst\u271d : IsEmpty \u03b9\nf : \u03b9 \u2192 \u211d\u22650\n\u22a2 \u2a05 (i : \u03b9), f i = 0\n[PROOFSTEP]\nrw [iInf_of_empty', sInf_empty]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\n\u03b1 : Sort u_2\n\u22a2 \u2a05 (x : \u03b1), 0 = 0\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, coe_iInf]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\n\u03b1 : Sort u_2\n\u22a2 \u2a05 (i : \u03b1), \u21910 = \u21910\n[PROOFSTEP]\nexact Real.ciInf_const_zero\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 iInf f * a = \u2a05 (i : \u03b9), f i * a\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, NNReal.coe_mul, coe_iInf, coe_iInf]\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 (\u2a05 (i : \u03b9), \u2191(f i)) * \u2191a = \u2a05 (i : \u03b9), \u2191(f i * a)\n[PROOFSTEP]\nexact Real.iInf_mul_of_nonneg (NNReal.coe_nonneg _) _\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 a * iInf f = \u2a05 (i : \u03b9), a * f i\n[PROOFSTEP]\nsimpa only [mul_comm] using iInf_mul f a\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 a * \u2a06 (i : \u03b9), f i = \u2a06 (i : \u03b9), a * f i\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq, NNReal.coe_mul, NNReal.coe_iSup, NNReal.coe_iSup]\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 \u2191a * \u2a06 (i : \u03b9), \u2191(f i) = \u2a06 (i : \u03b9), \u2191(a * f i)\n[PROOFSTEP]\nexact Real.mul_iSup_of_nonneg (NNReal.coe_nonneg _) _\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 (\u2a06 (i : \u03b9), f i) * a = \u2a06 (i : \u03b9), f i * a\n[PROOFSTEP]\nrw [mul_comm, mul_iSup]\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 \u2a06 (i : \u03b9), a * f i = \u2a06 (i : \u03b9), f i * a\n[PROOFSTEP]\nsimp_rw [mul_comm]\n[GOAL]\n\u03b9 : Sort u_1\nf\u271d f : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\n\u22a2 (\u2a06 (i : \u03b9), f i) / a = \u2a06 (i : \u03b9), f i / a\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, iSup_mul]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\na g : \u211d\u22650\nh : \u03b9 \u2192 \u211d\u22650\nH : \u2200 (j : \u03b9), g * h j \u2264 a\n\u22a2 g * iSup h \u2264 a\n[PROOFSTEP]\nrw [mul_iSup]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\na g : \u211d\u22650\nh : \u03b9 \u2192 \u211d\u22650\nH : \u2200 (j : \u03b9), g * h j \u2264 a\n\u22a2 \u2a06 (i : \u03b9), g * h i \u2264 a\n[PROOFSTEP]\nexact ciSup_le' H\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\ng : \u03b9 \u2192 \u211d\u22650\nh : \u211d\u22650\nH : \u2200 (i : \u03b9), g i * h \u2264 a\n\u22a2 iSup g * h \u2264 a\n[PROOFSTEP]\nrw [iSup_mul]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\na : \u211d\u22650\ng : \u03b9 \u2192 \u211d\u22650\nh : \u211d\u22650\nH : \u2200 (i : \u03b9), g i * h \u2264 a\n\u22a2 \u2a06 (i : \u03b9), g i * h \u2264 a\n[PROOFSTEP]\nexact ciSup_le' H\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ninst\u271d : Nonempty \u03b9\na g : \u211d\u22650\nh : \u03b9 \u2192 \u211d\u22650\nH : \u2200 (j : \u03b9), a \u2264 g * h j\n\u22a2 a \u2264 g * iInf h\n[PROOFSTEP]\nrw [mul_iInf]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ninst\u271d : Nonempty \u03b9\na g : \u211d\u22650\nh : \u03b9 \u2192 \u211d\u22650\nH : \u2200 (j : \u03b9), a \u2264 g * h j\n\u22a2 a \u2264 \u2a05 (i : \u03b9), g * h i\n[PROOFSTEP]\nexact le_ciInf H\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ninst\u271d : Nonempty \u03b9\na : \u211d\u22650\ng : \u03b9 \u2192 \u211d\u22650\nh : \u211d\u22650\nH : \u2200 (i : \u03b9), a \u2264 g i * h\n\u22a2 a \u2264 iInf g * h\n[PROOFSTEP]\nrw [iInf_mul]\n[GOAL]\n\u03b9 : Sort u_1\nf : \u03b9 \u2192 \u211d\u22650\ninst\u271d : Nonempty \u03b9\na : \u211d\u22650\ng : \u03b9 \u2192 \u211d\u22650\nh : \u211d\u22650\nH : \u2200 (i : \u03b9), a \u2264 g i * h\n\u22a2 a \u2264 \u2a05 (i : \u03b9), g i * h\n[PROOFSTEP]\nexact le_ciInf H\n[GOAL]\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\n\u22a2 OrdConnected (toNNReal '' s)\n[PROOFSTEP]\nrefine' \u27e8ball_image_iff.2 fun x hx => ball_image_iff.2 fun y hy z hz => _\u27e9\n[GOAL]\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\n\u22a2 z \u2208 toNNReal '' s\n[PROOFSTEP]\ncases' le_total y 0 with hy\u2080 hy\u2080\n[GOAL]\ncase inl\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\nhy\u2080 : y \u2264 0\n\u22a2 z \u2208 toNNReal '' s\n[PROOFSTEP]\nrw [mem_Icc, Real.toNNReal_of_nonpos hy\u2080, nonpos_iff_eq_zero] at hz \n[GOAL]\ncase inl\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : toNNReal x \u2264 z \u2227 z = 0\nhy\u2080 : y \u2264 0\n\u22a2 z \u2208 toNNReal '' s\n[PROOFSTEP]\nexact \u27e8y, hy, (toNNReal_of_nonpos hy\u2080).trans hz.2.symm\u27e9\n[GOAL]\ncase inr\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\nhy\u2080 : 0 \u2264 y\n\u22a2 z \u2208 toNNReal '' s\n[PROOFSTEP]\nlift y to \u211d\u22650 using hy\u2080\n[GOAL]\ncase inr.intro\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\nz y : \u211d\u22650\nhy : \u2191y \u2208 s\nhz : z \u2208 Icc (toNNReal x) (toNNReal \u2191y)\n\u22a2 z \u2208 toNNReal '' s\n[PROOFSTEP]\nrw [toNNReal_coe] at hz \n[GOAL]\ncase inr.intro\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\nz y : \u211d\u22650\nhy : \u2191y \u2208 s\nhz : z \u2208 Icc (toNNReal x) y\n\u22a2 z \u2208 toNNReal '' s\n[PROOFSTEP]\nexact \u27e8z, h.out hx hy \u27e8toNNReal_le_iff_le_coe.1 hz.1, hz.2\u27e9, toNNReal_coe\u27e9\n[GOAL]\n\u22a2 (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u22a2 \u2191((fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0) = \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 ZeroHom.toFun\n      { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n        map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n      1 =\n    1\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u22a2 \u2191(ZeroHom.toFun\n        { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n          map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n        1) =\n    \u21911\n[PROOFSTEP]\nsimp\n[GOAL]\nx y : \u211d\n\u22a2 ZeroHom.toFun\n      { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n        map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n      (x * y) =\n    ZeroHom.toFun\n        { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n          map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n        x *\n      ZeroHom.toFun\n        { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n          map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n        y\n[PROOFSTEP]\next\n[GOAL]\ncase a\nx y : \u211d\n\u22a2 \u2191(ZeroHom.toFun\n        { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n          map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n        (x * y)) =\n    \u2191(ZeroHom.toFun\n          { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n            map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n          x *\n        ZeroHom.toFun\n          { toFun := fun x => { val := |x|, property := (_ : 0 \u2264 |x|) },\n            map_zero' := (_ : (fun x => { val := |x|, property := (_ : 0 \u2264 |x|) }) 0 = 0) }\n          y)\n[PROOFSTEP]\nsimp [abs_mul]\n[GOAL]\nx : \u211d\nh : 0 \u2264 x\n\u22a2 \u2191nnabs x = toNNReal x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nx : \u211d\nh : 0 \u2264 x\n\u22a2 \u2191(\u2191nnabs x) = \u2191(toNNReal x)\n[PROOFSTEP]\nrw [coe_toNNReal x h, coe_nnabs, abs_of_nonneg h]\n[GOAL]\nx : \u211d\u22650\n\u22a2 \u2191nnabs \u2191x = x\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2124\n\u22a2 \u2191(Int.natAbs n) = \u2191nnabs \u2191n\n[PROOFSTEP]\next\n[GOAL]\ncase a\nn : \u2124\n\u22a2 \u2191\u2191(Int.natAbs n) = \u2191(\u2191nnabs \u2191n)\n[PROOFSTEP]\nrw [NNReal.coe_nat_cast, Int.cast_natAbs, Real.coe_nnabs, Int.cast_abs]\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.NNReal", "llama_tokens": 10032, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527631, "lm_q2_score": 0.6859494550081925, "lm_q1q2_score": 0.5542932647465065}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : Zero R\nn : R\n\u22a2 \u00acNeZero n \u2194 n = 0\n[PROOFSTEP]\nsimp [neZero_iff]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.NeZero", "llama_tokens": 52, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8418256472515684, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.5542727385720987}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\nA B : Set (Finset \u03b1)\nr : \u2115\nf : \u03b9 \u2192 Set (Finset \u03b1)\n\u22a2 Sized r (\u22c3 (i : \u03b9), f i) \u2194 \u2200 (i : \u03b9), Sized r (f i)\n[PROOFSTEP]\nsimp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\nA B : Set (Finset \u03b1)\nr : \u2115\nf : \u03b9 \u2192 Set (Finset \u03b1)\n\u22a2 (\u2200 \u2983x : Finset \u03b1\u2984 (x_1 : \u03b9), x \u2208 f x_1 \u2192 card x = r) \u2194 \u2200 (i : \u03b9) \u2983x : Finset \u03b1\u2984, x \u2208 f i \u2192 card x = r\n[PROOFSTEP]\nexact forall_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\nA B : Set (Finset \u03b1)\nr : \u2115\nf : (i : \u03b9) \u2192 \u03ba i \u2192 Set (Finset \u03b1)\n\u22a2 Sized r (\u22c3 (i : \u03b9) (j : \u03ba i), f i j) \u2194 \u2200 (i : \u03b9) (j : \u03ba i), Sized r (f i j)\n[PROOFSTEP]\nsimp only [Set.sized_iUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nr : \u2115\nA : Finset \u03b1\n\u22a2 A \u2208 \ud835\udc9c \u2192 A \u2208 powersetLen r univ \u2194 A \u2208 \u2191\ud835\udc9c \u2192 card A = r\n[PROOFSTEP]\nrw [mem_powerset_len_univ_iff, mem_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\n\u22a2 card \ud835\udc9c \u2264 Nat.choose (Fintype.card \u03b1) r\n[PROOFSTEP]\nrw [Fintype.card, \u2190 card_powersetLen]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns : Finset \u03b1\nr : \u2115\nh\ud835\udc9c : Set.Sized r \u2191\ud835\udc9c\n\u22a2 card \ud835\udc9c \u2264 card (powersetLen r univ)\n[PROOFSTEP]\nexact card_le_of_subset (subset_powersetLen_univ_iff.mpr h\ud835\udc9c)\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\n\ud835\udc9c : Finset (Finset \u03b1)\nA A\u2081 A\u2082 : Finset \u03b1\nr r\u2081 r\u2082 : \u2115\ninst\u271d : Fintype \u03b1\n\u22a2 \u2211 r in Iic (Fintype.card \u03b1), card (\ud835\udc9c # r) = card \ud835\udc9c\n[PROOFSTEP]\nletI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\n\ud835\udc9c : Finset (Finset \u03b1)\nA A\u2081 A\u2082 : Finset \u03b1\nr r\u2081 r\u2082 : \u2115\ninst\u271d : Fintype \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 \u2211 r in Iic (Fintype.card \u03b1), card (\ud835\udc9c # r) = card \ud835\udc9c\n[PROOFSTEP]\nrw [\u2190 card_biUnion, biUnion_slice]\n[GOAL]\n\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\n\ud835\udc9c : Finset (Finset \u03b1)\nA A\u2081 A\u2082 : Finset \u03b1\nr r\u2081 r\u2082 : \u2115\ninst\u271d : Fintype \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 \u2200 (x : \u2115), x \u2208 Iic (Fintype.card \u03b1) \u2192 \u2200 (y : \u2115), y \u2208 Iic (Fintype.card \u03b1) \u2192 x \u2260 y \u2192 Disjoint (\ud835\udc9c # x) (\ud835\udc9c # y)\n[PROOFSTEP]\nexact Finset.pairwiseDisjoint_slice.subset (Set.subset_univ _)\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Slice", "llama_tokens": 1233, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.553744878619252}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\n\u22a2 EquitableOn s f \u2194 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nrefine' \u27e8_, fun \u27e8b, hb\u27e9 x y hx hy => (hb x hx).2.trans (add_le_add_right (hb y hy).1 _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\n\u22a2 EquitableOn s f \u2192 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nobtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u2115\n\u22a2 EquitableOn \u2205 f \u2192 \u2203 b, \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\n\u22a2 EquitableOn s f \u2192 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nintro hs\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\n\u22a2 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nby_cases h : \u2200 y \u2208 s, f x \u2264 f y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\nh : \u2200 (y : \u03b1), y \u2208 s \u2192 f x \u2264 f y\n\u22a2 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nexact \u27e8f x, fun y hy => \u27e8h _ hy, hs hy hx\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\nh : \u00ac\u2200 (y : \u03b1), y \u2208 s \u2192 f x \u2264 f y\n\u22a2 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\nh : \u2203 y, y \u2208 s \u2227 f y < f x\n\u22a2 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nobtain \u27e8w, hw, hwx\u27e9 := h\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\nw : \u03b1\nhw : w \u2208 s\nhwx : f w < f x\n\u22a2 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\nrefine' \u27e8f w, fun y hy => \u27e8Nat.le_of_succ_le_succ _, hs hy hw\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\nw : \u03b1\nhw : w \u2208 s\nhwx : f w < f x\ny : \u03b1\nhy : y \u2208 s\n\u22a2 Nat.succ (f w) \u2264 Nat.succ (f y)\n[PROOFSTEP]\nrw [(Nat.succ_le_of_lt hwx).antisymm (hs hx hw)]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\nx : \u03b1\nhx : x \u2208 s\nhs : EquitableOn s f\nw : \u03b1\nhw : w \u2208 s\nhwx : f w < f x\ny : \u03b1\nhy : y \u2208 s\n\u22a2 f x \u2264 Nat.succ (f y)\n[PROOFSTEP]\nexact hs hx hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\n\u22a2 EquitableOn s f \u2194 \u2203 b, f '' s \u2286 Icc b (b + 1)\n[PROOFSTEP]\nsimpa only [image_subset_iff] using equitableOn_iff_exists_le_le_add_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set \u03b1\nf : \u03b1 \u2192 \u2115\n\u22a2 EquitableOn s f \u2194 \u2203 b, \u2200 (a : \u03b1), a \u2208 s \u2192 f a = b \u2228 f a = b + 1\n[PROOFSTEP]\nsimp_rw [equitableOn_iff_exists_le_le_add_one, Nat.le_and_le_add_one_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : OrderedSemiring \u03b2\ns : Set \u03b1\nhs : Set.Subsingleton s\nf : \u03b1 \u2192 \u03b2\ni j : \u03b1\nhi : i \u2208 s\nhj : j \u2208 s\n\u22a2 f i \u2264 f j + 1\n[PROOFSTEP]\nrw [hs hi hj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : OrderedSemiring \u03b2\ns : Set \u03b1\nhs : Set.Subsingleton s\nf : \u03b1 \u2192 \u03b2\ni j : \u03b1\nhi : i \u2208 s\nhj : j \u2208 s\n\u22a2 f j \u2264 f j + 1\n[PROOFSTEP]\nexact le_add_of_nonneg_right zero_le_one\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\n\u22a2 EquitableOn (\u2191s) f \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nrw [Set.equitableOn_iff_exists_le_le_add_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\n\u22a2 (\u2203 b, \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1) \u2194\n    \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nrefine' \u27e8_, fun h => \u27e8_, h\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\n\u22a2 (\u2203 b, \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1) \u2192\n    \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nrintro \u27e8b, hb\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nby_cases h : \u2200 a \u2208 s, f a = b + 1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = b + 1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na\u271d : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = b + 1\na : \u03b1\nha : a \u2208 s\n\u22a2 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nrw [h _ ha, sum_const_nat h, Nat.mul_div_cancel_left _ (card_pos.2 \u27e8a, ha\u27e9)]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na\u271d : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 f a = b + 1\na : \u03b1\nha : a \u2208 s\n\u22a2 b + 1 \u2264 b + 1 \u2227 b + 1 \u2264 b + 1 + 1\n[PROOFSTEP]\nexact \u27e8le_rfl, Nat.le_succ _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nh : \u00ac\u2200 (a : \u03b1), a \u2208 s \u2192 f a = b + 1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nh : \u2203 a, a \u2208 s \u2227 f a \u2260 b + 1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nobtain \u27e8x, hx\u2081, hx\u2082\u27e9 := h\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nsuffices h : b = (\u2211 i in s, f i) / s.card\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\nh : b = (\u2211 i in s, f i) / card s\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 (\u2211 i in s, f i) / card s \u2264 f a \u2227 f a \u2264 (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nsimp_rw [\u2190 h]\n[GOAL]\ncase neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\nh : b = (\u2211 i in s, f i) / card s\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\n[PROOFSTEP]\napply hb\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\n\u22a2 b = (\u2211 i in s, f i) / card s\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\n\u22a2 (\u2211 i in s, f i) / card s = b\n[PROOFSTEP]\nrefine'\n  Nat.div_eq_of_lt_le (le_trans (by simp [mul_comm]) (sum_le_sum fun a ha => (hb a ha).1))\n    ((sum_lt_sum (fun a ha => (hb a ha).2) \u27e8_, hx\u2081, (hb _ hx\u2081).2.lt_of_ne hx\u2082\u27e9).trans_le _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\n\u22a2 b * card s \u2264 \u2211 i in s, b\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\n\u22a2 \u2211 i in s, (b + 1) \u2264 Nat.succ b * card s\n[PROOFSTEP]\nrw [mul_comm, sum_const_nat]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\nb : \u2115\nhb : \u2200 (a : \u03b1), a \u2208 \u2191s \u2192 b \u2264 f a \u2227 f a \u2264 b + 1\nx : \u03b1\nhx\u2081 : x \u2208 s\nhx\u2082 : f x \u2260 b + 1\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 b + 1 = Nat.succ b\n[PROOFSTEP]\nexact fun _ _ => rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\n\u22a2 EquitableOn (\u2191s) f \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 f a = (\u2211 i in s, f i) / card s \u2228 f a = (\u2211 i in s, f i) / card s + 1\n[PROOFSTEP]\nsimp_rw [equitableOn_iff_le_le_add_one, Nat.le_and_le_add_one_iff]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Equitable", "llama_tokens": 4614, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956580903722561, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.5537448718747908}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nx : E\nx\u271d : x \u2208 univ\n\u22a2 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1\n[PROOFSTEP]\nhave : 0 < 1 + \u2016x\u2016 ^ 2 := by positivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nx : E\nx\u271d : x \u2208 univ\n\u22a2 0 < 1 + \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nx : E\nx\u271d : x \u2208 univ\nthis : 0 < 1 + \u2016x\u2016 ^ 2\n\u22a2 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1\n[PROOFSTEP]\nrw [mem_ball_zero_iff, norm_smul, Real.norm_eq_abs, abs_inv, \u2190 _root_.div_eq_inv_mul,\n  div_lt_one (abs_pos.mpr <| Real.sqrt_ne_zero'.mpr this), \u2190 abs_norm x, \u2190 sq_lt_sq, abs_norm, Real.sq_sqrt this.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nx : E\nx\u271d : x \u2208 univ\nthis : 0 < 1 + \u2016x\u2016 ^ 2\n\u22a2 \u2016x\u2016 ^ 2 < 1 + \u2016x\u2016 ^ 2\n[PROOFSTEP]\nexact lt_one_add _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nx : E\nx\u271d : x \u2208 univ\n\u22a2 (fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y) ((fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x) = x\n[PROOFSTEP]\nfield_simp [norm_smul, smul_smul, (zero_lt_one_add_norm_sq x).ne', sq_abs, Real.sq_sqrt (zero_lt_one_add_norm_sq x).le,\n  \u2190 Real.sqrt_div (zero_lt_one_add_norm_sq x).le]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) ((fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y) y) = y\n[PROOFSTEP]\nhave : 0 < 1 - \u2016y\u2016 ^ 2 := by nlinarith [norm_nonneg y, mem_ball_zero_iff.1 hy]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 0 < 1 - \u2016y\u2016 ^ 2\n[PROOFSTEP]\nnlinarith [norm_nonneg y, mem_ball_zero_iff.1 hy]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\nthis : 0 < 1 - \u2016y\u2016 ^ 2\n\u22a2 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) ((fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y) y) = y\n[PROOFSTEP]\nfield_simp [norm_smul, smul_smul, this.ne', sq_abs, Real.sq_sqrt this.le, \u2190 Real.sqrt_div this.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\n\u22a2 ContinuousOn\n    \u2191{ toFun := fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x, invFun := fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y,\n        source := univ, target := ball 0 1,\n        map_source' := (_ : \u2200 (x : E), x \u2208 univ \u2192 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1),\n        map_target' := (_ : \u2200 (x : E), x \u2208 ball 0 1 \u2192 True),\n        left_inv' :=\n          (_ :\n            \u2200 (x : E),\n              x \u2208 univ \u2192\n                (Real.sqrt (1 - \u2016(Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x = x),\n        right_inv' :=\n          (_ :\n            \u2200 (y : E),\n              y \u2208 ball 0 1 \u2192\n                (Real.sqrt (1 + \u2016(Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y = y) }\n    { toFun := fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x, invFun := fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y,\n        source := univ, target := ball 0 1,\n        map_source' := (_ : \u2200 (x : E), x \u2208 univ \u2192 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1),\n        map_target' := (_ : \u2200 (x : E), x \u2208 ball 0 1 \u2192 True),\n        left_inv' :=\n          (_ :\n            \u2200 (x : E),\n              x \u2208 univ \u2192\n                (Real.sqrt (1 - \u2016(Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x = x),\n        right_inv' :=\n          (_ :\n            \u2200 (y : E),\n              y \u2208 ball 0 1 \u2192\n                (Real.sqrt (1 + \u2016(Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y =\n                  y) }.source\n[PROOFSTEP]\nsuffices Continuous fun (x : E) => (1 + \u2016x\u2016 ^ 2).sqrt\u207b\u00b9 from (this.smul continuous_id).continuousOn\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\n\u22a2 Continuous fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9\n[PROOFSTEP]\nrefine' Continuous.inv\u2080 _ fun x => Real.sqrt_ne_zero'.mpr (by positivity)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nx : E\n\u22a2 0 < 1 + \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\n\u22a2 Continuous fun x => Real.sqrt (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\ncontinuity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\n\u22a2 ContinuousOn\n    { toFun := fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x, invFun := fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y,\n        source := univ, target := ball 0 1,\n        map_source' := (_ : \u2200 (x : E), x \u2208 univ \u2192 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1),\n        map_target' := (_ : \u2200 (x : E), x \u2208 ball 0 1 \u2192 True),\n        left_inv' :=\n          (_ :\n            \u2200 (x : E),\n              x \u2208 univ \u2192\n                (Real.sqrt (1 - \u2016(Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x = x),\n        right_inv' :=\n          (_ :\n            \u2200 (y : E),\n              y \u2208 ball 0 1 \u2192\n                (Real.sqrt (1 + \u2016(Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y =\n                  y) }.invFun\n    { toFun := fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x, invFun := fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y,\n        source := univ, target := ball 0 1,\n        map_source' := (_ : \u2200 (x : E), x \u2208 univ \u2192 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1),\n        map_target' := (_ : \u2200 (x : E), x \u2208 ball 0 1 \u2192 True),\n        left_inv' :=\n          (_ :\n            \u2200 (x : E),\n              x \u2208 univ \u2192\n                (Real.sqrt (1 - \u2016(Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x = x),\n        right_inv' :=\n          (_ :\n            \u2200 (y : E),\n              y \u2208 ball 0 1 \u2192\n                (Real.sqrt (1 + \u2016(Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y =\n                  y) }.target\n[PROOFSTEP]\nhave : \u2200 y \u2208 ball (0 : E) 1, (1 - \u2016(y : E)\u2016 ^ 2).sqrt \u2260 0 := fun y hy \u21a6\n  by\n  rw [Real.sqrt_ne_zero']\n  nlinarith [norm_nonneg y, mem_ball_zero_iff.1 hy]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 Real.sqrt (1 - \u2016y\u2016 ^ 2) \u2260 0\n[PROOFSTEP]\nrw [Real.sqrt_ne_zero']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 0 < 1 - \u2016y\u2016 ^ 2\n[PROOFSTEP]\nnlinarith [norm_nonneg y, mem_ball_zero_iff.1 hy]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nthis : \u2200 (y : E), y \u2208 ball 0 1 \u2192 Real.sqrt (1 - \u2016y\u2016 ^ 2) \u2260 0\n\u22a2 ContinuousOn\n    { toFun := fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x, invFun := fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y,\n        source := univ, target := ball 0 1,\n        map_source' := (_ : \u2200 (x : E), x \u2208 univ \u2192 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1),\n        map_target' := (_ : \u2200 (x : E), x \u2208 ball 0 1 \u2192 True),\n        left_inv' :=\n          (_ :\n            \u2200 (x : E),\n              x \u2208 univ \u2192\n                (Real.sqrt (1 - \u2016(Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x = x),\n        right_inv' :=\n          (_ :\n            \u2200 (y : E),\n              y \u2208 ball 0 1 \u2192\n                (Real.sqrt (1 + \u2016(Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y =\n                  y) }.invFun\n    { toFun := fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x, invFun := fun y => (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y,\n        source := univ, target := ball 0 1,\n        map_source' := (_ : \u2200 (x : E), x \u2208 univ \u2192 (fun x => (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x) x \u2208 ball 0 1),\n        map_target' := (_ : \u2200 (x : E), x \u2208 ball 0 1 \u2192 True),\n        left_inv' :=\n          (_ :\n            \u2200 (x : E),\n              x \u2208 univ \u2192\n                (Real.sqrt (1 - \u2016(Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9 \u2022 x = x),\n        right_inv' :=\n          (_ :\n            \u2200 (y : E),\n              y \u2208 ball 0 1 \u2192\n                (Real.sqrt (1 + \u2016(Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y\u2016 ^ 2))\u207b\u00b9 \u2022 (Real.sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9 \u2022 y =\n                  y) }.target\n[PROOFSTEP]\nexact\n  ContinuousOn.smul (ContinuousOn.inv\u2080 (continuousOn_const.sub (continuous_norm.continuousOn.pow _)).sqrt this)\n    continuousOn_id\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\n\u22a2 \u2191univUnitBall 0 = 0\n[PROOFSTEP]\nsimp [LocalHomeomorph.univUnitBall_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\n\u22a2 \u2191(LocalHomeomorph.symm univUnitBall) 0 = 0\n[PROOFSTEP]\nsimp [LocalHomeomorph.univUnitBall_symm_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 \u2191(Homeomorph.trans (Homeomorph.smulOfNeZero r (_ : r \u2260 0))\n          (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c))) ''\n      ball 0 1 =\n    ball c r\n[PROOFSTEP]\nchange (IsometryEquiv.vaddConst c) \u2218 (r \u2022 \u00b7) '' ball (0 : E) 1 = ball c r\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 (\u2191(IsometryEquiv.vaddConst c) \u2218 fun x => r \u2022 x) '' ball 0 1 = ball c r\n[PROOFSTEP]\nrw [image_comp, image_smul, smul_unitBall hr.ne', IsometryEquiv.image_ball]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 ball (\u2191(IsometryEquiv.vaddConst c) 0) \u2016r\u2016 = ball c r\n[PROOFSTEP]\nsimp [abs_of_pos hr]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 (univBall c r).toLocalEquiv.source = univ\n[PROOFSTEP]\nunfold univBall\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 (if h : 0 < r then\n          LocalHomeomorph.trans' univUnitBall (unitBallBall c r h)\n            (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target)\n        else\n          Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c))).toLocalEquiv.source =\n    univ\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nh\u271d : 0 < r\n\u22a2 (LocalHomeomorph.trans' univUnitBall (unitBallBall c r h\u271d)\n          (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target)).toLocalEquiv.source =\n    univ\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nh\u271d : \u00ac0 < r\n\u22a2 (Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c))).toLocalEquiv.source = univ\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 (univBall c r).toLocalEquiv.target = ball c r\n[PROOFSTEP]\nrw [univBall, dif_pos hr]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 (LocalHomeomorph.trans' univUnitBall (unitBallBall c r hr)\n          (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target)).toLocalEquiv.target =\n    ball c r\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 ball c r \u2286 (univBall c r).toLocalEquiv.target\n[PROOFSTEP]\nby_cases hr : 0 < r\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 ball c r \u2286 (univBall c r).toLocalEquiv.target\n[PROOFSTEP]\nrw [univBall_target c hr]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : \u00ac0 < r\n\u22a2 ball c r \u2286 (univBall c r).toLocalEquiv.target\n[PROOFSTEP]\nrw [univBall, dif_neg hr]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : \u00ac0 < r\n\u22a2 ball c r \u2286 (Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c))).toLocalEquiv.target\n[PROOFSTEP]\nexact subset_univ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 \u2191(univBall c r) 0 = c\n[PROOFSTEP]\nunfold univBall\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 \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      0 =\n    c\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nh\u271d : 0 < r\n\u22a2 \u2191(LocalHomeomorph.trans' univUnitBall (unitBallBall c r h\u271d)\n          (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target))\n      0 =\n    c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nh\u271d : \u00ac0 < r\n\u22a2 \u2191(Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c))) 0 = c\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 \u2191(LocalHomeomorph.symm (univBall c r)) c = 0\n[PROOFSTEP]\nhave : 0 \u2208 (univBall c r).source := by simp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 0 \u2208 (univBall c r).toLocalEquiv.source\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nthis : 0 \u2208 (univBall c r).toLocalEquiv.source\n\u22a2 \u2191(LocalHomeomorph.symm (univBall c r)) c = 0\n[PROOFSTEP]\nsimpa only [univBall_apply_zero] using (univBall c r).left_inv this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 Continuous \u2191(univBall c r)\n[PROOFSTEP]\nsimpa [continuous_iff_continuousOn_univ] using (univBall c r).continuousOn\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.HomeomorphBall", "llama_tokens": 7227, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.849971175657575, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.5536328715954588}}
{"text": "[GOAL]\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 HEq ((\u03c0.map f).map (Quotient.mk (Path.Homotopic.setoid x\u2080 x\u2081) p))\n    ((\u03c0.map g).map (Quotient.mk (Path.Homotopic.setoid x\u2082 x\u2083) q))\n[PROOFSTEP]\nsimp only [map_eq, \u2190 Path.Homotopic.map_lift]\n[GOAL]\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 HEq (Quotient.mk (Path.Homotopic.setoid (\u2191f x\u2080) (\u2191f x\u2081)) (Path.map p (_ : Continuous \u2191f)))\n    (Quotient.mk (Path.Homotopic.setoid (\u2191g x\u2082) (\u2191g x\u2083)) (Path.map q (_ : Continuous \u2191g)))\n[PROOFSTEP]\napply Path.Homotopic.hpath_hext\n[GOAL]\ncase hp\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 \u2200 (t : \u2191I), \u2191(Path.map p (_ : Continuous \u2191f)) t = \u2191(Path.map q (_ : Continuous \u2191g)) t\n[PROOFSTEP]\nexact hfg\n[GOAL]\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 \u2191f x\u2080 = \u2191g x\u2082\n[PROOFSTEP]\nconvert hfg 0\n[GOAL]\ncase h.e'_2.h.e'_6\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 x\u2080 = \u2191p 0\n[PROOFSTEP]\nsimp only [Path.source]\n[GOAL]\ncase h.e'_3.h.e'_6\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 x\u2082 = \u2191q 0\n[PROOFSTEP]\nsimp only [Path.source]\n[GOAL]\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 \u2191f x\u2081 = \u2191g x\u2083\n[PROOFSTEP]\nconvert hfg 1\n[GOAL]\ncase h.e'_2.h.e'_6\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 x\u2081 = \u2191p 1\n[PROOFSTEP]\nsimp only [Path.target]\n[GOAL]\ncase h.e'_3.h.e'_6\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 x\u2083 = \u2191q 1\n[PROOFSTEP]\nsimp only [Path.target]\n[GOAL]\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 (\u03c0.map f).map (Quotient.mk (Path.Homotopic.setoid x\u2080 x\u2081) p) =\n    hcast (_ : \u2191f x\u2080 = \u2191g x\u2082) \u226b (\u03c0.map g).map (Quotient.mk (Path.Homotopic.setoid x\u2082 x\u2083) q) \u226b hcast (_ : \u2191g x\u2083 = \u2191f x\u2081)\n[PROOFSTEP]\nrw [Functor.conj_eqToHom_iff_heq ((\u03c0\u2098 f).map \u27e6p\u27e7) ((\u03c0\u2098 g).map \u27e6q\u27e7) (start_path hfg) (end_path hfg)]\n[GOAL]\nX\u2081 X\u2082 Y : TopCat\nf : C(\u2191X\u2081, \u2191Y)\ng : C(\u2191X\u2082, \u2191Y)\nx\u2080 x\u2081 : \u2191X\u2081\nx\u2082 x\u2083 : \u2191X\u2082\np : Path x\u2080 x\u2081\nq : Path x\u2082 x\u2083\nhfg : \u2200 (t : \u2191I), \u2191f (\u2191p t) = \u2191g (\u2191q t)\n\u22a2 HEq ((\u03c0.map f).map (Quotient.mk (Path.Homotopic.setoid x\u2080 x\u2081) p))\n    ((\u03c0.map g).map (Quotient.mk (Path.Homotopic.setoid x\u2082 x\u2083) q))\n[PROOFSTEP]\nexact heq_path_of_eq_image hfg\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 (\u03c0.map f).map (Quotient.mk (Path.Homotopic.setoid (fromTop x\u2080) (fromTop x\u2081)) p') =\n    hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n      (\u03c0.map (uliftMap H)).map\n          (prodToProdTopI (\ud835\udfd9 (fromTop { down := 0 }))\n            (Quotient.mk (Path.Homotopic.setoid (fromTop x\u2080) (fromTop x\u2081)) p')) \u226b\n        hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081)\n[PROOFSTEP]\napply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 \u2200 (t : \u2191I), \u2191f (\u2191p' t) = \u2191(uliftMap H) (\u2191(Path.prod (Path.refl { down := 0 }) p') t)\n[PROOFSTEP]\nrw [Path.prod_coe]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 \u2200 (t : \u2191I), \u2191f (\u2191p' t) = \u2191(uliftMap H) ((fun t => (\u2191(Path.refl { down := 0 }) t, \u2191p' t)) t)\n[PROOFSTEP]\nsimp_rw [ulift_apply]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 \u2200 (t : \u2191I), \u2191f (\u2191p' t) = \u2191H ((\u2191(Path.refl { down := 0 }) t).down, \u2191p' t)\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 (\u03c0.map g).map (Quotient.mk (Path.Homotopic.setoid (fromTop x\u2080) (fromTop x\u2081)) p') =\n    hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080)) \u226b\n      (\u03c0.map (uliftMap H)).map\n          (prodToProdTopI (\ud835\udfd9 (fromTop { down := 1 }))\n            (Quotient.mk (Path.Homotopic.setoid (fromTop x\u2080) (fromTop x\u2081)) p')) \u226b\n        hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\n[PROOFSTEP]\napply @eq_path_of_eq_image _ _ _ _ H.uliftMap _ _ _ _ _ ((Path.refl (ULift.up _)).prod p')\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 \u2200 (t : \u2191I), \u2191g (\u2191p' t) = \u2191(uliftMap H) (\u2191(Path.prod (Path.refl { down := 1 }) p') t)\n[PROOFSTEP]\nrw [Path.prod_coe]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 \u2200 (t : \u2191I), \u2191g (\u2191p' t) = \u2191(uliftMap H) ((fun t => (\u2191(Path.refl { down := 1 }) t, \u2191p' t)) t)\n[PROOFSTEP]\nsimp_rw [ulift_apply]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\np' : Path (fromTop x\u2080) (fromTop x\u2081)\n\u22a2 \u2200 (t : \u2191I), \u2191g (\u2191p' t) = \u2191H ((\u2191(Path.refl { down := 1 }) t).down, \u2191p' t)\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\nx : \u2191X\n\u22a2 Quotient.mk (Path.Homotopic.setoid (fromTop (\u2191f x)) (fromTop (\u2191g x))) (evalAt H x) =\n    hcast (_ : \u2191f x = \u2191H (0, x)) \u226b\n      (\u03c0.map (uliftMap H)).map (prodToProdTopI uhpath01 (\ud835\udfd9 x)) \u226b hcast (_ : \u2191H (1, x) = \u2191g x)\n[PROOFSTEP]\ndsimp only [prodToProdTopI, uhpath01, hcast]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\nx : \u2191X\n\u22a2 Quotient.mk (Path.Homotopic.setoid (fromTop (\u2191f x)) (fromTop (\u2191g x))) (evalAt H x) =\n    eqToHom (_ : \u2191f x = \u2191H (0, x)) \u226b\n      (\u03c0.map (uliftMap H)).map\n          ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map\n            (Quotient.mk (Path.Homotopic.setoid (fromTop { down := 0 }) (fromTop { down := 1 })) unitInterval.upath01,\n              \ud835\udfd9 x)) \u226b\n        eqToHom (_ : \u2191H (1, x) = \u2191g x)\n[PROOFSTEP]\nrefine' (@Functor.conj_eqToHom_iff_heq (\u03c0\u2093 Y) _ _ _ _ _ _ _ _ (H.apply_one x).symm).mpr _\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\nx : \u2191X\n\u22a2 HEq (Quotient.mk (Path.Homotopic.setoid (fromTop (\u2191f x)) (fromTop (\u2191g x))) (evalAt H x))\n    ((\u03c0.map (uliftMap H)).map\n      ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map\n        (Quotient.mk (Path.Homotopic.setoid (fromTop { down := 0 }) (fromTop { down := 1 })) unitInterval.upath01,\n          \ud835\udfd9 x)))\n[PROOFSTEP]\nsimp only [id_eq_path_refl, prodToProdTop_map, Path.Homotopic.prod_lift, map_eq, \u2190 Path.Homotopic.map_lift]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\nx : \u2191X\n\u22a2 HEq (Quotient.mk (Path.Homotopic.setoid (fromTop (\u2191f x)) (fromTop (\u2191g x))) (evalAt H x))\n    (Path.Homotopic.Quotient.mapFn\n      ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map\n        (Quotient.mk (Path.Homotopic.setoid (fromTop { down := 0 }) (fromTop { down := 1 })) unitInterval.upath01,\n          Quotient.mk (Path.Homotopic.setoid x x) (Path.refl x)))\n      (uliftMap H))\n[PROOFSTEP]\napply Path.Homotopic.hpath_hext\n[GOAL]\ncase hp\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\nx : \u2191X\n\u22a2 \u2200 (t : \u2191I),\n    \u2191(evalAt H x) t =\n      \u2191((fun q => Path.map q (_ : Continuous \u2191(uliftMap H))) (Path.prod unitInterval.upath01 (Path.refl x))) t\n[PROOFSTEP]\nintro\n[GOAL]\ncase hp\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\nx : \u2191X\nt\u271d : \u2191I\n\u22a2 \u2191(evalAt H x) t\u271d =\n    \u2191((fun q => Path.map q (_ : Continuous \u2191(uliftMap H))) (Path.prod unitInterval.upath01 (Path.refl x))) t\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 (\u03c0.map f).map p \u226b Quotient.mk (Path.Homotopic.setoid ((\u03c0.map f).obj (fromTop x\u2081)) (\u2191g x\u2081)) (evalAt H x\u2081) =\n      diagonalPath' H p \u2227\n    Quotient.mk (Path.Homotopic.setoid (fromTop (\u2191f x\u2080)) (\u2191g x\u2080)) (evalAt H x\u2080) \u226b (\u03c0.map g).map p = diagonalPath' H p\n[PROOFSTEP]\nrw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI (\ud835\udfd9 (fromTop { down := 0 })) p) \u226b hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081)) \u226b\n        Quotient.mk (Path.Homotopic.setoid ((\u03c0.map f).obj (fromTop x\u2081)) (\u2191g x\u2081)) (evalAt H x\u2081) =\n      diagonalPath' H p \u2227\n    (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI uhpath01 (\ud835\udfd9 x\u2080)) \u226b hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080)) \u226b\n        hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI (\ud835\udfd9 (fromTop { down := 1 })) p) \u226b hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081) =\n      diagonalPath' H p\n[PROOFSTEP]\nerw [H.evalAt_eq]\n  -- Porting note: `rw` didn't work, so using `erw`\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI (\ud835\udfd9 (fromTop { down := 0 })) p) \u226b hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081)) \u226b\n        hcast (_ : \u2191f (fromTop x\u2081) = \u2191H (0, fromTop x\u2081)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI uhpath01 (\ud835\udfd9 (fromTop x\u2081))) \u226b\n            hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081)) =\n      diagonalPath' H p \u2227\n    (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI uhpath01 (\ud835\udfd9 x\u2080)) \u226b hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080)) \u226b\n        hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map (prodToProdTopI (\ud835\udfd9 (fromTop { down := 1 })) p) \u226b hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081) =\n      diagonalPath' H p\n[PROOFSTEP]\ndsimp only [prodToProdTopI]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n            hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081)) \u226b\n        hcast (_ : \u2191f (fromTop x\u2081) = \u2191H (0, fromTop x\u2081)) \u226b\n          (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081))) \u226b\n            hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081)) =\n      diagonalPath' H p \u2227\n    (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n            hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080)) \u226b\n        hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080)) \u226b\n          (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p)) \u226b\n            hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081) =\n      diagonalPath' H p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n        (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n          hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081)) \u226b\n      hcast (_ : \u2191f (fromTop x\u2081) = \u2191H (0, fromTop x\u2081)) \u226b\n        (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081))) \u226b\n          hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081)) =\n    diagonalPath' H p\n[PROOFSTEP]\nslice_lhs 2 4 =>\n  rw [eqToHom_trans, eqToHom_refl]\n    -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n    hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081) \u226b hcast (_ : \u2191f (fromTop x\u2081) = \u2191H (0, fromTop x\u2081))\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081))\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nrw [eqToHom_trans, eqToHom_refl]\n    -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n    hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081) \u226b hcast (_ : \u2191f (fromTop x\u2081) = \u2191H (0, fromTop x\u2081))\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081))\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nrw [eqToHom_trans, eqToHom_refl]\n    -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n    hcast (_ : \u2191H (0, x\u2081) = \u2191f x\u2081) \u226b hcast (_ : \u2191f (fromTop x\u2081) = \u2191H (0, fromTop x\u2081))\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081))\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nrw [eqToHom_trans, eqToHom_refl]\n  -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase left\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n      (((\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n            \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 0 }, x\u2081)))) \u226b\n          (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))) \u226b\n        hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081)) =\n    diagonalPath' H p\n[PROOFSTEP]\nslice_lhs 2 4 => simp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n    \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 0 }, x\u2081))) \u226b\n      (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081))\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nsimp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n    \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 0 }, x\u2081))) \u226b\n      (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081))\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nsimp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 0 }), p)) \u226b\n    \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 0 }, x\u2081))) \u226b\n      (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 (fromTop x\u2081)))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, fromTop x\u2081) = \u2191g (fromTop x\u2081))\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nsimp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase right\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 (hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n        (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n          hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080)) \u226b\n      hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080)) \u226b\n        (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p)) \u226b\n          hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081) =\n    diagonalPath' H p\n[PROOFSTEP]\nslice_lhs 2 4 =>\n  rw [eqToHom_trans, eqToHom_refl]\n    -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n    hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080) \u226b hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080))\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nrw [eqToHom_trans, eqToHom_refl]\n    -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n    hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080) \u226b hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080))\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nrw [eqToHom_trans, eqToHom_refl]\n    -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n    hcast (_ : \u2191H (1, x\u2080) = \u2191g x\u2080) \u226b hcast (_ : \u2191g x\u2080 = \u2191H (1, x\u2080))\ncase a.a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nrw [eqToHom_trans, eqToHom_refl]\n  -- Porting note: this \u2193 `simp` didn't do this\n[GOAL]\ncase right\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n\u22a2 hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080)) \u226b\n      (((\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n            \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 1 }, x\u2080)))) \u226b\n          (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))) \u226b\n        hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081) =\n    diagonalPath' H p\n[PROOFSTEP]\nslice_lhs 2 4 => simp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n    \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 1 }, x\u2080))) \u226b\n      (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nsimp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n    \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 1 }, x\u2080))) \u226b\n      (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nsimp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (uhpath01, \ud835\udfd9 x\u2080)) \u226b\n    \ud835\udfd9 ((\u03c0.map (uliftMap H)).obj ((prodToProdTop (TopCat.of (ULift \u2191I)) X).obj ({ down := 1 }, x\u2080))) \u226b\n      (\u03c0.map (uliftMap H)).map ((prodToProdTop (TopCat.of (ULift \u2191I)) X).map (\ud835\udfd9 (fromTop { down := 1 }), p))\ncase a.a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191H (1, x\u2081) = \u2191g x\u2081)\ncase a\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : Homotopy f g\nx\u2080 x\u2081 : \u2191X\np : fromTop x\u2080 \u27f6 fromTop x\u2081\n| hcast (_ : \u2191f x\u2080 = \u2191H (0, x\u2080))\n[PROOFSTEP]\nsimp [\u2190 CategoryTheory.Functor.map_comp]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nx y : \u2191(\u03c0.obj X)\np : x \u27f6 y\n\u22a2 (\u03c0.map f).map p \u226b\n      (fun x =>\n          Quotient.mk (Path.Homotopic.setoid ((\u03c0.map f).obj x) ((\u03c0.map g).obj x)) (ContinuousMap.Homotopy.evalAt H x))\n        y =\n    (fun x =>\n          Quotient.mk (Path.Homotopic.setoid ((\u03c0.map f).obj x) ((\u03c0.map g).obj x)) (ContinuousMap.Homotopy.evalAt H x))\n        x \u226b\n      (\u03c0.map g).map p\n[PROOFSTEP]\nerw [(H.eq_diag_path p).1, (H.eq_diag_path p).2]\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\n\u22a2 IsIso (homotopicMapsNatIso H)\n[PROOFSTEP]\napply NatIso.isIso_of_isIso_app\n[GOAL]\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \u2191(\u03c0.obj X) \u224c \u2191(\u03c0.obj Y)\n[PROOFSTEP]\napply CategoryTheory.Equivalence.mk (\u03c0\u2098 hequiv.toFun : \u03c0\u2093 X \u2964 \u03c0\u2093 Y) (\u03c0\u2098 hequiv.invFun : \u03c0\u2093 Y \u2964 \u03c0\u2093 X)\n[GOAL]\ncase \u03b7\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \ud835\udfed \u2191(\u03c0.obj X) \u2245 \u03c0.map hequiv.toFun \u22d9 \u03c0.map hequiv.invFun\n[PROOFSTEP]\nsimp only [Grpd.hom_to_functor, Grpd.id_to_functor]\n[GOAL]\ncase \u03b5\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \u03c0.map hequiv.invFun \u22d9 \u03c0.map hequiv.toFun \u2245 \ud835\udfed \u2191(\u03c0.obj Y)\n[PROOFSTEP]\nsimp only [Grpd.hom_to_functor, Grpd.id_to_functor]\n[GOAL]\ncase \u03b7\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \ud835\udfd9 (\u03c0.obj X) \u2245 \u03c0.map hequiv.toFun \u22d9 \u03c0.map hequiv.invFun\n[PROOFSTEP]\nconvert (asIso (homotopicMapsNatIso hequiv.left_inv.some)).symm\n[GOAL]\ncase h.e'_3\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \ud835\udfd9 (\u03c0.obj X) = \u03c0.map (ContinuousMap.id \u2191X)\ncase h.e'_4\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \u03c0.map hequiv.toFun \u22d9 \u03c0.map hequiv.invFun = \u03c0.map (ContinuousMap.comp hequiv.invFun hequiv.toFun)\n[PROOFSTEP]\nexacts [((\u03c0).map_id X).symm, ((\u03c0).map_comp _ _).symm]\n[GOAL]\ncase \u03b5\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \u03c0.map hequiv.invFun \u22d9 \u03c0.map hequiv.toFun \u2245 \ud835\udfd9 (\u03c0.obj Y)\n[PROOFSTEP]\nconvert asIso (homotopicMapsNatIso hequiv.right_inv.some)\n[GOAL]\ncase h.e'_3\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \u03c0.map hequiv.invFun \u22d9 \u03c0.map hequiv.toFun = \u03c0.map (ContinuousMap.comp hequiv.toFun hequiv.invFun)\ncase h.e'_4\nX Y : TopCat\nf g : C(\u2191X, \u2191Y)\nH : ContinuousMap.Homotopy f g\nhequiv : \u2191X \u2243\u2095 \u2191Y\n\u22a2 \ud835\udfd9 (\u03c0.obj Y) = \u03c0.map (ContinuousMap.id \u2191Y)\n[PROOFSTEP]\nexacts [((\u03c0).map_comp _ _).symm, ((\u03c0).map_id Y).symm]\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps", "llama_tokens": 13795, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430478583168, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.5530513598739326}}
{"text": "[GOAL]\nn k : \u2115\nx : Fin k \u2192 \u2115\n\u22a2 x \u2208 antidiagonalTuple k n \u2194 \u2211 i : Fin k, x i = n\n[PROOFSTEP]\ninduction x using Fin.consInduction generalizing n with\n| h0 =>\n  cases n\n  \u00b7 simp\n  \u00b7 simp [eq_comm]\n| h x\u2080 x ih =>\n  simp_rw [Fin.sum_cons]\n  rw [antidiagonalTuple]\n    -- porting note: simp_rw doesn't use the equation lemma properly\n  simp_rw [List.mem_bind, List.mem_map, List.Nat.mem_antidiagonal, Fin.cons_eq_cons, exists_eq_right_right, ih,\n    @eq_comm _ _ (Prod.snd _), and_comm (a := Prod.snd _ = _), \u2190 Prod.mk.inj_iff (a\u2081 := Prod.fst _), Prod.mk.eta,\n    exists_eq_right]\n[GOAL]\nn k : \u2115\nx : Fin k \u2192 \u2115\n\u22a2 x \u2208 antidiagonalTuple k n \u2194 \u2211 i : Fin k, x i = n\n[PROOFSTEP]\ninduction x using Fin.consInduction generalizing n with\n| h0 =>\n  cases n\n  \u00b7 simp\n  \u00b7 simp [eq_comm]\n| h x\u2080 x ih =>\n  simp_rw [Fin.sum_cons]\n  rw [antidiagonalTuple]\n    -- porting note: simp_rw doesn't use the equation lemma properly\n  simp_rw [List.mem_bind, List.mem_map, List.Nat.mem_antidiagonal, Fin.cons_eq_cons, exists_eq_right_right, ih,\n    @eq_comm _ _ (Prod.snd _), and_comm (a := Prod.snd _ = _), \u2190 Prod.mk.inj_iff (a\u2081 := Prod.fst _), Prod.mk.eta,\n    exists_eq_right]\n[GOAL]\ncase h0\nk n : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 n \u2194 \u2211 i : Fin 0, Fin.elim0 i = n\n[PROOFSTEP]\n\n| h0 =>\n  cases n\n  \u00b7 simp\n  \u00b7 simp [eq_comm]\n[GOAL]\ncase h0\nk n : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 n \u2194 \u2211 i : Fin 0, Fin.elim0 i = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase h0.zero\nk : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 Nat.zero \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h0.succ\nk n\u271d : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 (Nat.succ n\u271d) \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.succ n\u271d\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\ncase h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 \u2211 i : Fin (n\u271d + 1), Fin.cons x\u2080 x i = n\n[PROOFSTEP]\n\n| h x\u2080 x ih =>\n  simp_rw [Fin.sum_cons]\n  rw [antidiagonalTuple]\n    -- porting note: simp_rw doesn't use the equation lemma properly\n  simp_rw [List.mem_bind, List.mem_map, List.Nat.mem_antidiagonal, Fin.cons_eq_cons, exists_eq_right_right, ih,\n    @eq_comm _ _ (Prod.snd _), and_comm (a := Prod.snd _ = _), \u2190 Prod.mk.inj_iff (a\u2081 := Prod.fst _), Prod.mk.eta,\n    exists_eq_right]\n[GOAL]\ncase h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 \u2211 i : Fin (n\u271d + 1), Fin.cons x\u2080 x i = n\n[PROOFSTEP]\nsimp_rw [Fin.sum_cons]\n[GOAL]\ncase h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 x\u2080 + \u2211 i : Fin n\u271d, x i = n\n[PROOFSTEP]\nrw [antidiagonalTuple]\n  -- porting note: simp_rw doesn't use the equation lemma properly\n[GOAL]\ncase h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 (Fin.cons x\u2080 x \u2208\n      List.bind (antidiagonal n) fun ni => map (fun x => Fin.cons ni.fst x) (antidiagonalTuple n\u271d ni.snd)) \u2194\n    x\u2080 + \u2211 i : Fin n\u271d, x i = n\n[PROOFSTEP]\nsimp_rw [List.mem_bind, List.mem_map, List.Nat.mem_antidiagonal, Fin.cons_eq_cons, exists_eq_right_right, ih,\n  @eq_comm _ _ (Prod.snd _), and_comm (a := Prod.snd _ = _), \u2190 Prod.mk.inj_iff (a\u2081 := Prod.fst _), Prod.mk.eta,\n  exists_eq_right]\n[GOAL]\nk n : \u2115\n\u22a2 Nodup (antidiagonalTuple k n)\n[PROOFSTEP]\ninduction' k with k ih generalizing n\n[GOAL]\ncase zero\nn\u271d n : \u2115\n\u22a2 Nodup (antidiagonalTuple Nat.zero n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero.zero\nn : \u2115\n\u22a2 Nodup (antidiagonalTuple Nat.zero Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero.succ\nn n\u271d : \u2115\n\u22a2 Nodup (antidiagonalTuple Nat.zero (Nat.succ n\u271d))\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\ncase succ\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\n\u22a2 Nodup (antidiagonalTuple (Nat.succ k) n)\n[PROOFSTEP]\nsimp_rw [antidiagonalTuple, List.nodup_bind]\n[GOAL]\ncase succ\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\n\u22a2 (\u2200 (x : \u2115 \u00d7 \u2115), x \u2208 antidiagonal n \u2192 Nodup (map (fun x_1 => Fin.cons x.fst x_1) (antidiagonalTuple k x.snd))) \u2227\n    Pairwise\n      (fun a b =>\n        Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n          (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n      (antidiagonal n)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase succ.left\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115), x \u2208 antidiagonal n \u2192 Nodup (map (fun x_1 => Fin.cons x.fst x_1) (antidiagonalTuple k x.snd))\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase succ.left\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\ni : \u2115 \u00d7 \u2115\na\u271d : i \u2208 antidiagonal n\n\u22a2 Nodup (map (fun x => Fin.cons i.fst x) (antidiagonalTuple k i.snd))\n[PROOFSTEP]\nexact (ih i.snd).map (Fin.cons_right_injective (\u03b1 := fun _ => \u2115) i.fst)\n[GOAL]\ncase succ.right\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\n\u22a2 Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\n[PROOFSTEP]\ninduction' n with n n_ih\n[GOAL]\ncase succ.right.zero\nn k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\n\u22a2 Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal Nat.zero)\n[PROOFSTEP]\nexact List.pairwise_singleton _ _\n[GOAL]\ncase succ.right.succ\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal (Nat.succ n))\n[PROOFSTEP]\nrw [List.Nat.antidiagonal_succ]\n[GOAL]\ncase succ.right.succ\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    ((0, n + 1) :: map (Prod.map Nat.succ id) (antidiagonal n))\n[PROOFSTEP]\nrefine' List.Pairwise.cons (fun a ha x hx\u2081 hx\u2082 => _) (n_ih.map _ fun a b h x hx\u2081 hx\u2082 => _)\n[GOAL]\ncase succ.right.succ.refine'_1\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na : \u2115 \u00d7 \u2115\nha : a \u2208 map (Prod.map Nat.succ id) (antidiagonal n)\nx : Fin (Nat.succ k) \u2192 \u2115\nhx\u2081 : x \u2208 map (fun x => Fin.cons (0, n + 1).fst x) (antidiagonalTuple k (0, n + 1).snd)\nhx\u2082 : x \u2208 map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd)\n\u22a2 False\n[PROOFSTEP]\nrw [List.mem_map] at hx\u2081 hx\u2082 ha \n[GOAL]\ncase succ.right.succ.refine'_1\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na : \u2115 \u00d7 \u2115\nha : \u2203 a_1, a_1 \u2208 antidiagonal n \u2227 Prod.map Nat.succ id a_1 = a\nx : Fin (Nat.succ k) \u2192 \u2115\nhx\u2081 : \u2203 a, a \u2208 antidiagonalTuple k (0, n + 1).snd \u2227 Fin.cons (0, n + 1).fst a = x\nhx\u2082 : \u2203 a_1, a_1 \u2208 antidiagonalTuple k a.snd \u2227 Fin.cons a.fst a_1 = x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8a, -, rfl\u27e9, \u27e8x\u2081, -, rfl\u27e9, \u27e8x\u2082, -, h\u27e9\u27e9 := ha, hx\u2081, hx\u2082\n[GOAL]\ncase succ.right.succ.refine'_1.intro.intro.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na : \u2115 \u00d7 \u2115\nx\u2081 x\u2082 : Fin k \u2192 \u2115\nh : Fin.cons (Prod.map Nat.succ id a).fst x\u2082 = Fin.cons (0, n + 1).fst x\u2081\n\u22a2 False\n[PROOFSTEP]\nrw [Fin.cons_eq_cons] at h \n[GOAL]\ncase succ.right.succ.refine'_1.intro.intro.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na : \u2115 \u00d7 \u2115\nx\u2081 x\u2082 : Fin k \u2192 \u2115\nh : (Prod.map Nat.succ id a).fst = (0, n + 1).fst \u2227 x\u2082 = x\u2081\n\u22a2 False\n[PROOFSTEP]\ninjection h.1\n[GOAL]\ncase succ.right.succ.refine'_2\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd))\nx : Fin (Nat.succ k) \u2192 \u2115\nhx\u2081 : x \u2208 map (fun x => Fin.cons (Prod.map Nat.succ id a).fst x) (antidiagonalTuple k (Prod.map Nat.succ id a).snd)\nhx\u2082 : x \u2208 map (fun x => Fin.cons (Prod.map Nat.succ id b).fst x) (antidiagonalTuple k (Prod.map Nat.succ id b).snd)\n\u22a2 False\n[PROOFSTEP]\nrw [List.mem_map] at hx\u2081 hx\u2082 \n[GOAL]\ncase succ.right.succ.refine'_2\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd))\nx : Fin (Nat.succ k) \u2192 \u2115\nhx\u2081 : \u2203 a_1, a_1 \u2208 antidiagonalTuple k (Prod.map Nat.succ id a).snd \u2227 Fin.cons (Prod.map Nat.succ id a).fst a_1 = x\nhx\u2082 : \u2203 a, a \u2208 antidiagonalTuple k (Prod.map Nat.succ id b).snd \u2227 Fin.cons (Prod.map Nat.succ id b).fst a = x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8x\u2081, hx\u2081, rfl\u27e9, \u27e8x\u2082, hx\u2082, h\u2081\u2082\u27e9\u27e9 := hx\u2081, hx\u2082\n[GOAL]\ncase succ.right.succ.refine'_2.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd))\nx\u2081 : Fin k \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 antidiagonalTuple k (Prod.map Nat.succ id a).snd\nx\u2082 : Fin k \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id b).snd\nh\u2081\u2082 : Fin.cons (Prod.map Nat.succ id b).fst x\u2082 = Fin.cons (Prod.map Nat.succ id a).fst x\u2081\n\u22a2 False\n[PROOFSTEP]\ndsimp at h\u2081\u2082 \n[GOAL]\ncase succ.right.succ.refine'_2.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd))\nx\u2081 : Fin k \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 antidiagonalTuple k (Prod.map Nat.succ id a).snd\nx\u2082 : Fin k \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id b).snd\nh\u2081\u2082 : Fin.cons (Nat.succ b.fst) x\u2082 = Fin.cons (Nat.succ a.fst) x\u2081\n\u22a2 False\n[PROOFSTEP]\nrw [Fin.cons_eq_cons, Nat.succ_inj'] at h\u2081\u2082 \n[GOAL]\ncase succ.right.succ.refine'_2.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd))\nx\u2081 : Fin k \u2192 \u2115\nhx\u2081 : x\u2081 \u2208 antidiagonalTuple k (Prod.map Nat.succ id a).snd\nx\u2082 : Fin k \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id b).snd\nh\u2081\u2082 : b.fst = a.fst \u2227 x\u2082 = x\u2081\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8h\u2081\u2082, rfl\u27e9 := h\u2081\u2082\n[GOAL]\ncase succ.right.succ.refine'_2.intro.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd))\nx\u2082 : Fin k \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id b).snd\nh\u2081\u2082 : b.fst = a.fst\nhx\u2081 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id a).snd\n\u22a2 False\n[PROOFSTEP]\nrw [h\u2081\u2082] at h \n[GOAL]\ncase succ.right.succ.refine'_2.intro.intro.intro.intro.intro\nn\u271d k : \u2115\nih : \u2200 (n : \u2115), Nodup (antidiagonalTuple k n)\nn : \u2115\nn_ih :\n  Pairwise\n    (fun a b =>\n      Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n        (map (fun x => Fin.cons b.fst x) (antidiagonalTuple k b.snd)))\n    (antidiagonal n)\na b : \u2115 \u00d7 \u2115\nh :\n  Disjoint (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k a.snd))\n    (map (fun x => Fin.cons a.fst x) (antidiagonalTuple k b.snd))\nx\u2082 : Fin k \u2192 \u2115\nhx\u2082 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id b).snd\nh\u2081\u2082 : b.fst = a.fst\nhx\u2081 : x\u2082 \u2208 antidiagonalTuple k (Prod.map Nat.succ id a).snd\n\u22a2 False\n[PROOFSTEP]\nexact h (List.mem_map_of_mem _ hx\u2081) (List.mem_map_of_mem _ hx\u2082)\n[GOAL]\nk : \u2115\n\u22a2 antidiagonalTuple (k + 1) 0 = [0]\n[PROOFSTEP]\nrw [antidiagonalTuple, antidiagonal_zero, List.bind_singleton, antidiagonalTuple_zero_right k, List.map_singleton]\n[GOAL]\nk : \u2115\n\u22a2 [Fin.cons (0, 0).fst 0] = [0]\n[PROOFSTEP]\nexact congr_arg (fun x => [x]) Matrix.cons_zero_zero\n[GOAL]\nn : \u2115\n\u22a2 antidiagonalTuple 1 n = [![n]]\n[PROOFSTEP]\nsimp_rw [antidiagonalTuple, antidiagonal, List.range_succ, List.map_append, List.map_singleton, tsub_self,\n  List.append_bind, List.bind_singleton, List.map_bind]\n[GOAL]\nn : \u2115\n\u22a2 (List.bind (range n) fun a => map (fun x => Fin.cons a x) (antidiagonalTuple 0 (n - a))) ++\n      map (fun x => Fin.cons n x) (antidiagonalTuple 0 0) =\n    [![n]]\n[PROOFSTEP]\nconv_rhs => rw [\u2190 List.nil_append [![n]]]\n[GOAL]\nn : \u2115\n| [![n]]\n[PROOFSTEP]\nrw [\u2190 List.nil_append [![n]]]\n[GOAL]\nn : \u2115\n| [![n]]\n[PROOFSTEP]\nrw [\u2190 List.nil_append [![n]]]\n[GOAL]\nn : \u2115\n| [![n]]\n[PROOFSTEP]\nrw [\u2190 List.nil_append [![n]]]\n[GOAL]\nn : \u2115\n\u22a2 (List.bind (range n) fun a => map (fun x => Fin.cons a x) (antidiagonalTuple 0 (n - a))) ++\n      map (fun x => Fin.cons n x) (antidiagonalTuple 0 0) =\n    [] ++ [![n]]\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nn : \u2115\n\u22a2 (List.bind (range n) fun a => map (fun x => Fin.cons a x) (antidiagonalTuple 0 (n - a))) = []\n[PROOFSTEP]\nsimp_rw [List.bind_eq_nil, List.mem_range, List.map_eq_nil]\n[GOAL]\ncase e_a\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x < n \u2192 antidiagonalTuple 0 (n - x) = []\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase e_a\nn x : \u2115\nhx : x < n\n\u22a2 antidiagonalTuple 0 (n - x) = []\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := Nat.exists_eq_add_of_lt hx\n[GOAL]\ncase e_a.intro\nx m : \u2115\nhx : x < x + m + 1\n\u22a2 antidiagonalTuple 0 (x + m + 1 - x) = []\n[PROOFSTEP]\nrw [add_assoc, add_tsub_cancel_left, antidiagonalTuple_zero_succ]\n[GOAL]\nn : \u2115\n\u22a2 antidiagonalTuple 2 n = map (fun i => ![i.fst, i.snd]) (antidiagonal n)\n[PROOFSTEP]\nrw [antidiagonalTuple]\n[GOAL]\nn : \u2115\n\u22a2 (List.bind (antidiagonal n) fun ni => map (fun x => Fin.cons ni.fst x) (antidiagonalTuple 1 ni.snd)) =\n    map (fun i => ![i.fst, i.snd]) (antidiagonal n)\n[PROOFSTEP]\nsimp_rw [antidiagonalTuple_one, List.map_singleton]\n[GOAL]\nn : \u2115\n\u22a2 (List.bind (antidiagonal n) fun ni => [Fin.cons ni.fst ![ni.snd]]) = map (fun i => ![i.fst, i.snd]) (antidiagonal n)\n[PROOFSTEP]\nrw [List.map_eq_bind]\n[GOAL]\nn : \u2115\n\u22a2 (List.bind (antidiagonal n) fun ni => [Fin.cons ni.fst ![ni.snd]]) =\n    List.bind (antidiagonal n) fun x => [![x.fst, x.snd]]\n[PROOFSTEP]\nrfl\n[GOAL]\nk n : \u2115\n\u22a2 Pairwise (Pi.Lex (fun x x_1 => x < x_1) fun x x x_1 => x < x_1) (antidiagonalTuple (k + 1) n)\n[PROOFSTEP]\nsimp_rw [antidiagonalTuple, List.pairwise_bind, List.pairwise_map, List.mem_map, forall_exists_index, and_imp,\n  forall_apply_eq_imp_iff\u2082]\n[GOAL]\nk n : \u2115\n\u22a2 (\u2200 (a : \u2115 \u00d7 \u2115),\n      a \u2208 antidiagonal n \u2192\n        Pairwise\n          (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a.fst a_2) (Fin.cons a.fst b))\n          (antidiagonalTuple (Nat.add k 0) a.snd)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n                Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a\u2081.fst a) (Fin.cons a\u2082.fst a_2))\n      (antidiagonal n)\n[PROOFSTEP]\nsimp only [mem_antidiagonal, Prod.forall, and_imp, forall_apply_eq_imp_iff\u2082]\n[GOAL]\nk n : \u2115\n\u22a2 (\u2200 (a b : \u2115),\n      a + b = n \u2192\n        Pairwise (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a a_2) (Fin.cons a b))\n          (antidiagonalTuple (Nat.add k 0) b)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n                Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a\u2081.fst a) (Fin.cons a\u2082.fst a_2))\n      (antidiagonal n)\n[PROOFSTEP]\nsimp only [Fin.pi_lex_lt_cons_cons, eq_self_iff_true, true_and_iff, lt_self_iff_false, false_or_iff]\n[GOAL]\nk n : \u2115\n\u22a2 (\u2200 (a b : \u2115),\n      a + b = n \u2192\n        Pairwise (fun a b => Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a b)\n          (antidiagonalTuple (Nat.add k 0) b)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n                a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n      (antidiagonal n)\n[PROOFSTEP]\nrefine' \u27e8fun _ _ _ => antidiagonalTuple_pairwise_pi_lex k _, _\u27e9\n[GOAL]\nk n : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n[PROOFSTEP]\ninduction' n with n n_ih\n[GOAL]\ncase zero\nk : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal Nat.zero)\n[PROOFSTEP]\nrw [antidiagonal_zero]\n[GOAL]\ncase zero\nk : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    [(0, 0)]\n[PROOFSTEP]\nexact List.pairwise_singleton _ _\n[GOAL]\ncase succ\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal (Nat.succ n))\n[PROOFSTEP]\nrw [antidiagonal_succ, List.pairwise_cons, List.pairwise_map]\n[GOAL]\ncase succ\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 (\u2200 (a' : \u2115 \u00d7 \u2115),\n      a' \u2208 map (Prod.map Nat.succ id) (antidiagonal n) \u2192\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).snd \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a'.snd \u2192\n                (0, n + 1).fst < a'.fst \u2228\n                  (0, n + 1).fst = a'.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2) \u2227\n    Pairwise\n      (fun a b =>\n        \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115),\n          a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).snd \u2192\n            \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).snd \u2192\n                (Prod.map Nat.succ id a).fst < (Prod.map Nat.succ id b).fst \u2228\n                  (Prod.map Nat.succ id a).fst = (Prod.map Nat.succ id b).fst \u2227\n                    Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n      (antidiagonal n)\n[PROOFSTEP]\nrefine' \u27e8fun p hp x hx y hy => _, _\u27e9\n[GOAL]\ncase succ.refine'_1\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\np : \u2115 \u00d7 \u2115\nhp : p \u2208 map (Prod.map Nat.succ id) (antidiagonal n)\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).snd\ny : Fin (Nat.add k 0) \u2192 \u2115\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) p.snd\n\u22a2 (0, n + 1).fst < p.fst \u2228 (0, n + 1).fst = p.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y\n[PROOFSTEP]\nrw [List.mem_map, Prod.exists] at hp \n[GOAL]\ncase succ.refine'_1\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\np : \u2115 \u00d7 \u2115\nhp : \u2203 a b, (a, b) \u2208 antidiagonal n \u2227 Prod.map Nat.succ id (a, b) = p\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).snd\ny : Fin (Nat.add k 0) \u2192 \u2115\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) p.snd\n\u22a2 (0, n + 1).fst < p.fst \u2228 (0, n + 1).fst = p.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y\n[PROOFSTEP]\nobtain \u27e8a, b, _, rfl : (Nat.succ a, b) = p\u27e9 := hp\n[GOAL]\ncase succ.refine'_1.intro.intro.intro\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).snd\ny : Fin (Nat.add k 0) \u2192 \u2115\na b : \u2115\nleft\u271d : (a, b) \u2208 antidiagonal n\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) (Nat.succ a, b).snd\n\u22a2 (0, n + 1).fst < (Nat.succ a, b).fst \u2228\n    (0, n + 1).fst = (Nat.succ a, b).fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y\n[PROOFSTEP]\nexact Or.inl (Nat.zero_lt_succ _)\n[GOAL]\ncase succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).snd \u2192\n          \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).snd \u2192\n              (Prod.map Nat.succ id a).fst < (Prod.map Nat.succ id b).fst \u2228\n                (Prod.map Nat.succ id a).fst = (Prod.map Nat.succ id b).fst \u2227\n                  Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin k \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple k a.snd \u2192\n          \u2200 (a_3 : Fin k \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple k b.snd \u2192\n              Nat.succ a.fst < Nat.succ b.fst \u2228\n                Nat.succ a.fst = Nat.succ b.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)\n[PROOFSTEP]\nsimp_rw [Nat.succ_inj', Nat.succ_lt_succ_iff]\n[GOAL]\ncase succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.snd \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.snd \u2192\n              a\u2081.fst < a\u2082.fst \u2228 a\u2081.fst = a\u2082.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin k \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple k a.snd \u2192\n          \u2200 (a_3 : Fin k \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple k b.snd \u2192\n              a.fst < b.fst \u2228 a.fst = b.fst \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)\n[PROOFSTEP]\nexact n_ih\n", "meta": {"mathlib_filename": "Mathlib.Data.Fin.Tuple.NatAntidiagonal", "llama_tokens": 13535, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929104825007, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.5530299025446395}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2078 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u2077 : Category.{?u.28793, u_2} C\ninst\u271d\u2076 : Category.{?u.28797, u_3} D\ninst\u271d\u2075 : Preadditive C\ninst\u271d\u2074 : Linear R C\ninst\u271d\u00b3 : Preadditive D\ninst\u271d\u00b2 : Linear R D\ne : C \u224c D\ninst\u271d\u00b9 : Functor.Additive e.functor\ninst\u271d : Functor.Linear R e.functor\nX\u271d Y\u271d : D\nr : X\u271d \u27f6 Y\u271d\nf : R\n\u22a2 e.inverse.map (f \u2022 r) = f \u2022 e.inverse.map r\n[PROOFSTEP]\napply e.functor.map_injective\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2078 : Semiring R\nC : Type u_2\nD : Type u_3\ninst\u271d\u2077 : Category.{?u.28793, u_2} C\ninst\u271d\u2076 : Category.{?u.28797, u_3} D\ninst\u271d\u2075 : Preadditive C\ninst\u271d\u2074 : Linear R C\ninst\u271d\u00b3 : Preadditive D\ninst\u271d\u00b2 : Linear R D\ne : C \u224c D\ninst\u271d\u00b9 : Functor.Additive e.functor\ninst\u271d : Functor.Linear R e.functor\nX\u271d Y\u271d : D\nr : X\u271d \u27f6 Y\u271d\nf : R\n\u22a2 e.functor.map (e.inverse.map (f \u2022 r)) = e.functor.map (f \u2022 e.inverse.map r)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Linear.LinearFunctor", "llama_tokens": 483, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339596505965, "lm_q2_score": 0.6584175072643415, "lm_q1q2_score": 0.5528955404783609}}
{"text": "[GOAL]\n\u03b1 \u03b2 : LatCat\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 : LatCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget LatCat).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 : LatCat\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 : LatCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget LatCat).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.LatCat", "llama_tokens": 1341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339756938818, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.5528955397846407}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 1 \u2264 a\nh : m \u2264 n\n\u22a2 a ^ m \u2264 a ^ n\n[PROOFSTEP]\nhave ha\u2080 : 0 < a := one_pos.trans_le ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 1 \u2264 a\nh : m \u2264 n\nha\u2080 : 0 < a\n\u22a2 a ^ m \u2264 a ^ n\n[PROOFSTEP]\nlift n - m to \u2115 using sub_nonneg.2 h with k hk\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 1 \u2264 a\nh : m \u2264 n\nha\u2080 : 0 < a\nk : \u2115\nhk : \u2191k = n - m\n\u22a2 a ^ m \u2264 a ^ n\n[PROOFSTEP]\ncalc\n  a ^ m = a ^ m * 1 := (mul_one _).symm\n  _ \u2264 a ^ m * a ^ k := (mul_le_mul_of_nonneg_left (one_le_pow_of_one_le ha _) (zpow_nonneg ha\u2080.le _))\n  _ = a ^ n := by rw [\u2190 zpow_ofNat, \u2190 zpow_add\u2080 ha\u2080.ne', hk, add_sub_cancel'_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 1 \u2264 a\nh : m \u2264 n\nha\u2080 : 0 < a\nk : \u2115\nhk : \u2191k = n - m\n\u22a2 a ^ m * a ^ k = a ^ n\n[PROOFSTEP]\nrw [\u2190 zpow_ofNat, \u2190 zpow_add\u2080 ha\u2080.ne', hk, add_sub_cancel'_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n\u271d : \u2124\na : \u2115\nh : 0 < a\nn : \u2124\n\u22a2 0 < \u2191a ^ n\n[PROOFSTEP]\napply zpow_pos_of_pos\n[GOAL]\ncase ha\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n\u271d : \u2124\na : \u2115\nh : 0 < a\nn : \u2124\n\u22a2 0 < \u2191a\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh\u2080 : 0 < a\nh\u2081 : a \u2260 1\n\u22a2 Injective ((fun x x_1 => x ^ x_1) a)\n[PROOFSTEP]\nrcases h\u2081.lt_or_lt with (H | H)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh\u2080 : 0 < a\nh\u2081 : a \u2260 1\nH : a < 1\n\u22a2 Injective ((fun x x_1 => x ^ x_1) a)\n[PROOFSTEP]\nexact (zpow_strictAnti h\u2080 H).injective\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh\u2080 : 0 < a\nh\u2081 : a \u2260 1\nH : 1 < a\n\u22a2 Injective ((fun x x_1 => x ^ x_1) a)\n[PROOFSTEP]\nexact (zpow_strictMono H).injective\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b\u271d c\u271d d e : \u03b1\nm n : \u2124\nx : \u03b1\nhx : 1 < x\na b c : \u2124\n\u22a2 x ^ (-c) \u2264 max (x ^ (-a)) (x ^ (-b)) \u2194 min a b \u2264 c\n[PROOFSTEP]\nsimp_rw [le_max_iff, min_le_iff, zpow_le_iff_le hx, neg_le_neg_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\na : \u03b1\n\u22a2 0 \u2264 a ^ 2\n[PROOFSTEP]\nconvert zpow_bit0_nonneg a 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : a \u2260 0\n\u22a2 0 < a ^ 2\n[PROOFSTEP]\nconvert zpow_bit0_pos h 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhn : n \u2260 0\n\u22a2 0 < a ^ bit0 n \u2192 a \u2260 0\n[PROOFSTEP]\nrintro h rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\nhn : n \u2260 0\nh : 0 < 0 ^ bit0 n\n\u22a2 False\n[PROOFSTEP]\nrefine' (zero_zpow _ _).not_gt h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nb c d : \u03b1\nn : \u2124\nhn : n \u2260 0\nh : 0 < 0 ^ bit0 n\n\u22a2 bit0 n \u2260 0\n[PROOFSTEP]\nrwa [bit0_ne_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : a < 0\n\u22a2 a ^ bit1 n < 0\n[PROOFSTEP]\nrw [bit1, zpow_add_one\u2080 h.ne]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nh : a < 0\n\u22a2 a ^ bit0 n * a < 0\n[PROOFSTEP]\nexact mul_neg_of_pos_of_neg (zpow_bit0_pos h.ne _) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 a ^ bit1 n \u2264 0 \u2194 a \u2264 0\n[PROOFSTEP]\nrw [le_iff_lt_or_eq, le_iff_lt_or_eq, zpow_bit1_neg_iff, zpow_eq_zero_iff (Int.bit1_ne_zero n)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\nhn : Even n\na : \u03b1\n\u22a2 0 \u2264 a ^ n\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := hn\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d a : \u03b1\nk : \u2124\n\u22a2 0 \u2264 a ^ (k + k)\n[PROOFSTEP]\nexact zpow_bit0_nonneg _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhn : Even n\nh : n \u2260 0\n\u22a2 0 < a ^ n \u2194 a \u2260 0\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := hn\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nk : \u2124\nh : k + k \u2260 0\n\u22a2 0 < a ^ (k + k) \u2194 a \u2260 0\n[PROOFSTEP]\nexact zpow_bit0_pos_iff (by rintro rfl; simp at h )\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nk : \u2124\nh : k + k \u2260 0\n\u22a2 k \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nh : 0 + 0 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhn : Odd n\n\u22a2 a ^ n < 0 \u2194 a < 0\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn k : \u2124\nhk : n = 2 * k + 1\n\u22a2 a ^ n < 0 \u2194 a < 0\n[PROOFSTEP]\nsimpa only [hk, two_mul] using zpow_bit1_neg_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhn : Odd n\n\u22a2 0 \u2264 a ^ n \u2194 0 \u2264 a\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn k : \u2124\nhk : n = 2 * k + 1\n\u22a2 0 \u2264 a ^ n \u2194 0 \u2264 a\n[PROOFSTEP]\nsimpa only [hk, two_mul] using zpow_bit1_nonneg_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhn : Odd n\n\u22a2 a ^ n \u2264 0 \u2194 a \u2264 0\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn k : \u2124\nhk : n = 2 * k + 1\n\u22a2 a ^ n \u2264 0 \u2194 a \u2264 0\n[PROOFSTEP]\nsimpa only [hk, two_mul] using zpow_bit1_nonpos_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhn : Odd n\n\u22a2 0 < a ^ n \u2194 0 < a\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn k : \u2124\nhk : n = 2 * k + 1\n\u22a2 0 < a ^ n \u2194 0 < a\n[PROOFSTEP]\nsimpa only [hk, two_mul] using zpow_bit1_pos_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn p : \u2124\nhp : Even p\na : \u03b1\n\u22a2 |a| ^ p = a ^ p\n[PROOFSTEP]\ncases' abs_choice a with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn p : \u2124\nhp : Even p\na : \u03b1\nh : |a| = a\n\u22a2 |a| ^ p = a ^ p\n[PROOFSTEP]\nsimp only [h, hp.neg_zpow _]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn p : \u2124\nhp : Even p\na : \u03b1\nh : |a| = -a\n\u22a2 |a| ^ p = a ^ p\n[PROOFSTEP]\nsimp only [h, hp.neg_zpow _]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Field.Power", "llama_tokens": 3218, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396212, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.5528576726454961}}
{"text": "[GOAL]\n\u03b1 \u03b2 : Type u\nf : Type u \u2192 Type v\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\nh : \u03b1 \u2243 \u03b2\nx : f \u03b1\n\u22a2 \u2191h.symm <$> \u2191h <$> x = x\n[PROOFSTEP]\nsimp [map_map]\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : Type u \u2192 Type v\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\nh : \u03b1 \u2243 \u03b2\nx : f \u03b2\n\u22a2 \u2191h <$> \u2191h.symm <$> x = x\n[PROOFSTEP]\nsimp [map_map]\n[GOAL]\n\u03b1 \u03b2 : Type u\nf : Type u \u2192 Type v\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\n\u22a2 mapEquiv f (Equiv.refl \u03b1) = Equiv.refl (f \u03b1)\n[PROOFSTEP]\next x\n[GOAL]\ncase H\n\u03b1 \u03b2 : Type u\nf : Type u \u2192 Type v\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\nx : f \u03b1\n\u22a2 \u2191(mapEquiv f (Equiv.refl \u03b1)) x = \u2191(Equiv.refl (f \u03b1)) x\n[PROOFSTEP]\nsimp only [mapEquiv_apply, refl_apply]\n[GOAL]\ncase H\n\u03b1 \u03b2 : Type u\nf : Type u \u2192 Type v\ninst\u271d\u00b9 : Functor f\ninst\u271d : LawfulFunctor f\nx : f \u03b1\n\u22a2 \u2191(Equiv.refl \u03b1) <$> x = x\n[PROOFSTEP]\nexact LawfulFunctor.id_map x\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b1' \u03b2' : Type v\nF : Type u \u2192 Type v \u2192 Type w\ninst\u271d\u00b9 : Bifunctor F\ninst\u271d : LawfulBifunctor F\nh : \u03b1 \u2243 \u03b2\nh' : \u03b1' \u2243 \u03b2'\nx : F \u03b1 \u03b1'\n\u22a2 bimap (\u2191h.symm) (\u2191h'.symm) (bimap (\u2191h) (\u2191h') x) = x\n[PROOFSTEP]\nsimp [bimap_bimap, id_bimap]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b1' \u03b2' : Type v\nF : Type u \u2192 Type v \u2192 Type w\ninst\u271d\u00b9 : Bifunctor F\ninst\u271d : LawfulBifunctor F\nh : \u03b1 \u2243 \u03b2\nh' : \u03b1' \u2243 \u03b2'\nx : F \u03b2 \u03b2'\n\u22a2 bimap (\u2191h) (\u2191h') (bimap (\u2191h.symm) (\u2191h'.symm) x) = x\n[PROOFSTEP]\nsimp [bimap_bimap, id_bimap]\n[GOAL]\n\u03b1 \u03b2 : Type u\n\u03b1' \u03b2' : Type v\nF : Type u \u2192 Type v \u2192 Type w\ninst\u271d\u00b9 : Bifunctor F\ninst\u271d : LawfulBifunctor F\n\u22a2 mapEquiv F (Equiv.refl \u03b1) (Equiv.refl \u03b1') = Equiv.refl (F \u03b1 \u03b1')\n[PROOFSTEP]\next x\n[GOAL]\ncase H\n\u03b1 \u03b2 : Type u\n\u03b1' \u03b2' : Type v\nF : Type u \u2192 Type v \u2192 Type w\ninst\u271d\u00b9 : Bifunctor F\ninst\u271d : LawfulBifunctor F\nx : F \u03b1 \u03b1'\n\u22a2 \u2191(mapEquiv F (Equiv.refl \u03b1) (Equiv.refl \u03b1')) x = \u2191(Equiv.refl (F \u03b1 \u03b1')) x\n[PROOFSTEP]\nsimp [id_bimap]\n", "meta": {"mathlib_filename": "Mathlib.Logic.Equiv.Functor", "llama_tokens": 951, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396211, "lm_q2_score": 0.6926419831347361, "lm_q1q2_score": 0.552857672645496}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformInducing f \u2194 UniformContinuous f \u2227 comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nrw [uniformInducing_iff, UniformContinuous, tendsto_iff_comap, le_antisymm_iff, and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \ud835\udce4 \u03b1 \u2264 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2) \u2227 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1 \u2194\n    \ud835\udce4 \u03b1 \u2264 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2) \u2227 comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b9 : Sort u_1\n\u03b9' : Sort u_2\np : \u03b9 \u2192 Prop\np' : \u03b9' \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b1 \u00d7 \u03b1)\ns' : \u03b9' \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b1) p s\nh' : HasBasis (\ud835\udce4 \u03b2) p' s'\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformInducing f \u2194\n    (\u2200 (i : \u03b9'), p' i \u2192 \u2203 j, p j \u2227 \u2200 (x y : \u03b1), (x, y) \u2208 s j \u2192 (f x, f y) \u2208 s' i) \u2227\n      \u2200 (j : \u03b9), p j \u2192 \u2203 i, p' i \u2227 \u2200 (x y : \u03b1), (f x, f y) \u2208 s' i \u2192 (x, y) \u2208 s j\n[PROOFSTEP]\nsimp [uniformInducing_iff', h.uniformContinuous_iff h', (h'.comap _).le_basis_iff h, subset_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2194 \u2203 t, t \u2208 \ud835\udce4 \u03b2 \u2227 \u2200 (x y : \u03b1), (f x, f y) \u2208 t \u2192 (x, y) \u2208 s\n\u22a2 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2) = \ud835\udce4 \u03b1\n[PROOFSTEP]\nsimp [eq_comm, Filter.ext_iff, subset_def, h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u22a2 comap (fun x => (id x.fst, id x.snd)) (\ud835\udce4 \u03b1) = \ud835\udce4 \u03b1\n[PROOFSTEP]\nrw [\u2190 Prod.map_def, Prod.map_id, comap_id]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\nf : \u03b1 \u2192 \u03b2\nhf : UniformInducing f\n\u22a2 comap (fun x => ((g \u2218 f) x.fst, (g \u2218 f) x.snd)) (\ud835\udce4 \u03b3) = \ud835\udce4 \u03b1\n[PROOFSTEP]\nrw [\u2190 hf.1, \u2190 hg.1, comap_comap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\nf : \u03b1 \u2192 \u03b2\nhf : UniformInducing f\n\u22a2 comap (fun x => ((g \u2218 f) x.fst, (g \u2218 f) x.snd)) (\ud835\udce4 \u03b3) =\n    comap ((fun x => (g x.fst, g x.snd)) \u2218 fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b3)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : UniformInducing f\nF : Filter \u03b1\n\u22a2 Cauchy (map f F) \u2194 Cauchy F\n[PROOFSTEP]\nsimp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, \u2190 hf.comap_uniformity]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhf : UniformContinuous f\nhg : UniformContinuous g\nhgf : UniformInducing (g \u2218 f)\n\u22a2 UniformInducing f\n[PROOFSTEP]\nrefine' \u27e8le_antisymm _ hf.le_comap\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhf : UniformContinuous f\nhg : UniformContinuous g\nhgf : UniformInducing (g \u2218 f)\n\u22a2 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nrw [\u2190 hgf.1, \u2190 Prod.map_def, \u2190 Prod.map_def, \u2190 Prod.map_comp_map f f g g, \u2190 comap_comap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhf : UniformContinuous f\nhg : UniformContinuous g\nhgf : UniformInducing (g \u2218 f)\n\u22a2 comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 comap (Prod.map f f) (comap (Prod.map g g) (\ud835\udce4 \u03b3))\n[PROOFSTEP]\nexact comap_mono hg.le_comap\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\n\u22a2 UniformContinuous f \u2194 UniformContinuous (g \u2218 f)\n[PROOFSTEP]\ndsimp only [UniformContinuous, Tendsto]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\n\u22a2 map (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b1) \u2264 \ud835\udce4 \u03b2 \u2194 map (fun x => ((g \u2218 f) x.fst, (g \u2218 f) x.snd)) (\ud835\udce4 \u03b1) \u2264 \ud835\udce4 \u03b3\n[PROOFSTEP]\nrw [\u2190 hg.comap_uniformity, \u2190 map_le_iff_le_comap, Filter.map_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : UniformInducing g\n\u22a2 map ((fun x => (g x.fst, g x.snd)) \u2218 fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b1) \u2264 \ud835\udce4 \u03b3 \u2194\n    map (fun x => ((g \u2218 f) x.fst, (g \u2218 f) x.snd)) (\ud835\udce4 \u03b1) \u2264 \ud835\udce4 \u03b3\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nh : UniformInducing f\n\u22a2 Inducing f\n[PROOFSTEP]\nobtain rfl := h.comap_uniformSpace\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nh : UniformInducing f\n\u22a2 Inducing f\n[PROOFSTEP]\nexact inducing_induced f\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1' : Type u_1\n\u03b2' : Type u_2\ninst\u271d\u00b9 : UniformSpace \u03b1'\ninst\u271d : UniformSpace \u03b2'\ne\u2081 : \u03b1 \u2192 \u03b1'\ne\u2082 : \u03b2 \u2192 \u03b2'\nh\u2081 : UniformInducing e\u2081\nh\u2082 : UniformInducing e\u2082\n\u22a2 comap (fun x => ((e\u2081 x.fst.fst, e\u2082 x.fst.snd), e\u2081 x.snd.fst, e\u2082 x.snd.snd)) (\ud835\udce4 (\u03b1' \u00d7 \u03b2')) = \ud835\udce4 (\u03b1 \u00d7 \u03b2)\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7), uniformity_prod, \u2190 h\u2081.1, \u2190 h\u2082.1, comap_inf, comap_comap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformEmbedding f \u2194 Injective f \u2227 UniformContinuous f \u2227 comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nrw [uniformEmbedding_iff, and_comm, uniformInducing_iff']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b9 : Sort u_1\n\u03b9' : Sort u_2\np : \u03b9 \u2192 Prop\np' : \u03b9' \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b1 \u00d7 \u03b1)\ns' : \u03b9' \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b1) p s\nh' : HasBasis (\ud835\udce4 \u03b2) p' s'\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformEmbedding f \u2194\n    Injective f \u2227\n      (\u2200 (i : \u03b9'), p' i \u2192 \u2203 j, p j \u2227 \u2200 (x y : \u03b1), (x, y) \u2208 s j \u2192 (f x, f y) \u2208 s' i) \u2227\n        \u2200 (j : \u03b9), p j \u2192 \u2203 i, p' i \u2227 \u2200 (x y : \u03b1), (f x, f y) \u2208 s' i \u2192 (x, y) \u2208 s j\n[PROOFSTEP]\nrw [uniformEmbedding_iff, and_comm, h.uniformInducing_iff h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b9 : Sort u_1\n\u03b9' : Sort u_2\np : \u03b9 \u2192 Prop\np' : \u03b9' \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b1 \u00d7 \u03b1)\ns' : \u03b9' \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b1) p s\nh' : HasBasis (\ud835\udce4 \u03b2) p' s'\nf : \u03b1 \u2192 \u03b2\n\u22a2 UniformEmbedding f \u2194\n    Injective f \u2227 UniformContinuous f \u2227 \u2200 (j : \u03b9), p j \u2192 \u2203 i, p' i \u2227 \u2200 (x y : \u03b1), (f x, f y) \u2208 s' i \u2192 (x, y) \u2208 s j\n[PROOFSTEP]\nsimp only [h.uniformEmbedding_iff' h', h.uniformContinuous_iff h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns t : Set \u03b1\nhst : s \u2286 t\n\u22a2 comap (fun x => (inclusion hst x.fst, inclusion hst x.snd)) (\ud835\udce4 \u2191t) = \ud835\udce4 \u2191s\n[PROOFSTEP]\nrw [uniformity_subtype, uniformity_subtype, comap_comap]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns t : Set \u03b1\nhst : s \u2286 t\n\u22a2 comap ((fun q => (\u2191q.fst, \u2191q.snd)) \u2218 fun x => (inclusion hst x.fst, inclusion hst x.snd)) (\ud835\udce4 \u03b1) =\n    comap (fun q => (\u2191q.fst, \u2191q.snd)) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\u271d\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2243 \u03b2\nh\u2081 : UniformContinuous \u2191f\nh\u2082 : UniformContinuous \u2191f.symm\n\u22a2 comap (Prod.map \u2191f \u2191f) (\ud835\udce4 \u03b2) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nrwa [\u2190 Equiv.prodCongr_apply, \u2190 map_equiv_symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nx : \u03b1 \u00d7 \u03b1\nh : x \u2208 Prod.map Sum.inl Sum.inl \u207b\u00b9' (Prod.map Sum.inl Sum.inl '' s \u222a range (Prod.map Sum.inr Sum.inr))\n\u22a2 x \u2208 s\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nx : \u03b2 \u00d7 \u03b2\nh : x \u2208 Prod.map Sum.inr Sum.inr \u207b\u00b9' (range (Prod.map Sum.inl Sum.inl) \u222a Prod.map Sum.inr Sum.inr '' s)\n\u22a2 x \u2208 s\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\n\u22a2 comap (Prod.map f f) (\ud835\udce4 \u03b2) = \ud835\udcdf idRel\n[PROOFSTEP]\nrefine' le_antisymm _ (@refl_le_uniformity \u03b1 (UniformSpace.comap f _))\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\n\u22a2 comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 \ud835\udcdf idRel\n[PROOFSTEP]\ncalc\n  comap (Prod.map f f) (\ud835\udce4 \u03b2) \u2264 comap (Prod.map f f) (\ud835\udcdf s) := comap_mono (le_principal_iff.2 hs)\n  _ = \ud835\udcdf (Prod.map f f \u207b\u00b9' s) := comap_principal\n  _ \u2264 \ud835\udcdf idRel := principal_mono.2 ?_\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\n\u22a2 Prod.map f f \u207b\u00b9' s \u2286 idRel\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\nx y : \u03b1\n\u22a2 (x, y) \u2208 Prod.map f f \u207b\u00b9' s \u2192 (x, y) \u2208 idRel\n[PROOFSTEP]\nsimpa [not_imp_not] using @hf x y\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\n\u22a2 UniformEmbedding f\n[PROOFSTEP]\nlet _ : UniformSpace \u03b1 := \u22a5\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\nx\u271d : UniformSpace \u03b1 := \u22a5\n\u22a2 UniformEmbedding f\n[PROOFSTEP]\nhave := discreteTopology_bot \u03b1\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\nx\u271d : UniformSpace \u03b1 := \u22a5\nthis : DiscreteTopology \u03b1\n\u22a2 UniformEmbedding f\n[PROOFSTEP]\nexact UniformInducing.uniformEmbedding \u27e8comap_uniformity_of_spaced_out hs hf\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2075 : UniformSpace \u03b1\u271d\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : DiscreteTopology \u03b1\ninst\u271d : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\n\u22a2 ClosedEmbedding f\n[PROOFSTEP]\nrcases@DiscreteTopology.eq_bot \u03b1 _ _ with rfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\ninst\u271d : DiscreteTopology \u03b1\n\u22a2 ClosedEmbedding f\n[PROOFSTEP]\nlet _ : UniformSpace \u03b1 := \u22a5\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\u271d\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SeparatedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b2 \u00d7 \u03b2)\nhs : s \u2208 \ud835\udce4 \u03b2\nhf : Pairwise fun x y => \u00ac(f x, f y) \u2208 s\ninst\u271d : DiscreteTopology \u03b1\nx\u271d : UniformSpace \u03b1 := \u22a5\n\u22a2 ClosedEmbedding f\n[PROOFSTEP]\nexact { (uniformEmbedding_of_spaced_out hs hf).embedding with closed_range := isClosed_range_of_spaced_out hs hf }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 a, closure (e '' {a' | (a, a') \u2208 s}) \u2208 \ud835\udcdd b\n[PROOFSTEP]\nobtain \u27e8U, \u27e8hU, hUo, hsymm\u27e9, hs\u27e9 : \u2203 U, (U \u2208 \ud835\udce4 \u03b2 \u2227 IsOpen U \u2227 SymmetricRel U) \u2227 Prod.map e e \u207b\u00b9' U \u2286 s := by\n  rwa [\u2190 he\u2081.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 U, (U \u2208 \ud835\udce4 \u03b2 \u2227 IsOpen U \u2227 SymmetricRel U) \u2227 Prod.map e e \u207b\u00b9' U \u2286 s\n[PROOFSTEP]\nrwa [\u2190 he\u2081.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\n\u22a2 \u2203 a, closure (e '' {a' | (a, a') \u2208 s}) \u2208 \ud835\udcdd b\n[PROOFSTEP]\nrcases he\u2082.dense.mem_nhds (UniformSpace.ball_mem_nhds b hU) with \u27e8a, ha\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : \u03b1\nha : e a \u2208 UniformSpace.ball b U\n\u22a2 \u2203 a, closure (e '' {a' | (a, a') \u2208 s}) \u2208 \ud835\udcdd b\n[PROOFSTEP]\nrefine \u27e8a, mem_of_superset ?_ (closure_mono <| image_subset _ <| ball_mono hs a)\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : \u03b1\nha : e a \u2208 UniformSpace.ball b U\n\u22a2 closure (e '' UniformSpace.ball a (Prod.map e e \u207b\u00b9' U)) \u2208 \ud835\udcdd b\n[PROOFSTEP]\nhave ho : IsOpen (UniformSpace.ball (e a) U) := UniformSpace.isOpen_ball (e a) hUo\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : \u03b1\nha : e a \u2208 UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\n\u22a2 closure (e '' UniformSpace.ball a (Prod.map e e \u207b\u00b9' U)) \u2208 \ud835\udcdd b\n[PROOFSTEP]\nrefine mem_of_superset (ho.mem_nhds <| (mem_ball_symmetry hsymm).2 ha) fun y hy => ?_\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : \u03b1\nha : e a \u2208 UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : \u03b2\nhy : y \u2208 UniformSpace.ball (e a) U\n\u22a2 y \u2208 closure (e '' UniformSpace.ball a (Prod.map e e \u207b\u00b9' U))\n[PROOFSTEP]\nrefine mem_closure_iff_nhds.2 fun V hV => ?_\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : \u03b1\nha : e a \u2208 UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : \u03b2\nhy : y \u2208 UniformSpace.ball (e a) U\nV : Set \u03b2\nhV : V \u2208 \ud835\udcdd y\n\u22a2 Set.Nonempty (V \u2229 e '' UniformSpace.ball a (Prod.map e e \u207b\u00b9' U))\n[PROOFSTEP]\nrcases he\u2082.dense.mem_nhds (inter_mem hV (ho.mem_nhds hy)) with \u27e8x, hxV, hxU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set (\u03b1 \u00d7 \u03b1)\ne : \u03b1 \u2192 \u03b2\nb : \u03b2\nhe\u2081 : UniformInducing e\nhe\u2082 : DenseInducing e\nhs\u271d : s \u2208 \ud835\udce4 \u03b1\nU : Set (\u03b2 \u00d7 \u03b2)\nhs : Prod.map e e \u207b\u00b9' U \u2286 s\nhU : U \u2208 \ud835\udce4 \u03b2\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : \u03b1\nha : e a \u2208 UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : \u03b2\nhy : y \u2208 UniformSpace.ball (e a) U\nV : Set \u03b2\nhV : V \u2208 \ud835\udcdd y\nx : \u03b1\nhxV : e x \u2208 V\nhxU : e x \u2208 UniformSpace.ball (e a) U\n\u22a2 Set.Nonempty (V \u2229 e '' UniformSpace.ball a (Prod.map e e \u207b\u00b9' U))\n[PROOFSTEP]\nexact \u27e8e x, hxV, mem_image_of_mem e hxU\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nue : UniformEmbedding e\nde : DenseEmbedding e\n\u22a2 comap (fun x => (DenseEmbedding.subtypeEmb p e x.fst, DenseEmbedding.subtypeEmb p e x.snd))\n      (\ud835\udce4 { x // x \u2208 closure (e '' {x | p x}) }) =\n    \ud835\udce4 { x // p x }\n[PROOFSTEP]\nsimp [comap_comap, (\u00b7 \u2218 \u00b7), DenseEmbedding.subtypeEmb, uniformity_subtype, ue.comap_uniformity.symm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\n\u22a2 IsComplete s\n[PROOFSTEP]\nintro f hf hfs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\nf : Filter \u03b1\nhf : Cauchy f\nhfs : f \u2264 \ud835\udcdf s\n\u22a2 \u2203 x, x \u2208 s \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrw [le_principal_iff] at hfs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\nf : Filter \u03b1\nhf : Cauchy f\nhfs : s \u2208 f\n\u22a2 \u2203 x, x \u2208 s \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nobtain \u27e8_, \u27e8x, hx, rfl\u27e9, hyf\u27e9 : \u2203 y \u2208 m '' s, map m f \u2264 \ud835\udcdd y\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\nf : Filter \u03b1\nhf : Cauchy f\nhfs : s \u2208 f\n\u22a2 \u2203 y, y \u2208 m '' s \u2227 map m f \u2264 \ud835\udcdd y\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\nf : Filter \u03b1\nhf : Cauchy f\nhfs : s \u2208 f\nx : \u03b1\nhx : x \u2208 s\nhyf : map m f \u2264 \ud835\udcdd (m x)\n\u22a2 \u2203 x, x \u2208 s \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact hs (f.map m) (hf.map hm.uniformContinuous) (le_principal_iff.2 (image_mem_map hfs))\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\nf : Filter \u03b1\nhf : Cauchy f\nhfs : s \u2208 f\nx : \u03b1\nhx : x \u2208 s\nhyf : map m f \u2264 \ud835\udcdd (m x)\n\u22a2 \u2203 x, x \u2208 s \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrw [map_le_iff_le_comap, \u2190 nhds_induced, \u2190 hm.inducing.induced] at hyf \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nhs : IsComplete (m '' s)\nf : Filter \u03b1\nhf : Cauchy f\nhfs : s \u2208 f\nx : \u03b1\nhx : x \u2208 s\nhyf : f \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 s \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact \u27e8x, hx, hyf\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set \u03b1\nhs : IsComplete s\n\u22a2 IsComplete (Subtype.val '' univ)\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\n\u22a2 IsComplete (m '' s) \u2194 IsComplete s\n[PROOFSTEP]\nrefine' \u27e8isComplete_of_complete_image hm, fun c => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\n\u22a2 IsComplete (m '' s)\n[PROOFSTEP]\nhaveI : CompleteSpace s := c.completeSpace_coe\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\n\u22a2 IsComplete (m '' s)\n[PROOFSTEP]\nset m' : s \u2192 \u03b2 := m \u2218 (\u2191)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\n\u22a2 IsComplete (m '' s)\n[PROOFSTEP]\nsuffices IsComplete (range m') by rwa [range_comp, Subtype.range_coe] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis\u271d : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nthis : IsComplete (range m')\n\u22a2 IsComplete (m '' s)\n[PROOFSTEP]\nrwa [range_comp, Subtype.range_coe] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\n\u22a2 IsComplete (range m')\n[PROOFSTEP]\nhave hm' : UniformInducing m' := hm.comp uniformEmbedding_subtype_val.toUniformInducing\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nhm' : UniformInducing m'\n\u22a2 IsComplete (range m')\n[PROOFSTEP]\nintro f hf hfm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nhm' : UniformInducing m'\nf : Filter \u03b2\nhf : Cauchy f\nhfm : f \u2264 \ud835\udcdf (range m')\n\u22a2 \u2203 x, x \u2208 range m' \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrw [Filter.le_principal_iff] at hfm \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nhm' : UniformInducing m'\nf : Filter \u03b2\nhf : Cauchy f\nhfm : range m' \u2208 f\n\u22a2 \u2203 x, x \u2208 range m' \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nhave cf' : Cauchy (comap m' f) := hf.comap' hm'.comap_uniformity.le (NeBot.comap_of_range_mem hf.1 hfm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nhm' : UniformInducing m'\nf : Filter \u03b2\nhf : Cauchy f\nhfm : range m' \u2208 f\ncf' : Cauchy (comap m' f)\n\u22a2 \u2203 x, x \u2208 range m' \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrcases CompleteSpace.complete cf' with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nhm' : UniformInducing m'\nf : Filter \u03b2\nhf : Cauchy f\nhfm : range m' \u2208 f\ncf' : Cauchy (comap m' f)\nx : \u2191s\nhx : comap m' f \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 range m' \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrw [hm'.inducing.nhds_eq_comap, comap_le_comap_iff hfm] at hx \n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhm : UniformInducing m\nc : IsComplete s\nthis : CompleteSpace \u2191s\nm' : \u2191s \u2192 \u03b2 := m \u2218 Subtype.val\nhm' : UniformInducing m'\nf : Filter \u03b2\nhf : Cauchy f\nhfm : range m' \u2208 f\ncf' : Cauchy (comap m' f)\nx : \u2191s\nhx : f \u2264 \ud835\udcdd (m' x)\n\u22a2 \u2203 x, x \u2208 range m' \u2227 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact \u27e8m' x, mem_range_self _, hx\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : UniformInducing f\n\u22a2 CompleteSpace \u03b1 \u2194 IsComplete (range f)\n[PROOFSTEP]\nrw [completeSpace_iff_isComplete_univ, \u2190 isComplete_image_iff hf, image_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ne : \u03b1 \u2243 \u03b2\nhe : UniformEmbedding \u2191e\n\u22a2 CompleteSpace \u03b1 \u2194 CompleteSpace \u03b2\n[PROOFSTEP]\nrw [completeSpace_iff_isComplete_range he.toUniformInducing, e.range_eq_univ, completeSpace_iff_isComplete_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\ns : Set \u03b1\n\u22a2 IsComplete (range Subtype.val) \u2194 IsComplete s\n[PROOFSTEP]\nrw [Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nm : \u03b2 \u2192 \u03b1\nhm : UniformInducing m\ndense : DenseRange m\nh : \u2200 (f : Filter \u03b2), Cauchy f \u2192 \u2203 x, map m f \u2264 \ud835\udcdd x\nf : Filter \u03b1\nhf : Cauchy f\np : Set (\u03b1 \u00d7 \u03b1) \u2192 Set \u03b1 \u2192 Set \u03b1 := fun s t => {y | \u2203 x, x \u2208 t \u2227 (x, y) \u2208 s}\ng : Filter \u03b1 := Filter.lift (\ud835\udce4 \u03b1) fun s => Filter.lift' f (p s)\nmp\u2080 : Monotone p\nmp\u2081 : \u2200 {s : Set (\u03b1 \u00d7 \u03b1)}, Monotone (p s)\nthis\u271d\u2075 : f \u2264 g\nthis\u271d\u2074 : NeBot g\nthis\u271d\u00b3 : NeBot (comap m g)\nthis\u271d\u00b2 : Cauchy g\nthis\u271d\u00b9 : Cauchy (comap m g)\nx : \u03b1\nhx : map m (comap m g) \u2264 \ud835\udcdd x\nthis\u271d : ClusterPt x (map m (comap m g))\nthis : ClusterPt x g\n\u22a2 f \u2264 g\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 t_1, Set.Finite t_1 \u2227 f \u207b\u00b9' s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t_1), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nrw [\u2190 hf.comap_uniformity] at ht \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\n\u22a2 \u2203 t_1, Set.Finite t_1 \u2227 f \u207b\u00b9' s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t_1), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nrcases mem_comap.2 ht with \u27e8t', ht', ts\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\n\u22a2 \u2203 t_1, Set.Finite t_1 \u2227 f \u207b\u00b9' s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t_1), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nrcases totallyBounded_iff_subset.1 (totallyBounded_subset (image_preimage_subset f s) hs) _ ht' with \u27e8c, cs, hfc, hct\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\nc : Set \u03b2\ncs : c \u2286 f '' (f \u207b\u00b9' s)\nhfc : Set.Finite c\nhct : f '' (f \u207b\u00b9' s) \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\n\u22a2 \u2203 t_1, Set.Finite t_1 \u2227 f \u207b\u00b9' s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t_1), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nrefine' \u27e8f \u207b\u00b9' c, hfc.preimage (hf.inj.injOn _), fun x h => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\nc : Set \u03b2\ncs : c \u2286 f '' (f \u207b\u00b9' s)\nhfc : Set.Finite c\nhct : f '' (f \u207b\u00b9' s) \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\nx : \u03b1\nh : x \u2208 f \u207b\u00b9' s\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 f \u207b\u00b9' c), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nhave := hct (mem_image_of_mem f h)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\nc : Set \u03b2\ncs : c \u2286 f '' (f \u207b\u00b9' s)\nhfc : Set.Finite c\nhct : f '' (f \u207b\u00b9' s) \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\nx : \u03b1\nh : x \u2208 f \u207b\u00b9' s\nthis : f x \u2208 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\n\u22a2 x \u2208 \u22c3 (y : \u03b1) (_ : y \u2208 f \u207b\u00b9' c), {x | (x, y) \u2208 t}\n[PROOFSTEP]\nsimp at this \u22a2\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\nc : Set \u03b2\ncs : c \u2286 f '' (f \u207b\u00b9' s)\nhfc : Set.Finite c\nhct : f '' (f \u207b\u00b9' s) \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\nx : \u03b1\nh : x \u2208 f \u207b\u00b9' s\nthis : \u2203 i, i \u2208 c \u2227 (f x, i) \u2208 t'\n\u22a2 \u2203 i, f i \u2208 c \u2227 (x, i) \u2208 t\n[PROOFSTEP]\nrcases this with \u27e8z, zc, zt\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\nc : Set \u03b2\ncs : c \u2286 f '' (f \u207b\u00b9' s)\nhfc : Set.Finite c\nhct : f '' (f \u207b\u00b9' s) \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\nx : \u03b1\nh : x \u2208 f \u207b\u00b9' s\nz : \u03b2\nzc : z \u2208 c\nzt : (f x, z) \u2208 t'\n\u22a2 \u2203 i, f i \u2208 c \u2227 (x, i) \u2208 t\n[PROOFSTEP]\nrcases cs zc with \u27e8y, -, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhf : UniformEmbedding f\nhs : TotallyBounded s\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 comap (fun x => (f x.fst, f x.snd)) (\ud835\udce4 \u03b2)\nt' : Set (\u03b2 \u00d7 \u03b2)\nht' : t' \u2208 \ud835\udce4 \u03b2\nts : (fun x => (f x.fst, f x.snd)) \u207b\u00b9' t' \u2286 t\nc : Set \u03b2\ncs : c \u2286 f '' (f \u207b\u00b9' s)\nhfc : Set.Finite c\nhct : f '' (f \u207b\u00b9' s) \u2286 \u22c3 (y : \u03b2) (_ : y \u2208 c), {x | (x, y) \u2208 t'}\nx : \u03b1\nh : x \u2208 f \u207b\u00b9' s\ny : \u03b1\nzc : f y \u2208 c\nzt : (f x, f y) \u2208 t'\n\u22a2 \u2203 i, f i \u2208 c \u2227 (x, i) \u2208 t\n[PROOFSTEP]\nexact \u27e8y, zc, ts zt\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : CompleteSpace \u03b2\n\u22a2 CompleteSpace (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nrw [completeSpace_iff_isComplete_univ, \u2190 range_inl_union_range_inr]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : CompleteSpace \u03b2\n\u22a2 IsComplete (range Sum.inl \u222a range Sum.inr)\n[PROOFSTEP]\nexact\n  uniformEmbedding_inl.toUniformInducing.isComplete_range.union uniformEmbedding_inr.toUniformInducing.isComplete_range\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave de : DenseEmbedding e := he.denseEmbedding hd\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave de' : DenseEmbedding (DenseEmbedding.subtypeEmb p e) := de.subtype p\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave ue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e) := uniformEmbedding_subtypeEmb _ he de\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave : b \u2208 closure (e '' {x | p x}) := (closure_mono <| monotone_image <| hp) (mem_of_mem_nhds hb)\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nlet \u27e8c, hc\u27e9 := uniformly_extend_exists ue'.toUniformInducing de'.dense hf \u27e8b, this\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\nc : \u03b3\nhc : Tendsto (fun x => f \u2191x) (comap (DenseEmbedding.subtypeEmb p e) (\ud835\udcdd { val := b, property := this })) (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nreplace hc : Tendsto (f \u2218 Subtype.val) (((\ud835\udcdd b).comap e).comap Subtype.val) (\ud835\udcdd c) := by\n  simpa only [nhds_subtype_eq_comap, comap_comap, DenseEmbedding.subtypeEmb_coe] using hc\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\nc : \u03b3\nhc : Tendsto (fun x => f \u2191x) (comap (DenseEmbedding.subtypeEmb p e) (\ud835\udcdd { val := b, property := this })) (\ud835\udcdd c)\n\u22a2 Tendsto (f \u2218 Subtype.val) (comap Subtype.val (comap e (\ud835\udcdd b))) (\ud835\udcdd c)\n[PROOFSTEP]\nsimpa only [nhds_subtype_eq_comap, comap_comap, DenseEmbedding.subtypeEmb_coe] using hc\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\nc : \u03b3\nhc : Tendsto (f \u2218 Subtype.val) (comap Subtype.val (comap e (\ud835\udcdd b))) (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nrefine \u27e8c, (tendsto_comap'_iff ?_).1 hc\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\nc : \u03b3\nhc : Tendsto (f \u2218 Subtype.val) (comap Subtype.val (comap e (\ud835\udcdd b))) (\ud835\udcdd c)\n\u22a2 range Subtype.val \u2208 comap e (\ud835\udcdd b)\n[PROOFSTEP]\nrw [Subtype.range_coe_subtype]\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\nc : \u03b3\nhc : Tendsto (f \u2218 Subtype.val) (comap Subtype.val (comap e (\ud835\udcdd b))) (\ud835\udcdd c)\n\u22a2 {x | p x} \u2208 comap e (\ud835\udcdd b)\n[PROOFSTEP]\nexact \u27e8_, hb, by rwa [\u2190 de.toInducing.closure_eq_preimage_closure_image, hs.closure_eq]\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne\u271d : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\u271d\nh_dense : DenseRange e\u271d\nf\u271d : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\u271d\ninst\u271d : CompleteSpace \u03b3\np : \u03b1 \u2192 Prop\ne : \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b3\nb : \u03b2\ns : Set \u03b1\nhf : UniformContinuous fun x => f \u2191x\nhe : UniformEmbedding e\nhd : \u2200 (x : \u03b2), x \u2208 closure (range e)\nhb : closure (e '' s) \u2208 \ud835\udcdd b\nhs : IsClosed s\nhp : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\nde : DenseEmbedding e\nde' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)\nue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)\nthis : b \u2208 closure (e '' {x | p x})\nc : \u03b3\nhc : Tendsto (f \u2218 Subtype.val) (comap Subtype.val (comap e (\ud835\udcdd b))) (\ud835\udcdd c)\n\u22a2 e \u207b\u00b9' closure (e '' s) \u2286 {x | p x}\n[PROOFSTEP]\nrwa [\u2190 de.toInducing.closure_eq_preimage_closure_image, hs.closure_eq]\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\na : \u03b1\n\u22a2 Tendsto f (comap e (\ud835\udcdd a)) (\ud835\udcdd (DenseInducing.extend (_ : DenseInducing e) f a))\n[PROOFSTEP]\nsimpa only [DenseInducing.extend] using tendsto_nhds_limUnder (uniformly_extend_exists h_e \u2039_\u203a h_f _)\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\n\u22a2 Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrwa [\u2190 h_e.comap_uniformity] at this \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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis\u271d : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\nthis : Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nts : Prod.map e e \u207b\u00b9' t \u2286 Prod.map f f \u207b\u00b9' s\nx\u271d : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\nhx_t : (x\u2081, x\u2082) \u2208 interior t\n\u22a2 (x\u2081, x\u2082) \u2208\n    Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) \u207b\u00b9' d\n[PROOFSTEP]\nhave : interior t \u2208 \ud835\udcdd (x\u2081, x\u2082) := isOpen_interior.mem_nhds hx_t\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis\u271d\u00b9 : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\nthis\u271d : Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nts : Prod.map e e \u207b\u00b9' t \u2286 Prod.map f f \u207b\u00b9' s\nx\u271d : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\nhx_t : (x\u2081, x\u2082) \u2208 interior t\nthis : interior t \u2208 \ud835\udcdd (x\u2081, x\u2082)\n\u22a2 (x\u2081, x\u2082) \u2208\n    Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) \u207b\u00b9' d\n[PROOFSTEP]\nlet \u27e8m\u2081, hm\u2081, m\u2082, hm\u2082, (hm : m\u2081 \u00d7\u02e2 m\u2082 \u2286 interior t)\u27e9 := mem_nhds_prod_iff.mp this\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\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis\u271d\u00b9 : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\nthis\u271d : Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nts : Prod.map e e \u207b\u00b9' t \u2286 Prod.map f f \u207b\u00b9' s\nx\u271d : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\nhx_t : (x\u2081, x\u2082) \u2208 interior t\nthis : interior t \u2208 \ud835\udcdd (x\u2081, x\u2082)\nm\u2081 : Set \u03b1\nhm\u2081 : m\u2081 \u2208 \ud835\udcdd x\u2081\nm\u2082 : Set \u03b1\nhm\u2082 : m\u2082 \u2208 \ud835\udcdd x\u2082\nhm : m\u2081 \u00d7\u02e2 m\u2082 \u2286 interior t\n\u22a2 (x\u2081, x\u2082) \u2208\n    Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) \u207b\u00b9' d\n[PROOFSTEP]\nobtain \u27e8_, \u27e8a, ha\u2081, rfl\u27e9, _, ha\u2082\u27e9 := h_pnt hm\u2081\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\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis\u271d\u00b9 : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\nthis\u271d : Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nts : Prod.map e e \u207b\u00b9' t \u2286 Prod.map f f \u207b\u00b9' s\nx\u271d : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\nhx_t : (x\u2081, x\u2082) \u2208 interior t\nthis : interior t \u2208 \ud835\udcdd (x\u2081, x\u2082)\nm\u2081 : Set \u03b1\nhm\u2081 : m\u2081 \u2208 \ud835\udcdd x\u2081\nm\u2082 : Set \u03b1\nhm\u2082 : m\u2082 \u2208 \ud835\udcdd x\u2082\nhm : m\u2081 \u00d7\u02e2 m\u2082 \u2286 interior t\na : \u03b2\nha\u2081 : a \u2208 e \u207b\u00b9' m\u2081\nleft\u271d : (f a, DenseInducing.extend (_ : DenseInducing e) f x\u2081) \u2208 s\nha\u2082 : (DenseInducing.extend (_ : DenseInducing e) f x\u2081, f a) \u2208 s\n\u22a2 (x\u2081, x\u2082) \u2208\n    Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) \u207b\u00b9' d\n[PROOFSTEP]\nobtain \u27e8_, \u27e8b, hb\u2081, rfl\u27e9, hb\u2082, _\u27e9 := h_pnt hm\u2082\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\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis\u271d\u00b9 : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\nthis\u271d : Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nts : Prod.map e e \u207b\u00b9' t \u2286 Prod.map f f \u207b\u00b9' s\nx\u271d : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\nhx_t : (x\u2081, x\u2082) \u2208 interior t\nthis : interior t \u2208 \ud835\udcdd (x\u2081, x\u2082)\nm\u2081 : Set \u03b1\nhm\u2081 : m\u2081 \u2208 \ud835\udcdd x\u2081\nm\u2082 : Set \u03b1\nhm\u2082 : m\u2082 \u2208 \ud835\udcdd x\u2082\nhm : m\u2081 \u00d7\u02e2 m\u2082 \u2286 interior t\na : \u03b2\nha\u2081 : a \u2208 e \u207b\u00b9' m\u2081\nleft\u271d : (f a, DenseInducing.extend (_ : DenseInducing e) f x\u2081) \u2208 s\nha\u2082 : (DenseInducing.extend (_ : DenseInducing e) f x\u2081, f a) \u2208 s\nb : \u03b2\nhb\u2081 : b \u2208 e \u207b\u00b9' m\u2082\nhb\u2082 : (f b, DenseInducing.extend (_ : DenseInducing e) f x\u2082) \u2208 s\nright\u271d : (DenseInducing.extend (_ : DenseInducing e) f x\u2082, f b) \u2208 s\n\u22a2 (x\u2081, x\u2082) \u2208\n    Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) \u207b\u00b9' d\n[PROOFSTEP]\nhave : Prod.map f f (a, b) \u2208 s := ts <| mem_preimage.2 <| interior_subset (@hm (e a, e b) \u27e8ha\u2081, hb\u2081\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\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\ninst\u271d\u00b9 : UniformSpace \u03b3\ne : \u03b2 \u2192 \u03b1\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : \u03b2 \u2192 \u03b3\nh_f : UniformContinuous f\ninst\u271d : CompleteSpace \u03b3\nd : Set (\u03b3 \u00d7 \u03b3)\nhd : d \u2208 \ud835\udce4 \u03b3\ns : Set (\u03b3 \u00d7 \u03b3)\nhs : s \u2208 \ud835\udce4 \u03b3\nhs_comp : s \u25cb (s \u25cb s) \u2286 d\nh_pnt :\n  \u2200 {a : \u03b1} {m : Set \u03b1},\n    m \u2208 \ud835\udcdd a \u2192\n      \u2203 c,\n        c \u2208 f '' (e \u207b\u00b9' m) \u2227\n          (c, DenseInducing.extend (_ : DenseInducing e) f a) \u2208 s \u2227\n            (DenseInducing.extend (_ : DenseInducing e) f a, c) \u2208 s\nthis\u271d\u00b2 : Prod.map f f \u207b\u00b9' s \u2208 \ud835\udce4 \u03b2\nthis\u271d\u00b9 : Prod.map f f \u207b\u00b9' s \u2208 comap (Prod.map e e) (\ud835\udce4 \u03b1)\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nts : Prod.map e e \u207b\u00b9' t \u2286 Prod.map f f \u207b\u00b9' s\nx\u271d : \u03b1 \u00d7 \u03b1\nx\u2081 x\u2082 : \u03b1\nhx_t : (x\u2081, x\u2082) \u2208 interior t\nthis\u271d : interior t \u2208 \ud835\udcdd (x\u2081, x\u2082)\nm\u2081 : Set \u03b1\nhm\u2081 : m\u2081 \u2208 \ud835\udcdd x\u2081\nm\u2082 : Set \u03b1\nhm\u2082 : m\u2082 \u2208 \ud835\udcdd x\u2082\nhm : m\u2081 \u00d7\u02e2 m\u2082 \u2286 interior t\na : \u03b2\nha\u2081 : a \u2208 e \u207b\u00b9' m\u2081\nleft\u271d : (f a, DenseInducing.extend (_ : DenseInducing e) f x\u2081) \u2208 s\nha\u2082 : (DenseInducing.extend (_ : DenseInducing e) f x\u2081, f a) \u2208 s\nb : \u03b2\nhb\u2081 : b \u2208 e \u207b\u00b9' m\u2082\nhb\u2082 : (f b, DenseInducing.extend (_ : DenseInducing e) f x\u2082) \u2208 s\nright\u271d : (DenseInducing.extend (_ : DenseInducing e) f x\u2082, f b) \u2208 s\nthis : Prod.map f f (a, b) \u2208 s\n\u22a2 (x\u2081, x\u2082) \u2208\n    Prod.map (DenseInducing.extend (_ : DenseInducing e) f) (DenseInducing.extend (_ : DenseInducing e) f) \u207b\u00b9' d\n[PROOFSTEP]\nexact hs_comp \u27e8f a, ha\u2082, \u27e8f b, this, hb\u2082\u27e9\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.UniformSpace.UniformEmbedding", "llama_tokens": 24935, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891392358014, "lm_q2_score": 0.6723316860482762, "lm_q1q2_score": 0.5527838102329873}}
{"text": "[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\n\u22a2 { x_1 // x_1 \u2208 Algebra.adjoin F {x} } \u2243\u2090[F] AdjoinRoot (minpoly F x)\n[PROOFSTEP]\nrefine AlgEquiv.symm ?_\n[GOAL]\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\n\u22a2 AdjoinRoot (minpoly F x) \u2243\u2090[F] { x_1 // x_1 \u2208 Algebra.adjoin F {x} }\n[PROOFSTEP]\nrefine AlgEquiv.ofBijective (AlgHom.codRestrict (AdjoinRoot.liftHom _ x <| minpoly.aeval F x) _ fun p => ?_) \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : AdjoinRoot (minpoly F x)\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p \u2208 Algebra.adjoin F {x}\n[PROOFSTEP]\ninduction p using AdjoinRoot.induction_on with\n| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval F x).symm \u25b8 (Polynomial.aeval _).mem_range.mpr \u27e8p, rfl\u27e9\n[GOAL]\ncase refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : AdjoinRoot (minpoly F x)\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p \u2208 Algebra.adjoin F {x}\n[PROOFSTEP]\ninduction p using AdjoinRoot.induction_on with\n| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval F x).symm \u25b8 (Polynomial.aeval _).mem_range.mpr \u27e8p, rfl\u27e9\n[GOAL]\ncase refine_1.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) \u2208\n    Algebra.adjoin F {x}\n[PROOFSTEP]\n\n| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval F x).symm \u25b8 (Polynomial.aeval _).mem_range.mpr \u27e8p, rfl\u27e9\n[GOAL]\ncase refine_1.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) \u2208\n    Algebra.adjoin F {x}\n[PROOFSTEP]\nexact (Algebra.adjoin_singleton_eq_range_aeval F x).symm \u25b8 (Polynomial.aeval _).mem_range.mpr \u27e8p, rfl\u27e9\n[GOAL]\ncase refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\n\u22a2 Function.Injective\n    \u2191(AlgHom.codRestrict (AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (Algebra.adjoin F {x})\n        (_ :\n          \u2200 (p : AdjoinRoot (minpoly F x)),\n            \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p \u2208 Algebra.adjoin F {x}))\n[PROOFSTEP]\napply (AlgHom.injective_codRestrict _ _ _).2\n[GOAL]\ncase refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\n\u22a2 Function.Injective \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0))\n[PROOFSTEP]\napply (injective_iff_map_eq_zero _).2\n[GOAL]\ncase refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\n\u22a2 \u2200 (a : AdjoinRoot (minpoly F x)),\n    \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro p\n[GOAL]\ncase refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : AdjoinRoot (minpoly F x)\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p = 0 \u2192 p = 0\n[PROOFSTEP]\ninduction p using AdjoinRoot.induction_on with\n| ih p =>\n  intro hp\n  apply Ideal.Quotient.eq_zero_iff_mem.2\n  apply Ideal.mem_span_singleton.2\n  apply minpoly.dvd F x hp\n[GOAL]\ncase refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : AdjoinRoot (minpoly F x)\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p = 0 \u2192 p = 0\n[PROOFSTEP]\ninduction p using AdjoinRoot.induction_on with\n| ih p =>\n  intro hp\n  apply Ideal.Quotient.eq_zero_iff_mem.2\n  apply Ideal.mem_span_singleton.2\n  apply minpoly.dvd F x hp\n[GOAL]\ncase refine_2.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) = 0 \u2192\n    \u2191(AdjoinRoot.mk (minpoly F x)) p = 0\n[PROOFSTEP]\n\n| ih p =>\n  intro hp\n  apply Ideal.Quotient.eq_zero_iff_mem.2\n  apply Ideal.mem_span_singleton.2\n  apply minpoly.dvd F x hp\n[GOAL]\ncase refine_2.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\n\u22a2 \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) = 0 \u2192\n    \u2191(AdjoinRoot.mk (minpoly F x)) p = 0\n[PROOFSTEP]\nintro hp\n[GOAL]\ncase refine_2.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\nhp : \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) = 0\n\u22a2 \u2191(AdjoinRoot.mk (minpoly F x)) p = 0\n[PROOFSTEP]\napply Ideal.Quotient.eq_zero_iff_mem.2\n[GOAL]\ncase refine_2.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\nhp : \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) = 0\n\u22a2 p \u2208 Ideal.span {minpoly F x}\n[PROOFSTEP]\napply Ideal.mem_span_singleton.2\n[GOAL]\ncase refine_2.ih\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\np : F[X]\nhp : \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (\u2191(AdjoinRoot.mk (minpoly F x)) p) = 0\n\u22a2 minpoly F x \u2223 p\n[PROOFSTEP]\napply minpoly.dvd F x hp\n[GOAL]\ncase refine_3\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\n\u22a2 Function.Surjective\n    \u2191(AlgHom.codRestrict (AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (Algebra.adjoin F {x})\n        (_ :\n          \u2200 (p : AdjoinRoot (minpoly F x)),\n            \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p \u2208 Algebra.adjoin F {x}))\n[PROOFSTEP]\nintro y\n[GOAL]\ncase refine_3\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\ny : { x_1 // x_1 \u2208 Algebra.adjoin F {x} }\n\u22a2 \u2203 a,\n    \u2191(AlgHom.codRestrict (AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (Algebra.adjoin F {x})\n            (_ :\n              \u2200 (p : AdjoinRoot (minpoly F x)),\n                \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p \u2208 Algebra.adjoin F {x}))\n        a =\n      y\n[PROOFSTEP]\nlet \u27e8p, hp\u27e9 := (SetLike.ext_iff.1 (Algebra.adjoin_singleton_eq_range_aeval F x) (y : R)).1 y.2\n[GOAL]\ncase refine_3\nF : Type u_1\ninst\u271d\u00b2 : Field F\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Algebra F R\nx : R\ny : { x_1 // x_1 \u2208 Algebra.adjoin F {x} }\np : F[X]\nhp : \u2191\u2191(aeval x) p = \u2191y\n\u22a2 \u2203 a,\n    \u2191(AlgHom.codRestrict (AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) (Algebra.adjoin F {x})\n            (_ :\n              \u2200 (p : AdjoinRoot (minpoly F x)),\n                \u2191(AdjoinRoot.liftHom (minpoly F x) x (_ : \u2191(aeval x) (minpoly F x) = 0)) p \u2208 Algebra.adjoin F {x}))\n        a =\n      y\n[PROOFSTEP]\nexact \u27e8AdjoinRoot.mk _ p, Subtype.eq hp\u27e9\n[GOAL]\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns : Finset K\n\u22a2 (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\n[PROOFSTEP]\nclassical\nrefine' Finset.induction_on s (fun _ => _) fun a s _ ih H => _\n\u00b7 rw [coe_empty, Algebra.adjoin_empty]\n  exact \u27e8(Algebra.ofId F L).comp (Algebra.botEquiv F K)\u27e9\nrw [forall_mem_insert] at H \nrcases H with \u27e8\u27e8H1, H2\u27e9, H3\u27e9\ncases' ih H3 with f\nchoose H3 _ using H3\nrw [coe_insert, Set.insert_eq, Set.union_comm, Algebra.adjoin_union_eq_adjoin_adjoin]\nletI := (f : Algebra.adjoin F (\u2191s : Set K) \u2192+* L).toAlgebra\nhaveI : FiniteDimensional F (Algebra.adjoin F (\u2191s : Set K)) :=\n  ((Submodule.fg_iff_finiteDimensional _).1 (FG_adjoin_of_finite s.finite_toSet H3)).of_subalgebra_toSubmodule\nletI := fieldOfFiniteDimensional F (Algebra.adjoin F (\u2191s : Set K))\nhave H5 : IsIntegral (Algebra.adjoin F (s : Set K)) a := isIntegral_of_isScalarTower H1\nhave H6 : (minpoly (Algebra.adjoin F (s : Set K)) a).Splits (algebraMap (Algebra.adjoin F (s : Set K)) L) :=\n  by\n  have : Polynomial.map (algebraMap F (Algebra.adjoin F (s : Set K))) (minpoly F a) \u2260 0 :=\n    Polynomial.map_ne_zero <| minpoly.ne_zero H1\n  refine' Polynomial.splits_of_splits_of_dvd _ this ((Polynomial.splits_map_iff _ _).2 _) (minpoly.dvd _ _ _)\n  \u00b7 rw [\u2190 IsScalarTower.algebraMap_eq]\n    exact H2\n  \u00b7 rw [Polynomial.aeval_map_algebraMap, minpoly.aeval]\nobtain \u27e8y, hy\u27e9 := Polynomial.exists_root_of_splits _ H6 (ne_of_lt (minpoly.degree_pos H5)).symm\nrefine' \u27e8Subalgebra.ofRestrictScalars F _ _\u27e9\nrefine' (AdjoinRoot.liftHom (minpoly (Algebra.adjoin F (\u2191s : Set K)) a) y hy).comp _\nexact (AlgEquiv.adjoinSingletonEquivAdjoinRootMinpoly (Algebra.adjoin F (\u2191s : Set K)) a).toAlgHom\n[GOAL]\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns : Finset K\n\u22a2 (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\n[PROOFSTEP]\nrefine' Finset.induction_on s (fun _ => _) fun a s _ ih H => _\n[GOAL]\ncase refine'_1\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns : Finset K\nx\u271d : \u2200 (x : K), x \u2208 \u2205 \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)\n\u22a2 Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191\u2205 } \u2192\u2090[F] L)\n[PROOFSTEP]\nrw [coe_empty, Algebra.adjoin_empty]\n[GOAL]\ncase refine'_1\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns : Finset K\nx\u271d : \u2200 (x : K), x \u2208 \u2205 \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)\n\u22a2 Nonempty ({ x // x \u2208 \u22a5 } \u2192\u2090[F] L)\n[PROOFSTEP]\nexact \u27e8(Algebra.ofId F L).comp (Algebra.botEquiv F K)\u27e9\n[GOAL]\ncase refine'_2\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH : \u2200 (x : K), x \u2208 insert a s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)\n\u22a2 Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191(insert a s) } \u2192\u2090[F] L)\n[PROOFSTEP]\nrw [forall_mem_insert] at H \n[GOAL]\ncase refine'_2\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH :\n  (IsIntegral F a \u2227 Splits (algebraMap F L) (minpoly F a)) \u2227\n    \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)\n\u22a2 Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191(insert a s) } \u2192\u2090[F] L)\n[PROOFSTEP]\nrcases H with \u27e8\u27e8H1, H2\u27e9, H3\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\n\u22a2 Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191(insert a s) } \u2192\u2090[F] L)\n[PROOFSTEP]\ncases' ih H3 with f\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\n\u22a2 Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191(insert a s) } \u2192\u2090[F] L)\n[PROOFSTEP]\nchoose H3 _ using H3\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\n\u22a2 Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191(insert a s) } \u2192\u2090[F] L)\n[PROOFSTEP]\nrw [coe_insert, Set.insert_eq, Set.union_comm, Algebra.adjoin_union_eq_adjoin_adjoin]\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nletI := (f : Algebra.adjoin F (\u2191s : Set K) \u2192+* L).toAlgebra\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nhaveI : FiniteDimensional F (Algebra.adjoin F (\u2191s : Set K)) :=\n  ((Submodule.fg_iff_finiteDimensional _).1 (FG_adjoin_of_finite s.finite_toSet H3)).of_subalgebra_toSubmodule\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nletI := fieldOfFiniteDimensional F (Algebra.adjoin F (\u2191s : Set K))\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nhave H5 : IsIntegral (Algebra.adjoin F (s : Set K)) a := isIntegral_of_isScalarTower H1\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nhave H6 : (minpoly (Algebra.adjoin F (s : Set K)) a).Splits (algebraMap (Algebra.adjoin F (s : Set K)) L) :=\n  by\n  have : Polynomial.map (algebraMap F (Algebra.adjoin F (s : Set K))) (minpoly F a) \u2260 0 :=\n    Polynomial.map_ne_zero <| minpoly.ne_zero H1\n  refine' Polynomial.splits_of_splits_of_dvd _ this ((Polynomial.splits_map_iff _ _).2 _) (minpoly.dvd _ _ _)\n  \u00b7 rw [\u2190 IsScalarTower.algebraMap_eq]\n    exact H2\n  \u00b7 rw [Polynomial.aeval_map_algebraMap, minpoly.aeval]\n[GOAL]\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\n\u22a2 Splits (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\n[PROOFSTEP]\nhave : Polynomial.map (algebraMap F (Algebra.adjoin F (s : Set K))) (minpoly F a) \u2260 0 :=\n  Polynomial.map_ne_zero <| minpoly.ne_zero H1\n[GOAL]\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b2 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis\u271d : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nthis : Polynomial.map (algebraMap F { x // x \u2208 Algebra.adjoin F \u2191s }) (minpoly F a) \u2260 0\n\u22a2 Splits (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\n[PROOFSTEP]\nrefine' Polynomial.splits_of_splits_of_dvd _ this ((Polynomial.splits_map_iff _ _).2 _) (minpoly.dvd _ _ _)\n[GOAL]\ncase refine'_1\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b2 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis\u271d : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nthis : Polynomial.map (algebraMap F { x // x \u2208 Algebra.adjoin F \u2191s }) (minpoly F a) \u2260 0\n\u22a2 Splits (RingHom.comp (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (algebraMap F { x // x \u2208 Algebra.adjoin F \u2191s }))\n    (minpoly F a)\n[PROOFSTEP]\nrw [\u2190 IsScalarTower.algebraMap_eq]\n[GOAL]\ncase refine'_1\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b2 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis\u271d : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nthis : Polynomial.map (algebraMap F { x // x \u2208 Algebra.adjoin F \u2191s }) (minpoly F a) \u2260 0\n\u22a2 Splits (algebraMap F L) (minpoly F a)\n[PROOFSTEP]\nexact H2\n[GOAL]\ncase refine'_2\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b2 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d\u00b9 : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis\u271d : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nthis : Polynomial.map (algebraMap F { x // x \u2208 Algebra.adjoin F \u2191s }) (minpoly F a) \u2260 0\n\u22a2 \u2191(aeval a) (Polynomial.map (algebraMap F { x // x \u2208 Algebra.adjoin F \u2191s }) (minpoly F a)) = 0\n[PROOFSTEP]\nrw [Polynomial.aeval_map_algebraMap, minpoly.aeval]\n[GOAL]\ncase refine'_2.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nH6 : Splits (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := Polynomial.exists_root_of_splits _ H6 (ne_of_lt (minpoly.degree_pos H5)).symm\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nH6 : Splits (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\ny : L\nhy : eval\u2082 (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) y (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a) = 0\n\u22a2 Nonempty ({ x // x \u2208 Subalgebra.restrictScalars F (Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a}) } \u2192\u2090[F] L)\n[PROOFSTEP]\nrefine' \u27e8Subalgebra.ofRestrictScalars F _ _\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nH6 : Splits (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\ny : L\nhy : eval\u2082 (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) y (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a) = 0\n\u22a2 { x // x \u2208 Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a} } \u2192\u2090[{ x // x \u2208 Algebra.adjoin F \u2191s }] L\n[PROOFSTEP]\nrefine' (AdjoinRoot.liftHom (minpoly (Algebra.adjoin F (\u2191s : Set K)) a) y hy).comp _\n[GOAL]\ncase refine'_2.intro.intro.intro.intro\nF\u271d : Type u_1\ninst\u271d\u2075 : Field F\u271d\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra F K\ninst\u271d : Algebra F L\ns\u271d : Finset K\na : K\ns : Finset K\nx\u271d : \u00aca \u2208 s\nih :\n  (\u2200 (x : K), x \u2208 s \u2192 IsIntegral F x \u2227 Splits (algebraMap F L) (minpoly F x)) \u2192\n    Nonempty ({ x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L)\nH1 : IsIntegral F a\nH2 : Splits (algebraMap F L) (minpoly F a)\nf : { x // x \u2208 Algebra.adjoin F \u2191s } \u2192\u2090[F] L\nH3 : \u2200 (x : K), x \u2208 s \u2192 IsIntegral F x\n\u271d : \u2200 (x : K), x \u2208 s \u2192 Splits (algebraMap F L) (minpoly F x)\nthis\u271d\u00b9 : Algebra { x // x \u2208 Algebra.adjoin F \u2191s } L := RingHom.toAlgebra \u2191f\nthis\u271d : FiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nthis : Field { x // x \u2208 Algebra.adjoin F \u2191s } := fieldOfFiniteDimensional F { x // x \u2208 Algebra.adjoin F \u2191s }\nH5 : IsIntegral { x // x \u2208 Algebra.adjoin F \u2191s } a\nH6 : Splits (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\ny : L\nhy : eval\u2082 (algebraMap { x // x \u2208 Algebra.adjoin F \u2191s } L) y (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a) = 0\n\u22a2 { x // x \u2208 Algebra.adjoin { x // x \u2208 Algebra.adjoin F \u2191s } {a} } \u2192\u2090[{ x // x \u2208 Algebra.adjoin F \u2191s }]\n    AdjoinRoot (minpoly { x // x \u2208 Algebra.adjoin F \u2191s } a)\n[PROOFSTEP]\nexact (AlgEquiv.adjoinSingletonEquivAdjoinRootMinpoly (Algebra.adjoin F (\u2191s : Set K)) a).toAlgHom\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Adjoin.Field", "llama_tokens": 14430, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951182587158, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.5526624704961153}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\ns : Set E\n\u22a2 EMetric.diam (c \u2022 s) = \u2016c\u2016\u208a \u2022 EMetric.diam s\n[PROOFSTEP]\nrefine' le_antisymm (ediam_smul_le c s) _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\ns : Set E\n\u22a2 \u2016c\u2016\u208a \u2022 EMetric.diam s \u2264 EMetric.diam (c \u2022 s)\n[PROOFSTEP]\nobtain rfl | hc := eq_or_ne c 0\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\ns : Set E\n\u22a2 \u20160\u2016\u208a \u2022 EMetric.diam s \u2264 EMetric.diam (0 \u2022 s)\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\n\u22a2 \u20160\u2016\u208a \u2022 EMetric.diam \u2205 \u2264 EMetric.diam (0 \u2022 \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\ns : Set E\nhs : Set.Nonempty s\n\u22a2 \u20160\u2016\u208a \u2022 EMetric.diam s \u2264 EMetric.diam (0 \u2022 s)\n[PROOFSTEP]\nsimp [zero_smul_set hs, \u2190 Set.singleton_zero]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\ns : Set E\nhc : c \u2260 0\n\u22a2 \u2016c\u2016\u208a \u2022 EMetric.diam s \u2264 EMetric.diam (c \u2022 s)\n[PROOFSTEP]\nhave := (lipschitzWith_smul c\u207b\u00b9).ediam_image_le (c \u2022 s)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\ns : Set E\nhc : c \u2260 0\nthis : EMetric.diam ((fun x x_1 => x \u2022 x_1) c\u207b\u00b9 '' (c \u2022 s)) \u2264 \u2191\u2016c\u207b\u00b9\u2016\u208a * EMetric.diam (c \u2022 s)\n\u22a2 \u2016c\u2016\u208a \u2022 EMetric.diam s \u2264 EMetric.diam (c \u2022 s)\n[PROOFSTEP]\nrwa [\u2190 smul_eq_mul, \u2190 ENNReal.smul_def, Set.image_smul, inv_smul_smul\u2080 hc s, nnnorm_inv,\n  ENNReal.le_inv_smul_iff (nnnorm_ne_zero_iff.mpr hc)] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nx : Set E\n\u22a2 diam (c \u2022 x) = \u2016c\u2016 * diam x\n[PROOFSTEP]\nsimp_rw [diam, ediam_smul\u2080, ENNReal.toReal_smul, NNReal.smul_def, coe_nnnorm, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\n\u22a2 EMetric.infEdist (c \u2022 x) (c \u2022 s) = \u2016c\u2016\u208a \u2022 EMetric.infEdist x s\n[PROOFSTEP]\nsimp_rw [EMetric.infEdist]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\n\u22a2 \u2a05 (y : E) (_ : y \u2208 c \u2022 s), edist (c \u2022 x) y = \u2016c\u2016\u208a \u2022 \u2a05 (y : E) (_ : y \u2208 s), edist x y\n[PROOFSTEP]\nhave : Function.Surjective ((c \u2022 \u00b7) : E \u2192 E) := Function.RightInverse.surjective (smul_inv_smul\u2080 hc)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\nthis : Function.Surjective fun x => c \u2022 x\n\u22a2 \u2a05 (y : E) (_ : y \u2208 c \u2022 s), edist (c \u2022 x) y = \u2016c\u2016\u208a \u2022 \u2a05 (y : E) (_ : y \u2208 s), edist x y\n[PROOFSTEP]\ntrans \u2a05 (y) (_ : y \u2208 s), \u2016c\u2016\u208a \u2022 edist x y\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\nthis : Function.Surjective fun x => c \u2022 x\n\u22a2 \u2a05 (y : E) (_ : y \u2208 c \u2022 s), edist (c \u2022 x) y = \u2a05 (y : E) (_ : y \u2208 s), \u2016c\u2016\u208a \u2022 edist x y\n[PROOFSTEP]\nrefine' (this.iInf_congr _ fun y => _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\nthis : Function.Surjective fun x => c \u2022 x\ny : E\n\u22a2 \u2a05 (_ : c \u2022 y \u2208 c \u2022 s), edist (c \u2022 x) (c \u2022 y) = \u2a05 (_ : y \u2208 s), \u2016c\u2016\u208a \u2022 edist x y\n[PROOFSTEP]\nsimp_rw [smul_mem_smul_set_iff\u2080 hc, edist_smul\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\nthis : Function.Surjective fun x => c \u2022 x\n\u22a2 \u2a05 (y : E) (_ : y \u2208 s), \u2016c\u2016\u208a \u2022 edist x y = \u2016c\u2016\u208a \u2022 \u2a05 (y : E) (_ : y \u2208 s), edist x y\n[PROOFSTEP]\nhave : (\u2016c\u2016\u208a : ENNReal) \u2260 0 := by simp [hc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\nthis : Function.Surjective fun x => c \u2022 x\n\u22a2 \u2191\u2016c\u2016\u208a \u2260 0\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\nthis\u271d : Function.Surjective fun x => c \u2022 x\nthis : \u2191\u2016c\u2016\u208a \u2260 0\n\u22a2 \u2a05 (y : E) (_ : y \u2208 s), \u2016c\u2016\u208a \u2022 edist x y = \u2016c\u2016\u208a \u2022 \u2a05 (y : E) (_ : y \u2208 s), edist x y\n[PROOFSTEP]\nsimp_rw [ENNReal.smul_def, smul_eq_mul, ENNReal.mul_iInf_of_ne this ENNReal.coe_ne_top]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set E\nx : E\n\u22a2 infDist (c \u2022 x) (c \u2022 s) = \u2016c\u2016 * infDist x s\n[PROOFSTEP]\nsimp_rw [Metric.infDist, infEdist_smul\u2080 hc s, ENNReal.toReal_smul, NNReal.smul_def, coe_nnnorm, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\n\u22a2 c \u2022 ball x r = ball (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n\u22a2 y \u2208 c \u2022 ball x r \u2194 y \u2208 ball (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nrw [mem_smul_set_iff_inv_smul_mem\u2080 hc]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n\u22a2 c\u207b\u00b9 \u2022 y \u2208 ball x r \u2194 y \u2208 ball (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n| c\u207b\u00b9 \u2022 y \u2208 ball x r\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n| c\u207b\u00b9 \u2022 y \u2208 ball x r\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n| c\u207b\u00b9 \u2022 y \u2208 ball x r\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n\u22a2 c\u207b\u00b9 \u2022 y \u2208 ball (c\u207b\u00b9 \u2022 c \u2022 x) r \u2194 y \u2208 ball (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nsimp [\u2190 div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hc), mul_comm _ r, dist_smul\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\n\u22a2 c \u2022 ball 0 1 = ball 0 \u2016c\u2016\n[PROOFSTEP]\nrw [_root_.smul_ball hc, smul_zero, mul_one]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\n\u22a2 c \u2022 sphere x r = sphere (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n\u22a2 y \u2208 c \u2022 sphere x r \u2194 y \u2208 sphere (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nrw [mem_smul_set_iff_inv_smul_mem\u2080 hc]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n\u22a2 c\u207b\u00b9 \u2022 y \u2208 sphere x r \u2194 y \u2208 sphere (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n| c\u207b\u00b9 \u2022 y \u2208 sphere x r\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n| c\u207b\u00b9 \u2022 y \u2208 sphere x r\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n| c\u207b\u00b9 \u2022 y \u2208 sphere x r\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul\u2080 hc x]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\ny : E\n\u22a2 c\u207b\u00b9 \u2022 y \u2208 sphere (c\u207b\u00b9 \u2022 c \u2022 x) r \u2194 y \u2208 sphere (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nsimp only [mem_sphere, dist_smul\u2080, norm_inv, \u2190 div_eq_inv_mul, div_eq_iff (norm_pos_iff.2 hc).ne', mul_comm r]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : c \u2260 0\nx : E\nr : \u211d\n\u22a2 c \u2022 closedBall x r = closedBall (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nsimp only [\u2190 ball_union_sphere, Set.smul_set_union, _root_.smul_ball hc, smul_sphere' hc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u22a2 \u2200\u1da0 (r : \ud835\udd5c) in \ud835\udcdd 0, {x} + r \u2022 s \u2286 u\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 closedBall x \u03b5 \u2286 u := nhds_basis_closedBall.mem_iff.1 hu\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\n\u22a2 \u2200\u1da0 (r : \ud835\udd5c) in \ud835\udcdd 0, {x} + r \u2022 s \u2286 u\n[PROOFSTEP]\nobtain \u27e8R, Rpos, hR\u27e9 : \u2203 R : \u211d, 0 < R \u2227 s \u2286 closedBall 0 R := hs.subset_ball_lt 0 0\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\n\u22a2 \u2200\u1da0 (r : \ud835\udd5c) in \ud835\udcdd 0, {x} + r \u2022 s \u2286 u\n[PROOFSTEP]\nhave : Metric.closedBall (0 : \ud835\udd5c) (\u03b5 / R) \u2208 \ud835\udcdd (0 : \ud835\udd5c) := closedBall_mem_nhds _ (div_pos \u03b5pos Rpos)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\n\u22a2 \u2200\u1da0 (r : \ud835\udd5c) in \ud835\udcdd 0, {x} + r \u2022 s \u2286 u\n[PROOFSTEP]\nfilter_upwards [this] with r hr\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\n\u22a2 {x} + r \u2022 s \u2286 u\n[PROOFSTEP]\nsimp only [image_add_left, singleton_add]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\n\u22a2 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s) \u2286 u\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\n\u22a2 y \u2208 u\n[PROOFSTEP]\nobtain \u27e8z, zs, hz\u27e9 : \u2203 z : E, z \u2208 s \u2227 r \u2022 z = -x + y := by simpa [mem_smul_set] using hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\n\u22a2 \u2203 z, z \u2208 s \u2227 r \u2022 z = -x + y\n[PROOFSTEP]\nsimpa [mem_smul_set] using hy\n[GOAL]\ncase h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\nz : E\nzs : z \u2208 s\nhz : r \u2022 z = -x + y\n\u22a2 y \u2208 u\n[PROOFSTEP]\nhave I : \u2016r \u2022 z\u2016 \u2264 \u03b5 :=\n  calc\n    \u2016r \u2022 z\u2016 = \u2016r\u2016 * \u2016z\u2016 := norm_smul _ _\n    _ \u2264 \u03b5 / R * R :=\n      (mul_le_mul (mem_closedBall_zero_iff.1 hr) (mem_closedBall_zero_iff.1 (hR zs)) (norm_nonneg _)\n        (div_pos \u03b5pos Rpos).le)\n    _ = \u03b5 := by field_simp [Rpos.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\nz : E\nzs : z \u2208 s\nhz : r \u2022 z = -x + y\n\u22a2 \u03b5 / R * R = \u03b5\n[PROOFSTEP]\nfield_simp [Rpos.ne']\n[GOAL]\ncase h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\nz : E\nzs : z \u2208 s\nhz : r \u2022 z = -x + y\nI : \u2016r \u2022 z\u2016 \u2264 \u03b5\n\u22a2 y \u2208 u\n[PROOFSTEP]\nhave : y = x + r \u2022 z := by simp only [hz, add_neg_cancel_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\nz : E\nzs : z \u2208 s\nhz : r \u2022 z = -x + y\nI : \u2016r \u2022 z\u2016 \u2264 \u03b5\n\u22a2 y = x + r \u2022 z\n[PROOFSTEP]\nsimp only [hz, add_neg_cancel_left]\n[GOAL]\ncase h.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis\u271d : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\nz : E\nzs : z \u2208 s\nhz : r \u2022 z = -x + y\nI : \u2016r \u2022 z\u2016 \u2264 \u03b5\nthis : y = x + r \u2022 z\n\u22a2 y \u2208 u\n[PROOFSTEP]\napply h\u03b5\n[GOAL]\ncase h.intro.intro.a\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u \u2208 \ud835\udcdd x\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : closedBall x \u03b5 \u2286 u\nR : \u211d\nRpos : 0 < R\nhR : s \u2286 closedBall 0 R\nthis\u271d : closedBall 0 (\u03b5 / R) \u2208 \ud835\udcdd 0\nr : \ud835\udd5c\nhr : r \u2208 closedBall 0 (\u03b5 / R)\ny : E\nhy : y \u2208 (fun x_1 => -x + x_1) \u207b\u00b9' (r \u2022 s)\nz : E\nzs : z \u2208 s\nhz : r \u2022 z = -x + y\nI : \u2016r \u2022 z\u2016 \u2264 \u03b5\nthis : y = x + r \u2022 z\n\u22a2 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nsimpa only [this, dist_eq_norm, add_sub_cancel', mem_closedBall] using I\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 r : \u211d\nhr : 0 < r\n\u22a2 r \u2022 ball 0 1 = ball 0 r\n[PROOFSTEP]\nrw [smul_unitBall hr.ne', Real.norm_of_nonneg hr.le]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nx z : E\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2203 y, dist x y = b * dist x z \u2227 dist y z = a * dist x z\n[PROOFSTEP]\nuse a \u2022 x + b \u2022 z\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nx z : E\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 dist x (a \u2022 x + b \u2022 z) = b * dist x z \u2227 dist (a \u2022 x + b \u2022 z) z = a * dist x z\n[PROOFSTEP]\nnth_rw 1 [\u2190 one_smul \u211d x]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nx z : E\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 dist (1 \u2022 x) (a \u2022 x + b \u2022 z) = b * dist x z \u2227 dist (a \u2022 x + b \u2022 z) z = a * dist x z\n[PROOFSTEP]\nnth_rw 4 [\u2190 one_smul \u211d z]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nx z : E\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 dist (1 \u2022 x) (a \u2022 x + b \u2022 z) = b * dist x z \u2227 dist (a \u2022 x + b \u2022 z) (1 \u2022 z) = a * dist x z\n[PROOFSTEP]\nsimp [dist_eq_norm, \u2190 hab, add_smul, \u2190 smul_sub, norm_smul_of_nonneg, ha, hb]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z \u2264 \u03b5 + \u03b4\n\u22a2 \u2203 y, dist x y \u2264 \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nobtain rfl | h\u03b5' := h\u03b5.eq_or_lt\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 0\nh : dist x z \u2264 0 + \u03b4\n\u22a2 \u2203 y, dist x y \u2264 \u03b4 \u2227 dist y z \u2264 0\n[PROOFSTEP]\nexact \u27e8z, by rwa [zero_add] at h , (dist_self _).le\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 0\nh : dist x z \u2264 0 + \u03b4\n\u22a2 dist x z \u2264 \u03b4\n[PROOFSTEP]\nrwa [zero_add] at h \n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z \u2264 \u03b5 + \u03b4\nh\u03b5' : 0 < \u03b5\n\u22a2 \u2203 y, dist x y \u2264 \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nhave h\u03b5\u03b4 := add_pos_of_pos_of_nonneg h\u03b5' h\u03b4\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z \u2264 \u03b5 + \u03b4\nh\u03b5' : 0 < \u03b5\nh\u03b5\u03b4 : 0 < \u03b5 + \u03b4\n\u22a2 \u2203 y, dist x y \u2264 \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nrefine'\n  (exists_dist_eq x z (div_nonneg h\u03b5 <| add_nonneg h\u03b5 h\u03b4) (div_nonneg h\u03b4 <| add_nonneg h\u03b5 h\u03b4) <| by\n        rw [\u2190 add_div, div_self h\u03b5\u03b4.ne']).imp\n    fun y hy => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z \u2264 \u03b5 + \u03b4\nh\u03b5' : 0 < \u03b5\nh\u03b5\u03b4 : 0 < \u03b5 + \u03b4\n\u22a2 \u03b5 / (\u03b5 + \u03b4) + \u03b4 / (\u03b5 + \u03b4) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self h\u03b5\u03b4.ne']\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z \u2264 \u03b5 + \u03b4\nh\u03b5' : 0 < \u03b5\nh\u03b5\u03b4 : 0 < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x y \u2264 \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nrw [hy.1, hy.2, div_mul_comm, div_mul_comm \u03b5]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z \u2264 \u03b5 + \u03b4\nh\u03b5' : 0 < \u03b5\nh\u03b5\u03b4 : 0 < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x z / (\u03b5 + \u03b4) * \u03b4 \u2264 \u03b4 \u2227 dist x z / (\u03b5 + \u03b4) * \u03b5 \u2264 \u03b5\n[PROOFSTEP]\nrw [\u2190 div_le_one h\u03b5\u03b4] at h \n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z / (\u03b5 + \u03b4) \u2264 1\nh\u03b5' : 0 < \u03b5\nh\u03b5\u03b4 : 0 < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x z / (\u03b5 + \u03b4) * \u03b4 \u2264 \u03b4 \u2227 dist x z / (\u03b5 + \u03b4) * \u03b5 \u2264 \u03b5\n[PROOFSTEP]\nexact \u27e8mul_le_of_le_one_left h\u03b4 h, mul_le_of_le_one_left h\u03b5 h\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\n\u22a2 \u2203 y, dist x y \u2264 \u03b4 \u2227 dist y z < \u03b5\n[PROOFSTEP]\nrefine'\n  (exists_dist_eq x z (div_nonneg h\u03b5.le <| add_nonneg h\u03b5.le h\u03b4) (div_nonneg h\u03b4 <| add_nonneg h\u03b5.le h\u03b4) <| by\n        rw [\u2190 add_div, div_self (add_pos_of_pos_of_nonneg h\u03b5 h\u03b4).ne']).imp\n    fun y hy => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\n\u22a2 \u03b5 / (\u03b5 + \u03b4) + \u03b4 / (\u03b5 + \u03b4) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self (add_pos_of_pos_of_nonneg h\u03b5 h\u03b4).ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x y \u2264 \u03b4 \u2227 dist y z < \u03b5\n[PROOFSTEP]\nrw [hy.1, hy.2, div_mul_comm, div_mul_comm \u03b5]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x z / (\u03b5 + \u03b4) * \u03b4 \u2264 \u03b4 \u2227 dist x z / (\u03b5 + \u03b4) * \u03b5 < \u03b5\n[PROOFSTEP]\nrw [\u2190 div_lt_one (add_pos_of_pos_of_nonneg h\u03b5 h\u03b4)] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z / (\u03b5 + \u03b4) < 1\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x z / (\u03b5 + \u03b4) * \u03b4 \u2264 \u03b4 \u2227 dist x z / (\u03b5 + \u03b4) * \u03b5 < \u03b5\n[PROOFSTEP]\nexact \u27e8mul_le_of_le_one_left h\u03b4 h.le, mul_lt_of_lt_one_left h\u03b5 h\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z < \u03b5 + \u03b4\n\u22a2 \u2203 y, dist x y < \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8y, yz, xy\u27e9 := exists_dist_le_lt h\u03b5 h\u03b4 (show dist z x < \u03b4 + \u03b5 by simpa only [dist_comm, add_comm] using h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z < \u03b5 + \u03b4\n\u22a2 dist z x < \u03b4 + \u03b5\n[PROOFSTEP]\nsimpa only [dist_comm, add_comm] using h\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z < \u03b5 + \u03b4\ny : E\nyz : dist z y \u2264 \u03b5\nxy : dist y x < \u03b4\n\u22a2 \u2203 y, dist x y < \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nexact \u27e8y, by simp [dist_comm x y, dist_comm y z, *]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : dist x z < \u03b5 + \u03b4\ny : E\nyz : dist z y \u2264 \u03b5\nxy : dist y x < \u03b4\n\u22a2 dist x y < \u03b4 \u2227 dist y z \u2264 \u03b5\n[PROOFSTEP]\nsimp [dist_comm x y, dist_comm y z, *]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\n\u22a2 \u2203 y, dist x y < \u03b4 \u2227 dist y z < \u03b5\n[PROOFSTEP]\nrefine'\n  (exists_dist_eq x z (div_nonneg h\u03b5.le <| add_nonneg h\u03b5.le h\u03b4.le) (div_nonneg h\u03b4.le <| add_nonneg h\u03b5.le h\u03b4.le) <| by\n        rw [\u2190 add_div, div_self (add_pos h\u03b5 h\u03b4).ne']).imp\n    fun y hy => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\n\u22a2 \u03b5 / (\u03b5 + \u03b4) + \u03b4 / (\u03b5 + \u03b4) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self (add_pos h\u03b5 h\u03b4).ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x y < \u03b4 \u2227 dist y z < \u03b5\n[PROOFSTEP]\nrw [hy.1, hy.2, div_mul_comm, div_mul_comm \u03b5]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z < \u03b5 + \u03b4\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x z / (\u03b5 + \u03b4) * \u03b4 < \u03b4 \u2227 dist x z / (\u03b5 + \u03b4) * \u03b5 < \u03b5\n[PROOFSTEP]\nrw [\u2190 div_lt_one (add_pos h\u03b5 h\u03b4)] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y\u271d z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : dist x z / (\u03b5 + \u03b4) < 1\ny : E\nhy : dist x y = \u03b4 / (\u03b5 + \u03b4) * dist x z \u2227 dist y z = \u03b5 / (\u03b5 + \u03b4) * dist x z\n\u22a2 dist x z / (\u03b5 + \u03b4) * \u03b4 < \u03b4 \u2227 dist x z / (\u03b5 + \u03b4) * \u03b5 < \u03b5\n[PROOFSTEP]\nexact \u27e8mul_lt_of_lt_one_left h\u03b4 h, mul_lt_of_lt_one_left h\u03b5 h\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\n\u22a2 Disjoint (ball x \u03b4) (ball y \u03b5) \u2194 \u03b4 + \u03b5 \u2264 dist x y\n[PROOFSTEP]\nrefine' \u27e8fun h => le_of_not_lt fun hxy => _, ball_disjoint_ball\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : Disjoint (ball x \u03b4) (ball y \u03b5)\nhxy : dist x y < \u03b4 + \u03b5\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm] at hxy \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : Disjoint (ball x \u03b4) (ball y \u03b5)\nhxy : dist x y < \u03b5 + \u03b4\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8z, hxz, hzy\u27e9 := exists_dist_lt_lt h\u03b4 h\u03b5 hxy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : Disjoint (ball x \u03b4) (ball y \u03b5)\nhxy : dist x y < \u03b5 + \u03b4\nz : E\nhxz : dist x z < \u03b4\nhzy : dist z y < \u03b5\n\u22a2 False\n[PROOFSTEP]\nrw [dist_comm] at hxz \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 < \u03b5\nh : Disjoint (ball x \u03b4) (ball y \u03b5)\nhxy : dist x y < \u03b5 + \u03b4\nz : E\nhxz : dist z x < \u03b4\nhzy : dist z y < \u03b5\n\u22a2 False\n[PROOFSTEP]\nexact h.le_bot \u27e8hxz, hzy\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 Disjoint (ball x \u03b4) (closedBall y \u03b5) \u2194 \u03b4 + \u03b5 \u2264 dist x y\n[PROOFSTEP]\nrefine' \u27e8fun h => le_of_not_lt fun hxy => _, ball_disjoint_closedBall\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (ball x \u03b4) (closedBall y \u03b5)\nhxy : dist x y < \u03b4 + \u03b5\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm] at hxy \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (ball x \u03b4) (closedBall y \u03b5)\nhxy : dist x y < \u03b5 + \u03b4\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8z, hxz, hzy\u27e9 := exists_dist_lt_le h\u03b4 h\u03b5 hxy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (ball x \u03b4) (closedBall y \u03b5)\nhxy : dist x y < \u03b5 + \u03b4\nz : E\nhxz : dist x z < \u03b4\nhzy : dist z y \u2264 \u03b5\n\u22a2 False\n[PROOFSTEP]\nrw [dist_comm] at hxz \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (ball x \u03b4) (closedBall y \u03b5)\nhxy : dist x y < \u03b5 + \u03b4\nz : E\nhxz : dist z x < \u03b4\nhzy : dist z y \u2264 \u03b5\n\u22a2 False\n[PROOFSTEP]\nexact h.le_bot \u27e8hxz, hzy\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 < \u03b5\n\u22a2 Disjoint (closedBall x \u03b4) (ball y \u03b5) \u2194 \u03b4 + \u03b5 \u2264 dist x y\n[PROOFSTEP]\nrw [disjoint_comm, disjoint_ball_closedBall_iff h\u03b5 h\u03b4, add_comm, dist_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\n\u22a2 Disjoint (closedBall x \u03b4) (closedBall y \u03b5) \u2194 \u03b4 + \u03b5 < dist x y\n[PROOFSTEP]\nrefine' \u27e8fun h => lt_of_not_ge fun hxy => _, closedBall_disjoint_closedBall\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (closedBall x \u03b4) (closedBall y \u03b5)\nhxy : \u03b4 + \u03b5 \u2265 dist x y\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm] at hxy \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (closedBall x \u03b4) (closedBall y \u03b5)\nhxy : \u03b5 + \u03b4 \u2265 dist x y\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8z, hxz, hzy\u27e9 := exists_dist_le_le h\u03b4 h\u03b5 hxy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (closedBall x \u03b4) (closedBall y \u03b5)\nhxy : \u03b5 + \u03b4 \u2265 dist x y\nz : E\nhxz : dist x z \u2264 \u03b4\nhzy : dist z y \u2264 \u03b5\n\u22a2 False\n[PROOFSTEP]\nrw [dist_comm] at hxz \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 \u2264 \u03b4\nh\u03b5 : 0 \u2264 \u03b5\nh : Disjoint (closedBall x \u03b4) (closedBall y \u03b5)\nhxy : \u03b5 + \u03b4 \u2265 dist x y\nz : E\nhxz : dist z x \u2264 \u03b4\nhzy : dist z y \u2264 \u03b5\n\u22a2 False\n[PROOFSTEP]\nexact h.le_bot \u27e8hxz, hzy\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\n\u22a2 infEdist x (thickening \u03b4 s) = infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nobtain hs | hs := lt_or_le (infEdist x s) (ENNReal.ofReal \u03b4)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : infEdist x s < ENNReal.ofReal \u03b4\n\u22a2 infEdist x (thickening \u03b4 s) = infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrw [infEdist_zero_of_mem, tsub_eq_zero_of_le hs.le]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : infEdist x s < ENNReal.ofReal \u03b4\n\u22a2 x \u2208 thickening \u03b4 s\n[PROOFSTEP]\nexact hs\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\n\u22a2 infEdist x (thickening \u03b4 s) = infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrefine' (tsub_le_iff_right.2 infEdist_le_infEdist_thickening_add).antisymm' _\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\n\u22a2 infEdist x (thickening \u03b4 s) \u2264 infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrefine' le_sub_of_add_le_right ofReal_ne_top _\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 \u2264 infEdist x s\n[PROOFSTEP]\nrefine' le_infEdist.2 fun z hz => le_of_forall_lt' fun r h => _\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : \u211d\u22650\u221e\nh : edist x z < r\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 < r\n[PROOFSTEP]\ncases' r with r\n[GOAL]\ncase inr.none\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nh : edist x z < none\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 < none\n[PROOFSTEP]\nexact add_lt_top.2 \u27e8lt_top_iff_ne_top.2 <| infEdist_ne_top \u27e8z, self_subset_thickening h\u03b4 _ hz\u27e9, ofReal_lt_top\u27e9\n[GOAL]\ncase inr.some\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : edist x z < Option.some r\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 < Option.some r\n[PROOFSTEP]\nhave hr : 0 < \u2191r - \u03b4 := by\n  refine' sub_pos_of_lt _\n  have := hs.trans_lt ((infEdist_le_edist_of_mem hz).trans_lt h)\n  rw [ofReal_eq_coe_nnreal h\u03b4.le, some_eq_coe] at this \n  exact_mod_cast this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : edist x z < Option.some r\n\u22a2 0 < \u2191r - \u03b4\n[PROOFSTEP]\nrefine' sub_pos_of_lt _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : edist x z < Option.some r\n\u22a2 \u03b4 < \u2191r\n[PROOFSTEP]\nhave := hs.trans_lt ((infEdist_le_edist_of_mem hz).trans_lt h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : edist x z < Option.some r\nthis : ENNReal.ofReal \u03b4 < Option.some r\n\u22a2 \u03b4 < \u2191r\n[PROOFSTEP]\nrw [ofReal_eq_coe_nnreal h\u03b4.le, some_eq_coe] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : edist x z < Option.some r\nthis : \u2191{ val := \u03b4, property := (_ : 0 \u2264 \u03b4) } < \u2191r\n\u22a2 \u03b4 < \u2191r\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\ncase inr.some\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : edist x z < Option.some r\nhr : 0 < \u2191r - \u03b4\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 < Option.some r\n[PROOFSTEP]\nrw [some_eq_coe, edist_lt_coe, \u2190 dist_lt_coe, \u2190 add_sub_cancel'_right \u03b4 \u2191r] at h \n[GOAL]\ncase inr.some\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : dist x z < \u03b4 + (\u2191r - \u03b4)\nhr : 0 < \u2191r - \u03b4\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 < Option.some r\n[PROOFSTEP]\nobtain \u27e8y, hxy, hyz\u27e9 := exists_dist_lt_lt hr h\u03b4 h\n[GOAL]\ncase inr.some.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y\u271d z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : dist x z < \u03b4 + (\u2191r - \u03b4)\nhr : 0 < \u2191r - \u03b4\ny : E\nhxy : dist x y < \u2191r - \u03b4\nhyz : dist y z < \u03b4\n\u22a2 infEdist x (thickening \u03b4 s) + ENNReal.ofReal \u03b4 < Option.some r\n[PROOFSTEP]\nrefine'\n  (ENNReal.add_lt_add_right ofReal_ne_top <|\n        infEdist_lt_iff.2 \u27e8_, mem_thickening_iff.2 \u27e8_, hz, hyz\u27e9, edist_lt_ofReal.2 hxy\u27e9).trans_le\n    _\n[GOAL]\ncase inr.some.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y\u271d z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : dist x z < \u03b4 + (\u2191r - \u03b4)\nhr : 0 < \u2191r - \u03b4\ny : E\nhxy : dist x y < \u2191r - \u03b4\nhyz : dist y z < \u03b4\n\u22a2 ENNReal.ofReal (\u2191r - \u03b4) + ENNReal.ofReal \u03b4 \u2264 Option.some r\n[PROOFSTEP]\nrw [\u2190 ofReal_add hr.le h\u03b4.le, sub_add_cancel, ofReal_coe_nnreal]\n[GOAL]\ncase inr.some.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y\u271d z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\nhs : ENNReal.ofReal \u03b4 \u2264 infEdist x s\nz : E\nhz : z \u2208 s\nr : NNReal\nh : dist x z < \u03b4 + (\u2191r - \u03b4)\nhr : 0 < \u2191r - \u03b4\ny : E\nhxy : dist x y < \u2191r - \u03b4\nhyz : dist y z < \u03b4\n\u22a2 \u2191r \u2264 Option.some r\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\n\u22a2 x \u2208 thickening (\u03b5 + \u03b4) s \u2192 x \u2208 thickening \u03b5 (thickening \u03b4 s)\n[PROOFSTEP]\nsimp_rw [mem_thickening_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\n\u22a2 (\u2203 z, z \u2208 s \u2227 dist x z < \u03b5 + \u03b4) \u2192 \u2203 z, (\u2203 z_1, z_1 \u2208 s \u2227 dist z z_1 < \u03b4) \u2227 dist x z < \u03b5\n[PROOFSTEP]\nrintro \u27e8z, hz, hxz\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx z : E\nhz : z \u2208 s\nhxz : dist x z < \u03b5 + \u03b4\n\u22a2 \u2203 z, (\u2203 z_1, z_1 \u2208 s \u2227 dist z z_1 < \u03b4) \u2227 dist x z < \u03b5\n[PROOFSTEP]\nrw [add_comm] at hxz \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx z : E\nhz : z \u2208 s\nhxz : dist x z < \u03b4 + \u03b5\n\u22a2 \u2203 z, (\u2203 z_1, z_1 \u2208 s \u2227 dist z z_1 < \u03b4) \u2227 dist x z < \u03b5\n[PROOFSTEP]\nobtain \u27e8y, hxy, hyz\u27e9 := exists_dist_lt_lt h\u03b5 h\u03b4 hxz\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y\u271d z\u271d : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx z : E\nhz : z \u2208 s\nhxz : dist x z < \u03b4 + \u03b5\ny : E\nhxy : dist x y < \u03b5\nhyz : dist y z < \u03b4\n\u22a2 \u2203 z, (\u2203 z_1, z_1 \u2208 s \u2227 dist z z_1 < \u03b4) \u2227 dist x z < \u03b5\n[PROOFSTEP]\nexact \u27e8y, \u27e8_, hz, hyz\u27e9, hxy\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\n\u22a2 x \u2208 cthickening (\u03b5 + \u03b4) s \u2192 x \u2208 cthickening \u03b5 (thickening \u03b4 s)\n[PROOFSTEP]\nsimp_rw [mem_cthickening_iff, ENNReal.ofReal_add h\u03b5 h\u03b4.le, infEdist_thickening h\u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 < \u03b4\ns : Set E\nx : E\n\u22a2 infEdist x s \u2264 ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4 \u2192 infEdist x s - ENNReal.ofReal \u03b4 \u2264 ENNReal.ofReal \u03b5\n[PROOFSTEP]\nexact tsub_le_iff_right.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b4 : 0 < \u03b4\ns : Set E\n\u22a2 closure (thickening \u03b4 s) = cthickening \u03b4 s\n[PROOFSTEP]\nrw [\u2190 cthickening_zero, cthickening_thickening le_rfl h\u03b4, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4\u271d \u03b5 \u03b4 : \u211d\ns : Set E\nx : E\n\u22a2 infEdist x (cthickening \u03b4 s) = infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nobtain h\u03b4 | h\u03b4 := le_or_lt \u03b4 0\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4\u271d \u03b5 \u03b4 : \u211d\ns : Set E\nx : E\nh\u03b4 : \u03b4 \u2264 0\n\u22a2 infEdist x (cthickening \u03b4 s) = infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrw [cthickening_of_nonpos h\u03b4, infEdist_closure, ofReal_of_nonpos h\u03b4, tsub_zero]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4\u271d \u03b5 \u03b4 : \u211d\ns : Set E\nx : E\nh\u03b4 : 0 < \u03b4\n\u22a2 infEdist x (cthickening \u03b4 s) = infEdist x s - ENNReal.ofReal \u03b4\n[PROOFSTEP]\nrw [\u2190 closure_thickening h\u03b4, infEdist_closure, infEdist_thickening h\u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 \u2264 \u03b4\ns : Set E\n\u22a2 thickening \u03b5 (cthickening \u03b4 s) = thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nobtain rfl | h\u03b4 := h\u03b4.eq_or_lt\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ns : Set E\nh\u03b4 : 0 \u2264 0\n\u22a2 thickening \u03b5 (cthickening 0 s) = thickening (\u03b5 + 0) s\n[PROOFSTEP]\nrw [cthickening_zero, thickening_closure, add_zero]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4\u271d : 0 \u2264 \u03b4\ns : Set E\nh\u03b4 : 0 < \u03b4\n\u22a2 thickening \u03b5 (cthickening \u03b4 s) = thickening (\u03b5 + \u03b4) s\n[PROOFSTEP]\nrw [\u2190 closure_thickening h\u03b4, thickening_closure, thickening_thickening h\u03b5 h\u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\ns : Set E\nx : E\n\u22a2 x \u2208 cthickening (\u03b5 + \u03b4) s \u2192 x \u2208 cthickening \u03b5 (cthickening \u03b4 s)\n[PROOFSTEP]\nsimp_rw [mem_cthickening_iff, ENNReal.ofReal_add h\u03b5 h\u03b4, infEdist_cthickening]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\ns : Set E\nx : E\n\u22a2 infEdist x s \u2264 ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4 \u2192 infEdist x s - ENNReal.ofReal \u03b4 \u2264 ENNReal.ofReal \u03b5\n[PROOFSTEP]\nexact tsub_le_iff_right.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\nx : E\n\u22a2 thickening \u03b5 (Metric.ball x \u03b4) = Metric.ball x (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [\u2190 thickening_singleton, thickening_thickening h\u03b5 h\u03b4, thickening_singleton]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 thickening \u03b5 (Metric.closedBall x \u03b4) = Metric.ball x (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [\u2190 cthickening_singleton _ h\u03b4, thickening_cthickening h\u03b5 h\u03b4, thickening_singleton]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 < \u03b4\nx : E\n\u22a2 cthickening \u03b5 (Metric.ball x \u03b4) = Metric.closedBall x (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [\u2190 thickening_singleton, cthickening_thickening h\u03b5 h\u03b4, cthickening_singleton _ (add_nonneg h\u03b5 h\u03b4.le)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 cthickening \u03b5 (Metric.closedBall x \u03b4) = Metric.closedBall x (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [\u2190 cthickening_singleton _ h\u03b4, cthickening_cthickening h\u03b5 h\u03b4, cthickening_singleton _ (add_nonneg h\u03b5 h\u03b4)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\na b : E\n\u22a2 Metric.ball a \u03b5 + Metric.ball b \u03b4 = Metric.ball (a + b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [ball_add, thickening_ball h\u03b5 h\u03b4 b, Metric.vadd_ball, vadd_eq_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 < \u03b4\na b : E\n\u22a2 Metric.ball a \u03b5 - Metric.ball b \u03b4 = Metric.ball (a - b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nsimp_rw [sub_eq_add_neg, neg_ball, ball_add_ball h\u03b5 h\u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 \u2264 \u03b4\na b : E\n\u22a2 Metric.ball a \u03b5 + Metric.closedBall b \u03b4 = Metric.ball (a + b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [ball_add, thickening_closedBall h\u03b5 h\u03b4 b, Metric.vadd_ball, vadd_eq_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b4 : 0 \u2264 \u03b4\na b : E\n\u22a2 Metric.ball a \u03b5 - Metric.closedBall b \u03b4 = Metric.ball (a - b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nsimp_rw [sub_eq_add_neg, neg_closedBall, ball_add_closedBall h\u03b5 h\u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 < \u03b4\na b : E\n\u22a2 Metric.closedBall a \u03b5 + Metric.ball b \u03b4 = Metric.ball (a + b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [add_comm, ball_add_closedBall h\u03b4 h\u03b5 b, add_comm, add_comm \u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 < \u03b4\na b : E\n\u22a2 Metric.closedBall a \u03b5 - Metric.ball b \u03b4 = Metric.ball (a - b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nsimp_rw [sub_eq_add_neg, neg_ball, closedBall_add_ball h\u03b5 h\u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\ninst\u271d : ProperSpace E\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\na b : E\n\u22a2 Metric.closedBall a \u03b5 + Metric.closedBall b \u03b4 = Metric.closedBall (a + b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [(isCompact_closedBall _ _).add_closedBall h\u03b4 b, cthickening_closedBall h\u03b4 h\u03b5 a, Metric.vadd_closedBall, vadd_eq_add,\n  add_comm, add_comm \u03b4]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\nx y z : E\n\u03b4 \u03b5 : \u211d\ninst\u271d : ProperSpace E\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 \u2264 \u03b4\na b : E\n\u22a2 Metric.closedBall a \u03b5 - Metric.closedBall b \u03b4 = Metric.closedBall (a - b) (\u03b5 + \u03b4)\n[PROOFSTEP]\nrw [sub_eq_add_neg, neg_closedBall, closedBall_add_closedBall h\u03b5 h\u03b4, sub_eq_add_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 c \u2022 closedBall x r = closedBall (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (rfl | hc)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 0 \u2022 closedBall x r = closedBall (0 \u2022 x) (\u20160\u2016 * r)\n[PROOFSTEP]\nsimp [hr, zero_smul_set, Set.singleton_zero, \u2190 nonempty_closedBall]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\nhc : c \u2260 0\n\u22a2 c \u2022 closedBall x r = closedBall (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nexact smul_closedBall' hc x r\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\n\u22a2 c \u2022 closedBall 0 1 = closedBall 0 \u2016c\u2016\n[PROOFSTEP]\nrw [smul_closedBall _ _ zero_le_one, smul_zero, mul_one]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 r \u2022 closedBall 0 1 = closedBall 0 r\n[PROOFSTEP]\nrw [smul_closedUnitBall, Real.norm_of_nonneg hr]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\n\u22a2 Set.Nonempty (sphere x r) \u2194 0 \u2264 r\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := exists_ne x\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\n\u22a2 Set.Nonempty (sphere x r) \u2194 0 \u2264 r\n[PROOFSTEP]\nrefine' \u27e8fun h => nonempty_closedBall.1 (h.mono sphere_subset_closedBall), fun hr => \u27e8r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x, _\u27e9\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\n\u22a2 r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x \u2208 sphere x r\n[PROOFSTEP]\nhave : \u2016y - x\u2016 \u2260 0 := by simpa [sub_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\n\u22a2 \u2016y - x\u2016 \u2260 0\n[PROOFSTEP]\nsimpa [sub_eq_zero]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\nthis : \u2016y - x\u2016 \u2260 0\n\u22a2 r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x \u2208 sphere x r\n[PROOFSTEP]\nsimp [norm_smul, this, Real.norm_of_nonneg hr]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\nthis : \u2016y - x\u2016 \u2260 0\n\u22a2 |r| * (\u2016y - x\u2016\u207b\u00b9 * \u2016y - x\u2016) = r\n[PROOFSTEP]\nrw [inv_mul_cancel this, mul_one, abs_eq_self.mpr hr]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 c \u2022 sphere x r = sphere (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (rfl | hc)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 0 \u2022 sphere x r = sphere (0 \u2022 x) (\u20160\u2016 * r)\n[PROOFSTEP]\nsimp [zero_smul_set, Set.singleton_zero, hr]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\nhc : c \u2260 0\n\u22a2 c \u2022 sphere x r = sphere (c \u2022 x) (\u2016c\u2016 * r)\n[PROOFSTEP]\nexact smul_sphere' hc x r\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nr : \u211d\nhr : 0 < r\nx : E\n\u22a2 x +\u1d65 r \u2022 ball 0 1 = ball x r\n[PROOFSTEP]\nrw [smul_unitBall_of_pos hr, vadd_ball_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nr : \u211d\nhr : 0 \u2264 r\nx : E\n\u22a2 x +\u1d65 r \u2022 closedBall 0 1 = closedBall x r\n[PROOFSTEP]\nrw [smul_closedUnitBall, Real.norm_of_nonneg hr, vadd_closedBall_zero]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Pointwise", "llama_tokens": 28838, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950868503681, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.5526624442428545}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b3\nf : \u03b3 \u2192 \u03b1\ng : \u03b3 \u2192 \u03b2\nthis : DecidableEq \u03b3\n\u22a2 \u2200 \u2983a : \u03b3\u2984 {s : Finset \u03b3},\n    \u00aca \u2208 s \u2192\n      (\u220f x in s, f x, \u220f x in s, g x) = \u220f x in s, (f x, g x) \u2192\n        (\u220f x in insert a s, f x, \u220f x in insert a s, g x) = \u220f x in insert a s, (f x, g x)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Prod.ext_iff]\n[GOAL]\nI : Type u_1\ninst\u271d\u00b2 : DecidableEq I\nZ : I \u2192 Type u_2\ninst\u271d\u00b9 : (i : I) \u2192 CommMonoid (Z i)\ninst\u271d : Fintype I\nf : (i : I) \u2192 Z i\n\u22a2 \u220f i : I, Pi.mulSingle i (f i) = f\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nI : Type u_1\ninst\u271d\u00b2 : DecidableEq I\nZ : I \u2192 Type u_2\ninst\u271d\u00b9 : (i : I) \u2192 CommMonoid (Z i)\ninst\u271d : Fintype I\nf : (i : I) \u2192 Z i\na : I\n\u22a2 Finset.prod univ (fun i => Pi.mulSingle i (f i)) a = f a\n[PROOFSTEP]\nsimp\n[GOAL]\nI : Type u_1\ninst\u271d\u00b3 : DecidableEq I\nZ : I \u2192 Type u_2\ninst\u271d\u00b2 : (i : I) \u2192 CommMonoid (Z i)\ninst\u271d\u00b9 : Finite I\nG : Type u_3\ninst\u271d : CommMonoid G\ng h : ((i : I) \u2192 Z i) \u2192* G\nH : \u2200 (i : I) (x : Z i), \u2191g (Pi.mulSingle i x) = \u2191h (Pi.mulSingle i x)\n\u22a2 g = h\n[PROOFSTEP]\ncases nonempty_fintype I\n[GOAL]\ncase intro\nI : Type u_1\ninst\u271d\u00b3 : DecidableEq I\nZ : I \u2192 Type u_2\ninst\u271d\u00b2 : (i : I) \u2192 CommMonoid (Z i)\ninst\u271d\u00b9 : Finite I\nG : Type u_3\ninst\u271d : CommMonoid G\ng h : ((i : I) \u2192 Z i) \u2192* G\nH : \u2200 (i : I) (x : Z i), \u2191g (Pi.mulSingle i x) = \u2191h (Pi.mulSingle i x)\nval\u271d : Fintype I\n\u22a2 g = h\n[PROOFSTEP]\next k\n[GOAL]\ncase intro.h\nI : Type u_1\ninst\u271d\u00b3 : DecidableEq I\nZ : I \u2192 Type u_2\ninst\u271d\u00b2 : (i : I) \u2192 CommMonoid (Z i)\ninst\u271d\u00b9 : Finite I\nG : Type u_3\ninst\u271d : CommMonoid G\ng h : ((i : I) \u2192 Z i) \u2192* G\nH : \u2200 (i : I) (x : Z i), \u2191g (Pi.mulSingle i x) = \u2191h (Pi.mulSingle i x)\nval\u271d : Fintype I\nk : (i : I) \u2192 Z i\n\u22a2 \u2191g k = \u2191h k\n[PROOFSTEP]\nrw [\u2190 Finset.univ_prod_mulSingle k, g.map_prod, h.map_prod]\n[GOAL]\ncase intro.h\nI : Type u_1\ninst\u271d\u00b3 : DecidableEq I\nZ : I \u2192 Type u_2\ninst\u271d\u00b2 : (i : I) \u2192 CommMonoid (Z i)\ninst\u271d\u00b9 : Finite I\nG : Type u_3\ninst\u271d : CommMonoid G\ng h : ((i : I) \u2192 Z i) \u2192* G\nH : \u2200 (i : I) (x : Z i), \u2191g (Pi.mulSingle i x) = \u2191h (Pi.mulSingle i x)\nval\u271d : Fintype I\nk : (i : I) \u2192 Z i\n\u22a2 \u220f x : I, \u2191g (Pi.mulSingle x (k x)) = \u220f x : I, \u2191h (Pi.mulSingle x (k x))\n[PROOFSTEP]\nsimp only [H]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.Pi", "llama_tokens": 1200, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624688140726, "lm_q2_score": 0.7057850278370112, "lm_q1q2_score": 0.5523914523389242}}
{"text": "[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 x ^ (p / 2) = 1\n[PROOFSTEP]\nby_cases hc : p = 2\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : p = 2\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 x ^ (p / 2) = 1\n[PROOFSTEP]\nsubst hc\n[GOAL]\ncase pos\ninst\u271d : Fact (Nat.Prime 2)\nx : (ZMod 2)\u02e3\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 x ^ (2 / 2) = 1\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton, exists_const]\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 x ^ (p / 2) = 1\n[PROOFSTEP]\nhave h\u2080 := FiniteField.unit_isSquare_iff (by rwa [ringChar_zmod_n]) x\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\n\u22a2 ringChar (ZMod p) \u2260 2\n[PROOFSTEP]\nrwa [ringChar_zmod_n]\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\nh\u2080 : IsSquare x \u2194 x ^ (Fintype.card (ZMod p) / 2) = 1\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 x ^ (p / 2) = 1\n[PROOFSTEP]\nhave hs : (\u2203 y : (ZMod p)\u02e3, y ^ 2 = x) \u2194 IsSquare x :=\n  by\n  rw [isSquare_iff_exists_sq x]\n  simp_rw [eq_comm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\nh\u2080 : IsSquare x \u2194 x ^ (Fintype.card (ZMod p) / 2) = 1\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 IsSquare x\n[PROOFSTEP]\nrw [isSquare_iff_exists_sq x]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\nh\u2080 : IsSquare x \u2194 x ^ (Fintype.card (ZMod p) / 2) = 1\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 \u2203 c, x = c ^ 2\n[PROOFSTEP]\nsimp_rw [eq_comm]\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\nh\u2080 : IsSquare x \u2194 x ^ (Fintype.card (ZMod p) / 2) = 1\nhs : (\u2203 y, y ^ 2 = x) \u2194 IsSquare x\n\u22a2 (\u2203 y, y ^ 2 = x) \u2194 x ^ (p / 2) = 1\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx : (ZMod p)\u02e3\nhc : \u00acp = 2\nh\u2080 : IsSquare x \u2194 x ^ (Fintype.card (ZMod p) / 2) = 1\nhs : (\u2203 y, y ^ 2 = x) \u2194 IsSquare x\n\u22a2 IsSquare x \u2194 x ^ (p / 2) = 1\n[PROOFSTEP]\nrwa [card p] at h\u2080 \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 IsSquare a \u2194 a ^ (p / 2) = 1\n[PROOFSTEP]\napply (iff_congr _ (by simp [Units.ext_iff])).mp (euler_criterion_units p (Units.mk0 a ha))\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 Units.mk0 a ha ^ (p / 2) = 1 \u2194 a ^ (p / 2) = 1\n[PROOFSTEP]\nsimp [Units.ext_iff]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 (\u2203 y, y ^ 2 = Units.mk0 a ha) \u2194 IsSquare a\n[PROOFSTEP]\nsimp only [Units.ext_iff, sq, Units.val_mk0, Units.val_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 (\u2203 y, \u2191y * \u2191y = a) \u2194 IsSquare a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 (\u2203 y, \u2191y * \u2191y = a) \u2192 IsSquare a\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9\n[GOAL]\ncase mp.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\ny : (ZMod p)\u02e3\nhy : \u2191y * \u2191y = a\n\u22a2 IsSquare a\n[PROOFSTEP]\nexact \u27e8y, hy.symm\u27e9\n[GOAL]\ncase mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 IsSquare a \u2192 \u2203 y, \u2191y * \u2191y = a\n[PROOFSTEP]\nrintro \u27e8y, rfl\u27e9\n[GOAL]\ncase mpr.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\ny : ZMod p\nha : y * y \u2260 0\n\u22a2 \u2203 y_1, \u2191y_1 * \u2191y_1 = y * y\n[PROOFSTEP]\nhave hy : y \u2260 0 := by\n  rintro rfl\n  simp [zero_pow, mul_zero, ne_eq, not_true] at ha \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\ny : ZMod p\nha : y * y \u2260 0\n\u22a2 y \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nha : 0 * 0 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp [zero_pow, mul_zero, ne_eq, not_true] at ha \n[GOAL]\ncase mpr.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\ny : ZMod p\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 \u2203 y_1, \u2191y_1 * \u2191y_1 = y * y\n[PROOFSTEP]\nrefine' \u27e8Units.mk0 y hy, _\u27e9\n[GOAL]\ncase mpr.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\ny : ZMod p\nha : y * y \u2260 0\nhy : y \u2260 0\n\u22a2 \u2191(Units.mk0 y hy) * \u2191(Units.mk0 y hy) = y * y\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 a ^ (p / 2) = 1 \u2228 a ^ (p / 2) = -1\n[PROOFSTEP]\ncases' Prime.eq_two_or_odd (@Fact.out p.Prime _) with hp2 hp_odd\n[GOAL]\ncase inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\nhp2 : p = 2\n\u22a2 a ^ (p / 2) = 1 \u2228 a ^ (p / 2) = -1\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase inl\ninst\u271d : Fact (Nat.Prime 2)\na : ZMod 2\nha : a \u2260 0\n\u22a2 a ^ (2 / 2) = 1 \u2228 a ^ (2 / 2) = -1\n[PROOFSTEP]\nrevert a ha\n[GOAL]\ncase inl\ninst\u271d : Fact (Nat.Prime 2)\n\u22a2 \u2200 {a : ZMod 2}, a \u2260 0 \u2192 a ^ (2 / 2) = 1 \u2228 a ^ (2 / 2) = -1\n[PROOFSTEP]\nintro a\n[GOAL]\ncase inl\ninst\u271d : Fact (Nat.Prime 2)\na : ZMod 2\n\u22a2 a \u2260 0 \u2192 a ^ (2 / 2) = 1 \u2228 a ^ (2 / 2) = -1\n[PROOFSTEP]\nfin_cases a\n[GOAL]\ncase inl.head\ninst\u271d : Fact (Nat.Prime 2)\n\u22a2 { val := 0, isLt := (_ : 0 < 1 + 1) } \u2260 0 \u2192\n    { val := 0, isLt := (_ : 0 < 1 + 1) } ^ (2 / 2) = 1 \u2228 { val := 0, isLt := (_ : 0 < 1 + 1) } ^ (2 / 2) = -1\ncase inl.tail.head\ninst\u271d : Fact (Nat.Prime 2)\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } \u2260 0 \u2192\n    { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } ^ (2 / 2) = 1 \u2228\n      { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } ^ (2 / 2) = -1\n[PROOFSTEP]\ntauto\n[GOAL]\ncase inl.tail.head\ninst\u271d : Fact (Nat.Prime 2)\n\u22a2 { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } \u2260 0 \u2192\n    { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } ^ (2 / 2) = 1 \u2228\n      { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } ^ (2 / 2) = -1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\nhp_odd : p % 2 = 1\n\u22a2 a ^ (p / 2) = 1 \u2228 a ^ (p / 2) = -1\n[PROOFSTEP]\nrw [\u2190 mul_self_eq_one_iff, \u2190 pow_add, \u2190 two_mul, two_mul_odd_div_two hp_odd]\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\nhp_odd : p % 2 = 1\n\u22a2 a ^ (p - 1) = 1\n[PROOFSTEP]\nexact pow_card_sub_one_eq_one ha\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\n\u22a2 \u2191(legendreSym p a) = \u2191a ^ (p / 2)\n[PROOFSTEP]\ncases' eq_or_ne (ringChar (ZMod p)) 2 with hc hc\n[GOAL]\ncase inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) = 2\n\u22a2 \u2191(legendreSym p a) = \u2191a ^ (p / 2)\n[PROOFSTEP]\nby_cases ha : (a : ZMod p) = 0\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) = 2\nha : \u2191a = 0\n\u22a2 \u2191(legendreSym p a) = \u2191a ^ (p / 2)\n[PROOFSTEP]\nrw [legendreSym, ha, quadraticChar_zero, zero_pow (Nat.div_pos (@Fact.out p.Prime).two_le (succ_pos 1))]\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) = 2\nha : \u2191a = 0\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) = 2\nha : \u00ac\u2191a = 0\n\u22a2 \u2191(legendreSym p a) = \u2191a ^ (p / 2)\n[PROOFSTEP]\nhave := (ringChar_zmod_n p).symm.trans hc\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) = 2\nha : \u00ac\u2191a = 0\nthis : p = 2\n\u22a2 \u2191(legendreSym p a) = \u2191a ^ (p / 2)\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase neg\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\nha : \u00ac\u2191a = 0\n\u22a2 \u2191(legendreSym 2 a) = \u2191a ^ (2 / 2)\n[PROOFSTEP]\nrw [legendreSym, quadraticChar_eq_one_of_char_two hc ha]\n[GOAL]\ncase neg\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\nha : \u00ac\u2191a = 0\n\u22a2 \u21911 = \u2191a ^ (2 / 2)\n[PROOFSTEP]\nrevert ha\n[GOAL]\ncase neg\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\n\u22a2 \u00ac\u2191a = 0 \u2192 \u21911 = \u2191a ^ (2 / 2)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase neg\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\n\u22a2 \u00ac\u2191a = 0 \u2192 1 = \u2191a ^ (2 / 2)\n[PROOFSTEP]\ngeneralize (a : ZMod 2) = b\n[GOAL]\ncase neg\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\nb : ZMod 2\n\u22a2 \u00acb = 0 \u2192 1 = b ^ (2 / 2)\n[PROOFSTEP]\nfin_cases b\n[GOAL]\ncase neg.head\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\n\u22a2 \u00ac{ val := 0, isLt := (_ : 0 < 1 + 1) } = 0 \u2192 1 = { val := 0, isLt := (_ : 0 < 1 + 1) } ^ (2 / 2)\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg.tail.head\na : \u2124\ninst\u271d : Fact (Nat.Prime 2)\nhc : ringChar (ZMod 2) = 2\n\u22a2 \u00ac{ val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } = 0 \u2192\n    1 = { val := 1, isLt := (_ : (fun a => a < 1 + 1) 1) } ^ (2 / 2)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) \u2260 2\n\u22a2 \u2191(legendreSym p a) = \u2191a ^ (p / 2)\n[PROOFSTEP]\nconvert quadraticChar_eq_pow_of_char_ne_two' hc (a : ZMod p)\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) \u2260 2\n\u22a2 p = Fintype.card (ZMod p)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) \u2260 2\n\u22a2 p = Fintype.card (ZMod p)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_3.h.e'_6.h.e'_5\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nhc : ringChar (ZMod p) \u2260 2\n\u22a2 p = Fintype.card (ZMod p)\n[PROOFSTEP]\nexact (card p).symm\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 legendreSym p 0 = 0\n[PROOFSTEP]\nrw [legendreSym, Int.cast_zero, MulChar.map_zero]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 legendreSym p 1 = 1\n[PROOFSTEP]\nrw [legendreSym, Int.cast_one, MulChar.map_one]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na b : \u2124\n\u22a2 legendreSym p (a * b) = legendreSym p a * legendreSym p b\n[PROOFSTEP]\nsimp [legendreSym, Int.cast_mul, map_mul, quadraticCharFun_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\n\u22a2 legendreSym p (a ^ 2) = 1\n[PROOFSTEP]\ndsimp only [legendreSym]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\n\u22a2 \u2191(quadraticChar (ZMod p)) \u2191(a ^ 2) = 1\n[PROOFSTEP]\nrw [Int.cast_pow]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\n\u22a2 \u2191(quadraticChar (ZMod p)) (\u2191a ^ 2) = 1\n[PROOFSTEP]\nexact quadraticChar_sq_one' ha\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\n\u22a2 legendreSym p a = legendreSym p (a % \u2191p)\n[PROOFSTEP]\nsimp only [legendreSym, int_cast_mod]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2115\nha0 : \u2191a \u2260 0\n\u22a2 legendreSym p \u2191a = 1 \u2194 IsSquare \u2191a\n[PROOFSTEP]\nrw [eq_one_iff]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2115\nha0 : \u2191a \u2260 0\n\u22a2 IsSquare \u2191\u2191a \u2194 IsSquare \u2191a\ncase ha0 p : \u2115 inst\u271d : Fact (Nat.Prime p) a : \u2115 ha0 : \u2191a \u2260 0 \u22a2 \u2191\u2191a \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase ha0\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2115\nha0 : \u2191a \u2260 0\n\u22a2 \u2191\u2191a \u2260 0\n[PROOFSTEP]\nexact_mod_cast ha0\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2115\n\u22a2 legendreSym p \u2191a = -1 \u2194 \u00acIsSquare \u2191a\n[PROOFSTEP]\nrw [eq_neg_one_iff]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2115\n\u22a2 \u00acIsSquare \u2191\u2191a \u2194 \u00acIsSquare \u2191a\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhy : y \u2260 0\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\n\u22a2 legendreSym p a = 1\n[PROOFSTEP]\napply_fun (\u00b7 * y\u207b\u00b9 ^ 2) at hxy \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhy : y \u2260 0\nhxy : (x ^ 2 - \u2191a * y ^ 2) * y\u207b\u00b9 ^ 2 = 0 * y\u207b\u00b9 ^ 2\n\u22a2 legendreSym p a = 1\n[PROOFSTEP]\nsimp only [zero_mul] at hxy \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhy : y \u2260 0\nhxy : (x ^ 2 - \u2191a * y ^ 2) * y\u207b\u00b9 ^ 2 = 0\n\u22a2 legendreSym p a = 1\n[PROOFSTEP]\nrw [(by ring : (x ^ 2 - \u2191a * y ^ 2) * y\u207b\u00b9 ^ 2 = (x * y\u207b\u00b9) ^ 2 - a * (y * y\u207b\u00b9) ^ 2), mul_inv_cancel hy, one_pow, mul_one,\n  sub_eq_zero, pow_two] at hxy \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhy : y \u2260 0\nhxy : (x ^ 2 - \u2191a * y ^ 2) * y\u207b\u00b9 ^ 2 = 0\n\u22a2 (x ^ 2 - \u2191a * y ^ 2) * y\u207b\u00b9 ^ 2 = (x * y\u207b\u00b9) ^ 2 - \u2191a * (y * y\u207b\u00b9) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhy : y \u2260 0\nhxy : x * y\u207b\u00b9 * (x * y\u207b\u00b9) = \u2191a\n\u22a2 legendreSym p a = 1\n[PROOFSTEP]\nexact (eq_one_iff p ha).mpr \u27e8x * y\u207b\u00b9, hxy.symm\u27e9\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhx : x \u2260 0\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\n\u22a2 legendreSym p a = 1\n[PROOFSTEP]\nhaveI hy : y \u2260 0 := by\n  rintro rfl\n  rw [zero_pow' 2 (by norm_num), mul_zero, sub_zero, pow_eq_zero_iff (by norm_num : 0 < 2)] at hxy \n  exact hx hxy\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhx : x \u2260 0\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\n\u22a2 y \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx : ZMod p\nhx : x \u2260 0\nhxy : x ^ 2 - \u2191a * 0 ^ 2 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [zero_pow' 2 (by norm_num), mul_zero, sub_zero, pow_eq_zero_iff (by norm_num : 0 < 2)] at hxy \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx : ZMod p\nhx : x \u2260 0\nhxy : x ^ 2 - \u2191a * 0 ^ 2 = 0\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx : ZMod p\nhx : x \u2260 0\nhxy : x ^ 2 = 0\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx : ZMod p\nhx : x \u2260 0\nhxy : x = 0\n\u22a2 False\n[PROOFSTEP]\nexact hx hxy\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nha : \u2191a \u2260 0\nx y : ZMod p\nhx : x \u2260 0\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nhy : y \u2260 0\n\u22a2 legendreSym p a = 1\n[PROOFSTEP]\nexact eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\n\u22a2 x = 0 \u2227 y = 0\n[PROOFSTEP]\nhave ha : (a : ZMod p) \u2260 0 := by\n  intro hf\n  rw [(eq_zero_iff p a).mpr hf] at h \n  exact Int.zero_ne_neg_of_ne zero_ne_one h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\n\u22a2 \u2191a \u2260 0\n[PROOFSTEP]\nintro hf\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nhf : \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [(eq_zero_iff p a).mpr hf] at h \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : 0 = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nhf : \u2191a = 0\n\u22a2 False\n[PROOFSTEP]\nexact Int.zero_ne_neg_of_ne zero_ne_one h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nha : \u2191a \u2260 0\n\u22a2 x = 0 \u2227 y = 0\n[PROOFSTEP]\nby_contra hf\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nha : \u2191a \u2260 0\nhf : \u00ac(x = 0 \u2227 y = 0)\n\u22a2 False\n[PROOFSTEP]\ncases' imp_iff_or_not.mp (not_and'.mp hf) with hx hy\n[GOAL]\ncase inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nha : \u2191a \u2260 0\nhf : \u00ac(x = 0 \u2227 y = 0)\nhx : \u00acx = 0\n\u22a2 False\n[PROOFSTEP]\nrw [eq_one_of_sq_sub_mul_sq_eq_zero' ha hx hxy, eq_neg_self_iff] at h \n[GOAL]\ncase inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : 1 = 0\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nha : \u2191a \u2260 0\nhf : \u00ac(x = 0 \u2227 y = 0)\nhx : \u00acx = 0\n\u22a2 False\n[PROOFSTEP]\nexact one_ne_zero h\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nha : \u2191a \u2260 0\nhf : \u00ac(x = 0 \u2227 y = 0)\nhy : \u00acy = 0\n\u22a2 False\n[PROOFSTEP]\nrw [eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy, eq_neg_self_iff] at h \n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : 1 = 0\nx y : ZMod p\nhxy : x ^ 2 - \u2191a * y ^ 2 = 0\nha : \u2191a \u2260 0\nhf : \u00ac(x = 0 \u2227 y = 0)\nhy : \u00acy = 0\n\u22a2 False\n[PROOFSTEP]\nexact one_ne_zero 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]\nsimp_rw [\u2190 ZMod.int_cast_zmod_eq_zero_iff_dvd] at hxy \u22a2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : \u2124\nhxy : \u2191(x ^ 2 - a * y ^ 2) = 0\n\u22a2 \u2191x = 0 \u2227 \u2191y = 0\n[PROOFSTEP]\npush_cast at hxy \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : \u2124\nhxy : \u2191x ^ 2 - \u2191a * \u2191y ^ 2 = 0\n\u22a2 \u2191x = 0 \u2227 \u2191y = 0\n[PROOFSTEP]\nexact eq_zero_mod_of_eq_neg_one h hxy\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 legendreSym p (-1) = \u2191\u03c7\u2084 \u2191p\n[PROOFSTEP]\nsimp only [legendreSym, card p, quadraticChar_neg_one ((ringChar_zmod_n p).substr hp), Int.cast_neg, Int.cast_one]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 IsSquare (-1) \u2194 p % 4 \u2260 3\n[PROOFSTEP]\nrw [FiniteField.isSquare_neg_one_iff, card p]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx y : ZMod p\nhy : y \u2260 0\nhxy : x ^ 2 = -y ^ 2\n\u22a2 (x / y) ^ 2 = -1\n[PROOFSTEP]\napply_fun fun z => z / y ^ 2 at hxy \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx y : ZMod p\nhy : y \u2260 0\nhxy : x ^ 2 / y ^ 2 = -y ^ 2 / y ^ 2\n\u22a2 (x / y) ^ 2 = -1\n[PROOFSTEP]\nrwa [neg_div, \u2190 div_pow, \u2190 div_pow, div_self hy, one_pow] at hxy \n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LegendreSymbol.Basic", "llama_tokens": 9193, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677506936878, "lm_q2_score": 0.651354857898194, "lm_q1q2_score": 0.5523279137553382}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 choose n 0 = 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 choose zero 0 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 choose (succ n\u271d) 0 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nn k : \u2115\nhk : n + 1 < k + 1\n\u22a2 choose (n + 1) (k + 1) = 0\n[PROOFSTEP]\nhave hnk : n < k := lt_of_succ_lt_succ hk\n[GOAL]\nn k : \u2115\nhk : n + 1 < k + 1\nhnk : n < k\n\u22a2 choose (n + 1) (k + 1) = 0\n[PROOFSTEP]\nhave hnk1 : n < k + 1 := lt_of_succ_lt hk\n[GOAL]\nn k : \u2115\nhk : n + 1 < k + 1\nhnk : n < k\nhnk1 : n < k + 1\n\u22a2 choose (n + 1) (k + 1) = 0\n[PROOFSTEP]\nrw [choose_succ_succ, choose_eq_zero_of_lt hnk, choose_eq_zero_of_lt hnk1]\n[GOAL]\nn : \u2115\n\u22a2 choose n n = 1\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u22a2 choose zero zero = 1\n[PROOFSTEP]\nsimp [*, choose, choose_eq_zero_of_lt (lt_succ_self _)]\n[GOAL]\ncase succ\nn\u271d : \u2115\nn_ih\u271d : choose n\u271d n\u271d = 1\n\u22a2 choose (succ n\u271d) (succ n\u271d) = 1\n[PROOFSTEP]\nsimp [*, choose, choose_eq_zero_of_lt (lt_succ_self _)]\n[GOAL]\nn : \u2115\n\u22a2 choose n 1 = n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u22a2 choose zero 1 = zero\n[PROOFSTEP]\nsimp [*, choose, add_comm]\n[GOAL]\ncase succ\nn\u271d : \u2115\nn_ih\u271d : choose n\u271d 1 = n\u271d\n\u22a2 choose (succ n\u271d) 1 = succ n\u271d\n[PROOFSTEP]\nsimp [*, choose, add_comm]\n[GOAL]\nn : \u2115\n\u22a2 (n + 1) * (n + 1 - 1) / 2 = n * (n - 1) / 2 + n\n[PROOFSTEP]\nrw [\u2190 add_mul_div_left, mul_comm 2 n, \u2190 mul_add, add_tsub_cancel_right, mul_comm]\n[GOAL]\nn : \u2115\n\u22a2 n * (n + 1) / 2 = n * (n - 1 + 2) / 2\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 zero * (zero + 1) / 2 = zero * (zero - 1 + 2) / 2\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 succ n\u271d * (succ n\u271d + 1) / 2 = succ n\u271d * (succ n\u271d - 1 + 2) / 2\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H\nn : \u2115\n\u22a2 0 < 2\n[PROOFSTEP]\napply zero_lt_succ\n[GOAL]\nn : \u2115\n\u22a2 choose n 2 = n * (n - 1) / 2\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u22a2 choose zero 2 = zero * (zero - 1) / 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\nih : choose n 2 = n * (n - 1) / 2\n\u22a2 choose (succ n) 2 = succ n * (succ n - 1) / 2\n[PROOFSTEP]\nrw [triangle_succ n, choose, ih]\n[GOAL]\ncase succ\nn : \u2115\nih : choose n 2 = n * (n - 1) / 2\n\u22a2 choose n 1 + n * (n - 1) / 2 = n * (n - 1) / 2 + n\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\nx\u271d : \u2115\nhk : x\u271d \u2264 0\n\u22a2 0 < choose 0 x\u271d\n[PROOFSTEP]\nrw [Nat.eq_zero_of_le_zero hk]\n[GOAL]\nx\u271d : \u2115\nhk : x\u271d \u2264 0\n\u22a2 0 < choose 0 0\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\nx\u271d : 0 \u2264 n + 1\n\u22a2 0 < choose (n + 1) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn k : \u2115\nhk : k + 1 \u2264 n + 1\n\u22a2 0 < choose (n + 1) (k + 1)\n[PROOFSTEP]\nrw [choose_succ_succ]\n[GOAL]\nn k : \u2115\nhk : k + 1 \u2264 n + 1\n\u22a2 0 < choose n k + choose n (succ k)\n[PROOFSTEP]\nexact add_pos_of_pos_of_nonneg (choose_pos (le_of_succ_le_succ hk)) (Nat.zero_le _)\n[GOAL]\n\u22a2 succ 0 * choose 0 0 = choose (succ 0) (succ 0) * succ 0\n[PROOFSTEP]\ndecide\n[GOAL]\nk : \u2115\n\u22a2 succ 0 * choose 0 (k + 1) = choose (succ 0) (succ (k + 1)) * succ (k + 1)\n[PROOFSTEP]\nsimp [choose]\n[GOAL]\nn : \u2115\n\u22a2 succ (n + 1) * choose (n + 1) 0 = choose (succ (n + 1)) (succ 0) * succ 0\n[PROOFSTEP]\nsimp [choose, mul_succ, succ_eq_add_one, add_comm]\n[GOAL]\nn k : \u2115\n\u22a2 succ (n + 1) * choose (n + 1) (k + 1) = choose (succ (n + 1)) (succ (k + 1)) * succ (k + 1)\n[PROOFSTEP]\nrw [choose_succ_succ (succ n) (succ k), add_mul, \u2190 succ_mul_choose_eq n, mul_succ, \u2190 succ_mul_choose_eq n,\n  add_right_comm, \u2190 mul_add, \u2190 choose_succ_succ, \u2190 succ_mul]\n[GOAL]\nx\u271d : \u2115\nhk : x\u271d \u2264 0\n\u22a2 choose 0 x\u271d * x\u271d! * (0 - x\u271d)! = 0!\n[PROOFSTEP]\nsimp [Nat.eq_zero_of_le_zero hk]\n[GOAL]\nn : \u2115\nx\u271d : 0 \u2264 n + 1\n\u22a2 choose (n + 1) 0 * 0! * (n + 1 - 0)! = (n + 1)!\n[PROOFSTEP]\nsimp\n[GOAL]\nn k : \u2115\nhk : succ k \u2264 n + 1\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\ncases' lt_or_eq_of_le hk with hk\u2081 hk\u2081\n[GOAL]\ncase inl\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\nhave h : choose n k * k.succ ! * (n - k)! = (k + 1) * n ! :=\n  by\n  rw [\u2190 choose_mul_factorial_mul_factorial (le_of_succ_le_succ hk)]\n  simp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]\n[GOAL]\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\n\u22a2 choose n k * (succ k)! * (n - k)! = (k + 1) * n !\n[PROOFSTEP]\nrw [\u2190 choose_mul_factorial_mul_factorial (le_of_succ_le_succ hk)]\n[GOAL]\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\n\u22a2 choose n k * (succ k)! * (n - k)! = (k + 1) * (choose n k * k ! * (n - k)!)\n[PROOFSTEP]\nsimp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]\n[GOAL]\ncase inl\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\nhave h\u2081 : (n - k)! = (n - k) * (n - k.succ)! := by rw [\u2190 succ_sub_succ, succ_sub (le_of_lt_succ hk\u2081), factorial_succ]\n[GOAL]\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\n\u22a2 (n - k)! = (n - k) * (n - succ k)!\n[PROOFSTEP]\nrw [\u2190 succ_sub_succ, succ_sub (le_of_lt_succ hk\u2081), factorial_succ]\n[GOAL]\ncase inl\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh\u2081 : (n - k)! = (n - k) * (n - succ k)!\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\nhave h\u2082 : choose n (succ k) * k.succ ! * ((n - k) * (n - k.succ)!) = (n - k) * n ! :=\n  by\n  rw [\u2190 choose_mul_factorial_mul_factorial (le_of_lt_succ hk\u2081)]\n  simp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]\n[GOAL]\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh\u2081 : (n - k)! = (n - k) * (n - succ k)!\n\u22a2 choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\n[PROOFSTEP]\nrw [\u2190 choose_mul_factorial_mul_factorial (le_of_lt_succ hk\u2081)]\n[GOAL]\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh\u2081 : (n - k)! = (n - k) * (n - succ k)!\n\u22a2 choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * (choose n (succ k) * (succ k)! * (n - succ k)!)\n[PROOFSTEP]\nsimp [factorial_succ, mul_comm, mul_left_comm, mul_assoc]\n[GOAL]\ncase inl\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh\u2081 : (n - k)! = (n - k) * (n - succ k)!\nh\u2082 : choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\nhave h\u2083 : k * n ! \u2264 n * n ! := Nat.mul_le_mul_right _ (le_of_succ_le_succ hk)\n[GOAL]\ncase inl\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k < n + 1\nh : choose n k * (succ k)! * (n - k)! = (k + 1) * n !\nh\u2081 : (n - k)! = (n - k) * (n - succ k)!\nh\u2082 : choose n (succ k) * (succ k)! * ((n - k) * (n - succ k)!) = (n - k) * n !\nh\u2083 : k * n ! \u2264 n * n !\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\nrw [choose_succ_succ, add_mul, add_mul, succ_sub_succ, h, h\u2081, h\u2082, add_mul, tsub_mul, factorial_succ, \u2190\n  add_tsub_assoc_of_le h\u2083, add_assoc, \u2190 add_mul, add_tsub_cancel_left, add_comm]\n[GOAL]\ncase inr\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k = n + 1\n\u22a2 choose (n + 1) (succ k) * (succ k)! * (n + 1 - succ k)! = (n + 1)!\n[PROOFSTEP]\nrw [hk\u2081]\n[GOAL]\ncase inr\nn k : \u2115\nhk : succ k \u2264 n + 1\nhk\u2081 : succ k = n + 1\n\u22a2 choose (n + 1) (n + 1) * (n + 1)! * (n + 1 - (n + 1))! = (n + 1)!\n[PROOFSTEP]\nsimp [hk\u2081, mul_comm, choose, tsub_self]\n[GOAL]\nn k s : \u2115\nhkn : k \u2264 n\nhsk : s \u2264 k\n\u22a2 (n - k)! * (k - s)! * s ! \u2260 0\n[PROOFSTEP]\napply_rules [factorial_ne_zero, mul_ne_zero]\n[GOAL]\nn k s : \u2115\nhkn : k \u2264 n\nhsk : s \u2264 k\nh : (n - k)! * (k - s)! * s ! \u2260 0\n\u22a2 choose n k * choose k s * ((n - k)! * (k - s)! * s !) = choose n k * (choose k s * s ! * (k - s)!) * (n - k)!\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, mul_assoc, mul_assoc _ s !, mul_assoc, mul_comm (n - k)!, mul_comm s !]\n[GOAL]\nn k s : \u2115\nhkn : k \u2264 n\nhsk : s \u2264 k\nh : (n - k)! * (k - s)! * s ! \u2260 0\n\u22a2 choose n k * (choose k s * s ! * (k - s)!) * (n - k)! = n !\n[PROOFSTEP]\nrw [choose_mul_factorial_mul_factorial hsk, choose_mul_factorial_mul_factorial hkn]\n[GOAL]\nn k s : \u2115\nhkn : k \u2264 n\nhsk : s \u2264 k\nh : (n - k)! * (k - s)! * s ! \u2260 0\n\u22a2 n ! = choose n s * s ! * (choose (n - s) (k - s) * (k - s)! * (n - s - (k - s))!)\n[PROOFSTEP]\nrw [choose_mul_factorial_mul_factorial (tsub_le_tsub_right hkn _), choose_mul_factorial_mul_factorial (hsk.trans hkn)]\n[GOAL]\nn k s : \u2115\nhkn : k \u2264 n\nhsk : s \u2264 k\nh : (n - k)! * (k - s)! * s ! \u2260 0\n\u22a2 choose n s * s ! * (choose (n - s) (k - s) * (k - s)! * (n - s - (k - s))!) =\n    choose n s * choose (n - s) (k - s) * ((n - k)! * (k - s)! * s !)\n[PROOFSTEP]\nrw [tsub_tsub_tsub_cancel_right hsk, mul_assoc, mul_left_comm s !, mul_assoc, mul_comm (k - s)!, mul_comm s !,\n  mul_right_comm, \u2190 mul_assoc]\n[GOAL]\nn k : \u2115\nhk : k \u2264 n\n\u22a2 choose n k = n ! / (k ! * (n - k)!)\n[PROOFSTEP]\nrw [\u2190 choose_mul_factorial_mul_factorial hk, mul_assoc]\n[GOAL]\nn k : \u2115\nhk : k \u2264 n\n\u22a2 choose n k = choose n k * (k ! * (n - k)!) / (k ! * (n - k)!)\n[PROOFSTEP]\nexact (mul_div_left _ (mul_pos (factorial_pos _) (factorial_pos _))).symm\n[GOAL]\ni j : \u2115\n\u22a2 choose (i + j) j = (i + j)! / (i ! * j !)\n[PROOFSTEP]\nrw [choose_eq_factorial_div_factorial (Nat.le_add_left j i), add_tsub_cancel_right, mul_comm]\n[GOAL]\ni j : \u2115\n\u22a2 choose (i + j) j * i ! * j ! = (i + j)!\n[PROOFSTEP]\nrw [\u2190 choose_mul_factorial_mul_factorial (Nat.le_add_left _ _), add_tsub_cancel_right, mul_right_comm]\n[GOAL]\nn k : \u2115\nhk : k \u2264 n\n\u22a2 k ! * (n - k)! \u2223 n !\n[PROOFSTEP]\nrw [\u2190 choose_mul_factorial_mul_factorial hk, mul_assoc]\n[GOAL]\nn k : \u2115\nhk : k \u2264 n\n\u22a2 k ! * (n - k)! \u2223 choose n k * (k ! * (n - k)!)\n[PROOFSTEP]\nexact dvd_mul_left _ _\n[GOAL]\ni j : \u2115\n\u22a2 i ! * j ! \u2223 (i + j)!\n[PROOFSTEP]\nsuffices : i ! * (i + j - i)! \u2223 (i + j)!\n[GOAL]\ni j : \u2115\nthis : i ! * (i + j - i)! \u2223 (i + j)!\n\u22a2 i ! * j ! \u2223 (i + j)!\n[PROOFSTEP]\nrwa [add_tsub_cancel_left i j] at this \n[GOAL]\ncase this\ni j : \u2115\n\u22a2 i ! * (i + j - i)! \u2223 (i + j)!\n[PROOFSTEP]\nexact factorial_mul_factorial_dvd_factorial (Nat.le_add_right _ _)\n[GOAL]\nn k : \u2115\nhk : k \u2264 n\n\u22a2 choose n (n - k) = choose n k\n[PROOFSTEP]\nrw [choose_eq_factorial_div_factorial hk, choose_eq_factorial_div_factorial (Nat.sub_le _ _), tsub_tsub_cancel_of_le hk,\n  mul_comm]\n[GOAL]\nn a b : \u2115\nh : n = a + b\n\u22a2 choose n a = choose n b\n[PROOFSTEP]\nsuffices : choose n (n - b) = choose n b\n[GOAL]\nn a b : \u2115\nh : n = a + b\nthis : choose n (n - b) = choose n b\n\u22a2 choose n a = choose n b\n[PROOFSTEP]\nrw [h, add_tsub_cancel_right] at this \n[GOAL]\nn a b : \u2115\nh : n = a + b\nthis : choose (a + b) a = choose (a + b) b\n\u22a2 choose n a = choose n b\n[PROOFSTEP]\nrwa [h]\n[GOAL]\ncase this\nn a b : \u2115\nh : n = a + b\n\u22a2 choose n (n - b) = choose n b\n[PROOFSTEP]\nexact choose_symm (h \u25b8 le_add_left _ _)\n[GOAL]\nm : \u2115\n\u22a2 choose (2 * m + 1) (m + 1) = choose (2 * m + 1) m\n[PROOFSTEP]\napply choose_symm_of_eq_add\n[GOAL]\ncase h\nm : \u2115\n\u22a2 2 * m + 1 = m + 1 + m\n[PROOFSTEP]\nrw [add_comm m 1, add_assoc 1 m m, add_comm (2 * m) 1, two_mul m]\n[GOAL]\nn k : \u2115\n\u22a2 choose n (k + 1) * (k + 1) = choose n k * (n - k)\n[PROOFSTEP]\nhave e : (n + 1) * choose n k = choose n k * (k + 1) + choose n (k + 1) * (k + 1)\n[GOAL]\ncase e\nn k : \u2115\n\u22a2 (n + 1) * choose n k = choose n k * (k + 1) + choose n (k + 1) * (k + 1)\nn k : \u2115\ne : (n + 1) * choose n k = choose n k * (k + 1) + choose n (k + 1) * (k + 1)\n\u22a2 choose n (k + 1) * (k + 1) = choose n k * (n - k)\n[PROOFSTEP]\nrw [\u2190 right_distrib, \u2190 choose_succ_succ, succ_mul_choose_eq]\n[GOAL]\nn k : \u2115\ne : (n + 1) * choose n k = choose n k * (k + 1) + choose n (k + 1) * (k + 1)\n\u22a2 choose n (k + 1) * (k + 1) = choose n k * (n - k)\n[PROOFSTEP]\nrw [\u2190 tsub_eq_of_eq_add_rev e, mul_comm, \u2190 mul_tsub, add_tsub_add_eq_tsub_right]\n[GOAL]\nn : \u2115\n\u22a2 choose (n + 1 + 1) (n + 1) = n + 1 + 1\n[PROOFSTEP]\nrw [choose_succ_succ, choose_succ_self_right n, choose_self]\n[GOAL]\nn k : \u2115\n\u22a2 choose n k * (n + 1) = choose (n + 1) k * (n + 1 - k)\n[PROOFSTEP]\ncases k with\n| zero => simp\n| succ k =>\n  obtain hk | hk := le_or_lt (k + 1) (n + 1)\n  \u00b7\n    rw [choose_succ_succ, add_mul, succ_sub_succ, \u2190 choose_succ_right_eq, \u2190 succ_sub_succ, mul_tsub,\n      add_tsub_cancel_of_le (Nat.mul_le_mul_left _ hk)]\n  \u00b7 rw [choose_eq_zero_of_lt hk, choose_eq_zero_of_lt (n.lt_succ_self.trans hk), zero_mul, zero_mul]\n[GOAL]\nn k : \u2115\n\u22a2 choose n k * (n + 1) = choose (n + 1) k * (n + 1 - k)\n[PROOFSTEP]\ncases k with\n| zero => simp\n| succ k =>\n  obtain hk | hk := le_or_lt (k + 1) (n + 1)\n  \u00b7\n    rw [choose_succ_succ, add_mul, succ_sub_succ, \u2190 choose_succ_right_eq, \u2190 succ_sub_succ, mul_tsub,\n      add_tsub_cancel_of_le (Nat.mul_le_mul_left _ hk)]\n  \u00b7 rw [choose_eq_zero_of_lt hk, choose_eq_zero_of_lt (n.lt_succ_self.trans hk), zero_mul, zero_mul]\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 choose n zero * (n + 1) = choose (n + 1) zero * (n + 1 - zero)\n[PROOFSTEP]\n\n| zero => simp\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 choose n zero * (n + 1) = choose (n + 1) zero * (n + 1 - zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn k : \u2115\n\u22a2 choose n (succ k) * (n + 1) = choose (n + 1) (succ k) * (n + 1 - succ k)\n[PROOFSTEP]\n\n| succ k =>\n  obtain hk | hk := le_or_lt (k + 1) (n + 1)\n  \u00b7\n    rw [choose_succ_succ, add_mul, succ_sub_succ, \u2190 choose_succ_right_eq, \u2190 succ_sub_succ, mul_tsub,\n      add_tsub_cancel_of_le (Nat.mul_le_mul_left _ hk)]\n  \u00b7 rw [choose_eq_zero_of_lt hk, choose_eq_zero_of_lt (n.lt_succ_self.trans hk), zero_mul, zero_mul]\n[GOAL]\ncase succ\nn k : \u2115\n\u22a2 choose n (succ k) * (n + 1) = choose (n + 1) (succ k) * (n + 1 - succ k)\n[PROOFSTEP]\nobtain hk | hk := le_or_lt (k + 1) (n + 1)\n[GOAL]\ncase succ.inl\nn k : \u2115\nhk : k + 1 \u2264 n + 1\n\u22a2 choose n (succ k) * (n + 1) = choose (n + 1) (succ k) * (n + 1 - succ k)\n[PROOFSTEP]\nrw [choose_succ_succ, add_mul, succ_sub_succ, \u2190 choose_succ_right_eq, \u2190 succ_sub_succ, mul_tsub,\n  add_tsub_cancel_of_le (Nat.mul_le_mul_left _ hk)]\n[GOAL]\ncase succ.inr\nn k : \u2115\nhk : n + 1 < k + 1\n\u22a2 choose n (succ k) * (n + 1) = choose (n + 1) (succ k) * (n + 1 - succ k)\n[PROOFSTEP]\nrw [choose_eq_zero_of_lt hk, choose_eq_zero_of_lt (n.lt_succ_self.trans hk), zero_mul, zero_mul]\n[GOAL]\nn k : \u2115\n\u22a2 ascFactorial n k = k ! * choose (n + k) k\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 ascFactorial n k = choose (n + k) k * k !\n[PROOFSTEP]\napply mul_right_cancel\u2080 (factorial_ne_zero (n + k - k))\n[GOAL]\nn k : \u2115\n\u22a2 ascFactorial n k * (n + k - k)! = choose (n + k) k * k ! * (n + k - k)!\n[PROOFSTEP]\nrw [choose_mul_factorial_mul_factorial, add_tsub_cancel_right, \u2190 factorial_mul_ascFactorial, mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 k \u2264 n + k\n[PROOFSTEP]\nexact Nat.le_add_left k n\n[GOAL]\nn k : \u2115\n\u22a2 choose (n + k) k = ascFactorial n k / k !\n[PROOFSTEP]\napply mul_left_cancel\u2080 (factorial_ne_zero k)\n[GOAL]\nn k : \u2115\n\u22a2 k ! * choose (n + k) k = k ! * (ascFactorial n k / k !)\n[PROOFSTEP]\nrw [\u2190 ascFactorial_eq_factorial_mul_choose]\n[GOAL]\nn k : \u2115\n\u22a2 ascFactorial n k = k ! * (ascFactorial n k / k !)\n[PROOFSTEP]\nexact (Nat.mul_div_cancel' <| factorial_dvd_ascFactorial _ _).symm\n[GOAL]\nn k : \u2115\n\u22a2 descFactorial n k = k ! * choose n k\n[PROOFSTEP]\nobtain h | h := Nat.lt_or_ge n k\n[GOAL]\ncase inl\nn k : \u2115\nh : n < k\n\u22a2 descFactorial n k = k ! * choose n k\n[PROOFSTEP]\nrw [descFactorial_eq_zero_iff_lt.2 h, choose_eq_zero_of_lt h, mul_zero]\n[GOAL]\ncase inr\nn k : \u2115\nh : n \u2265 k\n\u22a2 descFactorial n k = k ! * choose n k\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase inr\nn k : \u2115\nh : n \u2265 k\n\u22a2 descFactorial n k = choose n k * k !\n[PROOFSTEP]\napply mul_right_cancel\u2080 (factorial_ne_zero (n - k))\n[GOAL]\ncase inr\nn k : \u2115\nh : n \u2265 k\n\u22a2 descFactorial n k * (n - k)! = choose n k * k ! * (n - k)!\n[PROOFSTEP]\nrw [choose_mul_factorial_mul_factorial h, \u2190 factorial_mul_descFactorial h, mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 choose n k = descFactorial n k / k !\n[PROOFSTEP]\napply mul_left_cancel\u2080 (factorial_ne_zero k)\n[GOAL]\nn k : \u2115\n\u22a2 k ! * choose n k = k ! * (descFactorial n k / k !)\n[PROOFSTEP]\nrw [\u2190 descFactorial_eq_factorial_mul_choose]\n[GOAL]\nn k : \u2115\n\u22a2 descFactorial n k = k ! * (descFactorial n k / k !)\n[PROOFSTEP]\nexact (Nat.mul_div_cancel' <| factorial_dvd_descFactorial _ _).symm\n[GOAL]\nr n : \u2115\nh : r < n / 2\n\u22a2 choose n r \u2264 choose n (r + 1)\n[PROOFSTEP]\nrefine' le_of_mul_le_mul_right _ (lt_tsub_iff_left.mpr (lt_of_lt_of_le h (n.div_le_self 2)))\n[GOAL]\nr n : \u2115\nh : r < n / 2\n\u22a2 choose n r * (n - r) \u2264 choose n (r + 1) * (n - r)\n[PROOFSTEP]\nrw [\u2190 choose_succ_right_eq]\n[GOAL]\nr n : \u2115\nh : r < n / 2\n\u22a2 choose n (r + 1) * (r + 1) \u2264 choose n (r + 1) * (n - r)\n[PROOFSTEP]\napply Nat.mul_le_mul_left\n[GOAL]\ncase h\nr n : \u2115\nh : r < n / 2\n\u22a2 r + 1 \u2264 n - r\n[PROOFSTEP]\nrw [\u2190 Nat.lt_iff_add_one_le, lt_tsub_iff_left, \u2190 mul_two]\n[GOAL]\ncase h\nr n : \u2115\nh : r < n / 2\n\u22a2 r * 2 < n\n[PROOFSTEP]\nexact lt_of_lt_of_le (mul_lt_mul_of_pos_right h zero_lt_two) (n.div_mul_le_self 2)\n[GOAL]\nr n : \u2115\n\u22a2 choose n r \u2264 choose n (n / 2)\n[PROOFSTEP]\ncases' le_or_gt r n with b b\n[GOAL]\ncase inl\nr n : \u2115\nb : r \u2264 n\n\u22a2 choose n r \u2264 choose n (n / 2)\n[PROOFSTEP]\ncases' le_or_lt r (n / 2) with a h\n[GOAL]\ncase inl.inl\nr n : \u2115\nb : r \u2264 n\na : r \u2264 n / 2\n\u22a2 choose n r \u2264 choose n (n / 2)\n[PROOFSTEP]\napply choose_le_middle_of_le_half_left a\n[GOAL]\ncase inl.inr\nr n : \u2115\nb : r \u2264 n\nh : n / 2 < r\n\u22a2 choose n r \u2264 choose n (n / 2)\n[PROOFSTEP]\nrw [\u2190 choose_symm b]\n[GOAL]\ncase inl.inr\nr n : \u2115\nb : r \u2264 n\nh : n / 2 < r\n\u22a2 choose n (n - r) \u2264 choose n (n / 2)\n[PROOFSTEP]\napply choose_le_middle_of_le_half_left\n[GOAL]\ncase inl.inr.hr\nr n : \u2115\nb : r \u2264 n\nh : n / 2 < r\n\u22a2 n - r \u2264 n / 2\n[PROOFSTEP]\nrw [div_lt_iff_lt_mul' zero_lt_two] at h \n[GOAL]\ncase inl.inr.hr\nr n : \u2115\nb : r \u2264 n\nh : n < r * 2\n\u22a2 n - r \u2264 n / 2\n[PROOFSTEP]\nrw [le_div_iff_mul_le' zero_lt_two, tsub_mul, tsub_le_iff_tsub_le, mul_two, add_tsub_cancel_right]\n[GOAL]\ncase inl.inr.hr\nr n : \u2115\nb : r \u2264 n\nh : n < r * 2\n\u22a2 n \u2264 r * 2\n[PROOFSTEP]\nexact le_of_lt h\n[GOAL]\ncase inr\nr n : \u2115\nb : r > n\n\u22a2 choose n r \u2264 choose n (n / 2)\n[PROOFSTEP]\nrw [choose_eq_zero_of_lt b]\n[GOAL]\ncase inr\nr n : \u2115\nb : r > n\n\u22a2 0 \u2264 choose n (n / 2)\n[PROOFSTEP]\napply zero_le\n[GOAL]\na c : \u2115\n\u22a2 choose a c \u2264 choose (succ a) c\n[PROOFSTEP]\ncases c\n[GOAL]\ncase zero\na : \u2115\n\u22a2 choose a zero \u2264 choose (succ a) zero\n[PROOFSTEP]\nsimp [Nat.choose_succ_succ]\n[GOAL]\ncase succ\na n\u271d : \u2115\n\u22a2 choose a (succ n\u271d) \u2264 choose (succ a) (succ n\u271d)\n[PROOFSTEP]\nsimp [Nat.choose_succ_succ]\n[GOAL]\na b c : \u2115\n\u22a2 choose a c \u2264 choose (a + b) c\n[PROOFSTEP]\ninduction' b with b_n b_ih\n[GOAL]\ncase zero\na c : \u2115\n\u22a2 choose a c \u2264 choose (a + zero) c\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\na c b_n : \u2115\nb_ih : choose a c \u2264 choose (a + b_n) c\n\u22a2 choose a c \u2264 choose (a + succ b_n) c\n[PROOFSTEP]\nexact le_trans b_ih (choose_le_succ (a + b_n) c)\n[GOAL]\nn : \u2115\n\u22a2 multichoose n 0 = 1\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u22a2 multichoose zero 0 = 1\n[PROOFSTEP]\nsimp [multichoose]\n[GOAL]\ncase succ\nn\u271d : \u2115\n\u22a2 multichoose (succ n\u271d) 0 = 1\n[PROOFSTEP]\nsimp [multichoose]\n[GOAL]\nk : \u2115\n\u22a2 multichoose 0 (k + 1) = 0\n[PROOFSTEP]\nsimp [multichoose]\n[GOAL]\nn k : \u2115\n\u22a2 multichoose (n + 1) (k + 1) = multichoose n (k + 1) + multichoose (n + 1) k\n[PROOFSTEP]\nsimp [multichoose]\n[GOAL]\nk : \u2115\n\u22a2 multichoose 1 k = 1\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\n\u22a2 multichoose 1 zero = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nk : \u2115\nIH : multichoose 1 k = 1\n\u22a2 multichoose 1 (succ k) = 1\n[PROOFSTEP]\nsimp [multichoose_succ_succ 0 k, IH]\n[GOAL]\nk : \u2115\n\u22a2 multichoose 2 k = k + 1\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\n\u22a2 multichoose 2 zero = zero + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nk : \u2115\nIH : multichoose 2 k = k + 1\n\u22a2 multichoose 2 (succ k) = succ k + 1\n[PROOFSTEP]\nrw [multichoose, IH]\n[GOAL]\ncase succ\nk : \u2115\nIH : multichoose 2 k = k + 1\n\u22a2 multichoose 1 (k + 1) + (k + 1) = succ k + 1\n[PROOFSTEP]\nsimp [add_comm, succ_eq_add_one]\n[GOAL]\nn : \u2115\n\u22a2 multichoose n 1 = n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u22a2 multichoose zero 1 = zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn : \u2115\nIH : multichoose n 1 = n\n\u22a2 multichoose (succ n) 1 = succ n\n[PROOFSTEP]\nsimp [multichoose_succ_succ n 0, IH]\n[GOAL]\nx\u271d : \u2115\n\u22a2 multichoose x\u271d 0 = choose (x\u271d + 0 - 1) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nk : \u2115\n\u22a2 multichoose 0 (k + 1) = choose (0 + (k + 1) - 1) (k + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nn k : \u2115\n\u22a2 multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)\n[PROOFSTEP]\nhave : n + (k + 1) < (n + 1) + (k + 1) := add_lt_add_right (Nat.lt_succ_self _) _\n[GOAL]\nn k : \u2115\nthis : n + (k + 1) < n + 1 + (k + 1)\n\u22a2 multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)\n[PROOFSTEP]\nhave : (n + 1) + k < (n + 1) + (k + 1) := add_lt_add_left (Nat.lt_succ_self _) _\n[GOAL]\nn k : \u2115\nthis\u271d : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n\u22a2 multichoose (n + 1) (k + 1) = choose (n + 1 + (k + 1) - 1) (k + 1)\n[PROOFSTEP]\nerw [multichoose_succ_succ, add_comm, Nat.succ_add_sub_one, \u2190 add_assoc, Nat.choose_succ_succ]\n[GOAL]\nn k : \u2115\nthis\u271d : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n\u22a2 multichoose (n + 1) k + multichoose n (k + 1) = choose (n + k) k + choose (n + k) (succ k)\n[PROOFSTEP]\nsimp [multichoose_eq n (k + 1), multichoose_eq (n + 1) k]\n[GOAL]\nn k : \u2115\nx\u271d :\n  \u2200 (y : (_ : \u2115) \u00d7' \u2115),\n    (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 y\n        { fst := succ n, snd := succ k } \u2192\n      multichoose y.1 y.2 = choose (y.1 + y.2 - 1) y.2\nthis\u271d : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 { fst := n + 1, snd := k }\n    { fst := succ n, snd := succ k }\n[PROOFSTEP]\n{assumption\n}\n[GOAL]\nn k : \u2115\nx\u271d :\n  \u2200 (y : (_ : \u2115) \u00d7' \u2115),\n    (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 y\n        { fst := succ n, snd := succ k } \u2192\n      multichoose y.1 y.2 = choose (y.1 + y.2 - 1) y.2\nthis\u271d : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 { fst := n + 1, snd := k }\n    { fst := succ n, snd := succ k }\n[PROOFSTEP]\nassumption\n[GOAL]\nn k : \u2115\nx\u271d :\n  \u2200 (y : (_ : \u2115) \u00d7' \u2115),\n    (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 y\n        { fst := succ n, snd := succ k } \u2192\n      multichoose y.1 y.2 = choose (y.1 + y.2 - 1) y.2\nthis\u271d : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 { fst := n, snd := k + 1 }\n    { fst := succ n, snd := succ k }\n[PROOFSTEP]\n{assumption\n}\n[GOAL]\nn k : \u2115\nx\u271d :\n  \u2200 (y : (_ : \u2115) \u00d7' \u2115),\n    (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 y\n        { fst := succ n, snd := succ k } \u2192\n      multichoose y.1 y.2 = choose (y.1 + y.2 - 1) y.2\nthis\u271d : n + (k + 1) < n + 1 + (k + 1)\nthis : n + 1 + k < n + 1 + (k + 1)\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => a + snd) instWellFoundedRelation).1 { fst := n, snd := k + 1 }\n    { fst := succ n, snd := succ k }\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Choose.Basic", "llama_tokens": 11876, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245870332531, "lm_q2_score": 0.6619228825191871, "lm_q1q2_score": 0.5515966127231621}}
{"text": "[GOAL]\nk : \u2115\n\u22a2 bernoulliFun k 0 = \u2191(bernoulli k)\n[PROOFSTEP]\nrw [bernoulliFun, Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]\n[GOAL]\nk : \u2115\nhk : k \u2260 1\n\u22a2 bernoulliFun k 1 = bernoulliFun k 0\n[PROOFSTEP]\nrw [bernoulliFun_eval_zero, bernoulliFun, Polynomial.eval_one_map, Polynomial.bernoulli_eval_one,\n  bernoulli_eq_bernoulli'_of_ne_one hk, eq_ratCast]\n[GOAL]\nk : \u2115\n\u22a2 bernoulliFun k 1 = bernoulliFun k 0 + if k = 1 then 1 else 0\n[PROOFSTEP]\nrw [bernoulliFun, bernoulliFun_eval_zero, Polynomial.eval_one_map, Polynomial.bernoulli_eval_one]\n[GOAL]\nk : \u2115\n\u22a2 \u2191(algebraMap \u211a \u211d) (bernoulli' k) = \u2191(bernoulli k) + if k = 1 then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nk : \u2115\nh : k = 1\n\u22a2 \u2191(algebraMap \u211a \u211d) (bernoulli' k) = \u2191(bernoulli k) + 1\n[PROOFSTEP]\nrw [h, bernoulli_one, bernoulli'_one, eq_ratCast]\n[GOAL]\ncase pos\nk : \u2115\nh : k = 1\n\u22a2 \u2191(1 / 2) = \u2191(-1 / 2) + 1\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase pos\nk : \u2115\nh : k = 1\n\u22a2 1 / \u21912 = -1 / \u21912 + 1\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nk : \u2115\nh : \u00ack = 1\n\u22a2 \u2191(algebraMap \u211a \u211d) (bernoulli' k) = \u2191(bernoulli k) + 0\n[PROOFSTEP]\nrw [bernoulli_eq_bernoulli'_of_ne_one h, add_zero, eq_ratCast]\n[GOAL]\nk : \u2115\nx : \u211d\n\u22a2 HasDerivAt (bernoulliFun k) (\u2191k * bernoulliFun (k - 1) x) x\n[PROOFSTEP]\nconvert ((Polynomial.bernoulli k).map <| algebraMap \u211a \u211d).hasDerivAt x using 1\n[GOAL]\ncase h.e'_7\nk : \u2115\nx : \u211d\n\u22a2 \u2191k * bernoulliFun (k - 1) x =\n    Polynomial.eval x (\u2191Polynomial.derivative (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli k)))\n[PROOFSTEP]\nsimp only [bernoulliFun, Polynomial.derivative_map, Polynomial.derivative_bernoulli k, Polynomial.map_mul,\n  Polynomial.map_nat_cast, Polynomial.eval_mul, Polynomial.eval_nat_cast]\n[GOAL]\nk : \u2115\nx : \u211d\n\u22a2 HasDerivAt (fun x => bernoulliFun (k + 1) x / (\u2191k + 1)) (bernoulliFun k x) x\n[PROOFSTEP]\nconvert (hasDerivAt_bernoulliFun (k + 1) x).div_const _ using 1\n[GOAL]\ncase h.e'_7\nk : \u2115\nx : \u211d\n\u22a2 bernoulliFun k x = \u2191(k + 1) * bernoulliFun (k + 1 - 1) x / (\u2191k + 1)\n[PROOFSTEP]\nfield_simp [Nat.cast_add_one_ne_zero k]\n[GOAL]\ncase h.e'_7\nk : \u2115\nx : \u211d\n\u22a2 bernoulliFun k x * (\u2191k + 1) = (\u2191k + 1) * bernoulliFun k x\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 \u222b (x : \u211d) in 0 ..1, bernoulliFun k x = 0\n[PROOFSTEP]\nrw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)\n    ((Polynomial.continuous _).intervalIntegrable _ _)]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 bernoulliFun (k + 1) 1 / (\u2191k + 1) - bernoulliFun (k + 1) 0 / (\u2191k + 1) = 0\n[PROOFSTEP]\nrw [bernoulliFun_eval_one]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 (bernoulliFun (k + 1) 0 + if k + 1 = 1 then 1 else 0) / (\u2191k + 1) - bernoulliFun (k + 1) 0 / (\u2191k + 1) = 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nk : \u2115\nhk : k \u2260 0\nh : k + 1 = 1\n\u22a2 (bernoulliFun (k + 1) 0 + 1) / (\u2191k + 1) - bernoulliFun (k + 1) 0 / (\u2191k + 1) = 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nk : \u2115\nhk : k \u2260 0\nh : k + 1 = 1\n\u22a2 False\n[PROOFSTEP]\nexact hk (Nat.succ_inj'.mp h)\n[GOAL]\ncase neg\nk : \u2115\nhk : k \u2260 0\nh : \u00ack + 1 = 1\n\u22a2 (bernoulliFun (k + 1) 0 + 0) / (\u2191k + 1) - bernoulliFun (k + 1) 0 / (\u2191k + 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 bernoulliFourierCoeff k n = 1 / (-2 * \u2191\u03c0 * I * \u2191n) * ((if k = 1 then 1 else 0) - \u2191k * bernoulliFourierCoeff (k - 1) n)\n[PROOFSTEP]\nunfold bernoulliFourierCoeff\n[GOAL]\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 fourierCoeffOn bernoulliFourierCoeff.proof_1 (fun x => \u2191(bernoulliFun k x)) n =\n    1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      ((if k = 1 then 1 else 0) -\n        \u2191k * fourierCoeffOn bernoulliFourierCoeff.proof_1 (fun x => \u2191(bernoulliFun (k - 1) x)) n)\n[PROOFSTEP]\nrw [fourierCoeffOn_of_hasDerivAt zero_lt_one hn (fun x _ => (hasDerivAt_bernoulliFun k x).ofReal_comp)\n    ((continuous_ofReal.comp <| continuous_const.mul <| Polynomial.continuous _).intervalIntegrable _ _)]\n[GOAL]\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      (\u2191(fourier (-n)) \u21910 * (\u2191(bernoulliFun k 1) - \u2191(bernoulliFun k 0)) -\n        (\u21911 - \u21910) * fourierCoeffOn (_ : 0 < 1) (fun x => \u2191(\u2191k * bernoulliFun (k - 1) x)) n) =\n    1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      ((if k = 1 then 1 else 0) -\n        \u2191k * fourierCoeffOn bernoulliFourierCoeff.proof_1 (fun x => \u2191(bernoulliFun (k - 1) x)) n)\n[PROOFSTEP]\nsimp_rw [ofReal_one, ofReal_zero, sub_zero, one_mul]\n[GOAL]\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      (\u2191(fourier (-n)) \u21910 * (\u2191(bernoulliFun k 1) - \u2191(bernoulliFun k 0)) -\n        fourierCoeffOn (_ : 0 < 1) (fun x => \u2191(\u2191k * bernoulliFun (k - 1) x)) n) =\n    1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      ((if k = 1 then 1 else 0) -\n        \u2191k * fourierCoeffOn bernoulliFourierCoeff.proof_1 (fun x => \u2191(bernoulliFun (k - 1) x)) n)\n[PROOFSTEP]\nrw [QuotientAddGroup.mk_zero, fourier_eval_zero, one_mul, \u2190 ofReal_sub, bernoulliFun_eval_one, add_sub_cancel']\n[GOAL]\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      (\u2191(if k = 1 then 1 else 0) - fourierCoeffOn (_ : 0 < 1) (fun x => \u2191(\u2191k * bernoulliFun (k - 1) x)) n) =\n    1 / (-2 * \u2191\u03c0 * I * \u2191n) *\n      ((if k = 1 then 1 else 0) -\n        \u2191k * fourierCoeffOn bernoulliFourierCoeff.proof_1 (fun x => \u2191(bernoulliFun (k - 1) x)) n)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 \u2191(if k = 1 then 1 else 0) = if k = 1 then 1 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nk : \u2115\nn : \u2124\nhn : n \u2260 0\nh\u271d : k = 1\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nsimp only [ofReal_one, ofReal_zero, one_mul]\n[GOAL]\ncase neg\nk : \u2115\nn : \u2124\nhn : n \u2260 0\nh\u271d : \u00ack = 1\n\u22a2 \u21910 = 0\n[PROOFSTEP]\nsimp only [ofReal_one, ofReal_zero, one_mul]\n[GOAL]\ncase e_a.e_a\nk : \u2115\nn : \u2124\nhn : n \u2260 0\n\u22a2 fourierCoeffOn (_ : 0 < 1) (fun x => \u2191(\u2191k * bernoulliFun (k - 1) x)) n =\n    \u2191k * fourierCoeffOn bernoulliFourierCoeff.proof_1 (fun x => \u2191(bernoulliFun (k - 1) x)) n\n[PROOFSTEP]\nsimp_rw [ofReal_mul, ofReal_nat_cast, fourierCoeffOn.const_mul]\n[GOAL]\nn : \u2124\nhn : n \u2260 0\n\u22a2 bernoulliFourierCoeff 0 n = 0\n[PROOFSTEP]\nsimpa using bernoulliFourierCoeff_recurrence 0 hn\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 bernoulliFourierCoeff k 0 = 0\n[PROOFSTEP]\nsimp_rw [bernoulliFourierCoeff, fourierCoeffOn_eq_integral, neg_zero, fourier_zero, sub_zero, div_one, one_smul,\n  intervalIntegral.integral_ofReal, integral_bernoulliFun_eq_zero hk, ofReal_zero]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nn : \u2124\n\u22a2 bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\nk : \u2115\nhk : k \u2260 0\n\u22a2 bernoulliFourierCoeff k 0 = -\u2191k ! / (2 * \u2191\u03c0 * I * \u21910) ^ k\n[PROOFSTEP]\nrw [bernoulliFourierCoeff_zero hk, Int.cast_zero, mul_zero, zero_pow' _ hk, div_zero]\n[GOAL]\ncase inr\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nhn : n \u2260 0\n\u22a2 bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nrefine' Nat.le_induction _ (fun k hk h'k => _) k (Nat.one_le_iff_ne_zero.mpr hk)\n[GOAL]\ncase inr.refine'_1\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nhn : n \u2260 0\n\u22a2 bernoulliFourierCoeff 1 n = -\u21911! / (2 * \u2191\u03c0 * I * \u2191n) ^ 1\n[PROOFSTEP]\nrw [bernoulliFourierCoeff_recurrence 1 hn]\n[GOAL]\ncase inr.refine'_1\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nhn : n \u2260 0\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) * ((if 1 = 1 then 1 else 0) - \u21911 * bernoulliFourierCoeff (1 - 1) n) =\n    -\u21911! / (2 * \u2191\u03c0 * I * \u2191n) ^ 1\n[PROOFSTEP]\nsimp only [Nat.cast_one, tsub_self, neg_mul, one_mul, eq_self_iff_true, if_true, Nat.factorial_one, pow_one, inv_I,\n  mul_neg]\n[GOAL]\ncase inr.refine'_1\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nhn : n \u2260 0\n\u22a2 1 / -(2 * \u2191\u03c0 * I * \u2191n) * (1 - bernoulliFourierCoeff 0 n) = -1 / (2 * \u2191\u03c0 * I * \u2191n)\n[PROOFSTEP]\nrw [bernoulli_zero_fourier_coeff hn, sub_zero, mul_one, div_neg, neg_div]\n[GOAL]\ncase inr.refine'_2\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n\u22a2 bernoulliFourierCoeff (k + 1) n = -\u2191(k + 1)! / (2 * \u2191\u03c0 * I * \u2191n) ^ (k + 1)\n[PROOFSTEP]\nrw [bernoulliFourierCoeff_recurrence (k + 1) hn, Nat.add_sub_cancel k 1]\n[GOAL]\ncase inr.refine'_2\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) * ((if k + 1 = 1 then 1 else 0) - \u2191(k + 1) * bernoulliFourierCoeff k n) =\n    -\u2191(k + 1)! / (2 * \u2191\u03c0 * I * \u2191n) ^ (k + 1)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nh : k + 1 = 1\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) * (1 - \u2191(k + 1) * bernoulliFourierCoeff k n) = -\u2191(k + 1)! / (2 * \u2191\u03c0 * I * \u2191n) ^ (k + 1)\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase pos.h\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nh : k + 1 = 1\n\u22a2 False\n[PROOFSTEP]\nexact (ne_of_gt (Nat.lt_succ_iff.mpr hk)) h\n[GOAL]\ncase neg\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nh : \u00ack + 1 = 1\n\u22a2 1 / (-2 * \u2191\u03c0 * I * \u2191n) * (0 - \u2191(k + 1) * bernoulliFourierCoeff k n) = -\u2191(k + 1)! / (2 * \u2191\u03c0 * I * \u2191n) ^ (k + 1)\n[PROOFSTEP]\nrw [h'k, Nat.factorial_succ, zero_sub, Nat.cast_mul, pow_add, pow_one, neg_div, mul_neg, mul_neg, mul_neg, neg_neg,\n  neg_mul, neg_mul, neg_mul, div_neg]\n[GOAL]\ncase neg\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nh : \u00ack + 1 = 1\n\u22a2 -(1 / (2 * \u2191\u03c0 * I * \u2191n)) * (\u2191(k + 1) * (\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k)) =\n    -(\u2191(k + 1) * \u2191k !) / ((2 * \u2191\u03c0 * I * \u2191n) ^ k * (2 * \u2191\u03c0 * I * \u2191n))\n[PROOFSTEP]\nfield_simp [Int.cast_ne_zero.mpr hn, I_ne_zero]\n[GOAL]\ncase neg\nk\u271d : \u2115\nhk\u271d : k\u271d \u2260 0\nn : \u2124\nhn : n \u2260 0\nk : \u2115\nhk : 1 \u2264 k\nh'k : bernoulliFourierCoeff k n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nh : \u00ack + 1 = 1\n\u22a2 -((\u2191k + 1) * \u2191k !) / (2 * \u2191\u03c0 * I * \u2191n * (2 * \u2191\u03c0 * I * \u2191n) ^ k) =\n    -((\u2191k + 1) * \u2191k !) / ((2 * \u2191\u03c0 * I * \u2191n) ^ k * (2 * \u2191\u03c0 * I * \u2191n))\n[PROOFSTEP]\nring_nf\n[GOAL]\nk : \u2115\nhk : k \u2260 1\n\u22a2 bernoulliFun k 0 = bernoulliFun k 1\n[PROOFSTEP]\nexact_mod_cast (bernoulliFun_endpoints_eq_of_ne_one hk).symm\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nn : \u2124\n\u22a2 fourierCoeff (ofReal' \u2218 periodizedBernoulli k) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nhave : ((\u2191) \u2218 periodizedBernoulli k : \ud835\udd4c \u2192 \u2102) = AddCircle.liftIco 1 0 ((\u2191) \u2218 bernoulliFun k) := by ext1 x; rfl\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nn : \u2124\n\u22a2 ofReal' \u2218 periodizedBernoulli k = AddCircle.liftIco 1 0 (ofReal' \u2218 bernoulliFun k)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nx : \ud835\udd4c\n\u22a2 (ofReal' \u2218 periodizedBernoulli k) x = AddCircle.liftIco 1 0 (ofReal' \u2218 bernoulliFun k) x\n[PROOFSTEP]\nrfl\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nthis : ofReal' \u2218 periodizedBernoulli k = AddCircle.liftIco 1 0 (ofReal' \u2218 bernoulliFun k)\n\u22a2 fourierCoeff (ofReal' \u2218 periodizedBernoulli k) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nrw [this, fourierCoeff_liftIco_eq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nn : \u2124\nthis : ofReal' \u2218 periodizedBernoulli k = AddCircle.liftIco 1 0 (ofReal' \u2218 bernoulliFun k)\n\u22a2 fourierCoeffOn (_ : 0 < 0 + 1) (ofReal' \u2218 bernoulliFun k) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nsimpa only [zero_add] using bernoulliFourierCoeff_eq hk n\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\n\u22a2 Summable fun n => -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nhave : \u2200 n : \u2124, -(k ! : \u2102) / (2 * \u03c0 * I * n) ^ k = -k ! / (2 * \u03c0 * I) ^ k * (1 / (n : \u2102) ^ k) := by intro n;\n  rw [mul_one_div, div_div, \u2190 mul_pow]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\n\u22a2 \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n[PROOFSTEP]\nintro n\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nn : \u2124\n\u22a2 -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n[PROOFSTEP]\nrw [mul_one_div, div_div, \u2190 mul_pow]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n\u22a2 Summable fun n => -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n\u22a2 Summable fun n => -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n[PROOFSTEP]\napply Summable.mul_left\n[GOAL]\ncase hf\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n\u22a2 Summable fun i => 1 / \u2191i ^ k\n[PROOFSTEP]\nrw [\u2190 summable_norm_iff]\n[GOAL]\ncase hf\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n\u22a2 Summable fun x => \u20161 / \u2191x ^ k\u2016\n[PROOFSTEP]\nhave : (fun x : \u2124 => \u20161 / (x : \u2102) ^ k\u2016) = fun x : \u2124 => |1 / (x : \u211d) ^ k| :=\n  by\n  ext1 x\n  rw [norm_eq_abs, \u2190 Complex.abs_ofReal]\n  congr 1\n  norm_cast\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\n\u22a2 (fun x => \u20161 / \u2191x ^ k\u2016) = fun x => |1 / \u2191x ^ k|\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\nx : \u2124\n\u22a2 \u20161 / \u2191x ^ k\u2016 = |1 / \u2191x ^ k|\n[PROOFSTEP]\nrw [norm_eq_abs, \u2190 Complex.abs_ofReal]\n[GOAL]\ncase h\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\nx : \u2124\n\u22a2 \u2191Complex.abs (1 / \u2191x ^ k) = \u2191Complex.abs \u2191(1 / \u2191x ^ k)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.e_6.h\nk : \u2115\nhk : 2 \u2264 k\nthis : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\nx : \u2124\n\u22a2 1 / \u2191x ^ k = \u2191(1 / \u2191x ^ k)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hf\nk : \u2115\nhk : 2 \u2264 k\nthis\u271d : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\nthis : (fun x => \u20161 / \u2191x ^ k\u2016) = fun x => |1 / \u2191x ^ k|\n\u22a2 Summable fun x => \u20161 / \u2191x ^ k\u2016\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\ncase hf\nk : \u2115\nhk : 2 \u2264 k\nthis\u271d : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\nthis : (fun x => \u20161 / \u2191x ^ k\u2016) = fun x => |1 / \u2191x ^ k|\n\u22a2 Summable fun x => |1 / \u2191x ^ k|\n[PROOFSTEP]\nrw [summable_abs_iff]\n[GOAL]\ncase hf\nk : \u2115\nhk : 2 \u2264 k\nthis\u271d : \u2200 (n : \u2124), -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k = -\u2191k ! / (2 * \u2191\u03c0 * I) ^ k * (1 / \u2191n ^ k)\nthis : (fun x => \u20161 / \u2191x ^ k\u2016) = fun x => |1 / \u2191x ^ k|\n\u22a2 Summable fun x => 1 / \u2191x ^ k\n[PROOFSTEP]\nexact Real.summable_one_div_int_pow.mpr (one_lt_two.trans_le hk)\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191x) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x))\n[PROOFSTEP]\nsuffices\n  \u2200 {y : \u211d},\n    y \u2208 Ico (0 : \u211d) 1 \u2192\n      HasSum (\u03bb (n : \u2124) => 1 / (n : \u2102) ^ k * fourier n y) (-(2 * (\u03c0 : \u2102) * I) ^ k / k ! * bernoulliFun k y)\n  by\n  rw [\u2190 Ico_insert_right (zero_le_one' \u211d), mem_insert_iff, or_comm] at hx \n  rcases hx with (hx | rfl)\n  \u00b7 exact this hx\n  \u00b7 convert this (left_mem_Ico.mpr zero_lt_one) using 1\n    \u00b7 rw [AddCircle.coe_period, QuotientAddGroup.mk_zero]\n    \u00b7 rw [bernoulliFun_endpoints_eq_of_ne_one (by linarith : k \u2260 1)]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191x) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x))\n[PROOFSTEP]\nrw [\u2190 Ico_insert_right (zero_le_one' \u211d), mem_insert_iff, or_comm] at hx \n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx\u271d : x \u2208 Icc 0 1\nhx : x \u2208 Ico 0 1 \u2228 x = 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191x) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x))\n[PROOFSTEP]\nrcases hx with (hx | rfl)\n[GOAL]\ncase inl\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx\u271d : x \u2208 Icc 0 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\nhx : x \u2208 Ico 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191x) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x))\n[PROOFSTEP]\nexact this hx\n[GOAL]\ncase inr\nk : \u2115\nhk : 2 \u2264 k\nhx : 1 \u2208 Icc 0 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u21911) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k 1))\n[PROOFSTEP]\nconvert this (left_mem_Ico.mpr zero_lt_one) using 1\n[GOAL]\ncase h.e'_5\nk : \u2115\nhk : 2 \u2264 k\nhx : 1 \u2208 Icc 0 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n\u22a2 (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u21911) = fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u21910\n[PROOFSTEP]\nrw [AddCircle.coe_period, QuotientAddGroup.mk_zero]\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : 2 \u2264 k\nhx : 1 \u2208 Icc 0 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n\u22a2 -(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k 1) = -(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k 0)\n[PROOFSTEP]\nrw [bernoulliFun_endpoints_eq_of_ne_one (by linarith : k \u2260 1)]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nhx : 1 \u2208 Icc 0 1\nthis :\n  \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n\u22a2 k \u2260 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 \u2200 {y : \u211d},\n    y \u2208 Ico 0 1 \u2192 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n[PROOFSTEP]\nintro y hy\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n[PROOFSTEP]\nlet B : C(\ud835\udd4c, \u2102) :=\n  ContinuousMap.mk ((\u2191) \u2218 periodizedBernoulli k) (continuous_ofReal.comp (periodizedBernoulli.continuous (by linarith)))\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\n\u22a2 k \u2260 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n[PROOFSTEP]\nhave step1 : \u2200 n : \u2124, fourierCoeff B n = -k ! / (2 * \u03c0 * I * n) ^ k := by rw [ContinuousMap.coe_mk];\n  exact fourierCoeff_bernoulli_eq (by linarith : k \u2260 0)\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\n\u22a2 \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\n\u22a2 \u2200 (n : \u2124), fourierCoeff (ofReal' \u2218 periodizedBernoulli k) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n[PROOFSTEP]\nexact fourierCoeff_bernoulli_eq (by linarith : k \u2260 0)\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\n\u22a2 k \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n[PROOFSTEP]\nhave step2 :=\n  has_pointwise_sum_fourier_series_of_summable ((summable_bernoulli_fourier hk).congr fun n => (step1 n).symm) y\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 : HasSum (fun i => fourierCoeff (\u2191B) i \u2022 \u2191(fourier i) \u2191y) (\u2191B \u2191y)\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n[PROOFSTEP]\nsimp_rw [step1] at step2 \n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * \u2191(fourier n) \u2191y) (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k y))\n[PROOFSTEP]\nconvert step2.mul_left (-(2 * \u2191\u03c0 * I) ^ k / (k ! : \u2102)) using 2 with n\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\nn : \u2124\n\u22a2 1 / \u2191n ^ k * \u2191(fourier n) \u2191y = -(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k) \u2022 \u2191(fourier n) \u2191y\ncase h.e'_6.h.e'_6\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\n\u22a2 \u2191(bernoulliFun k y) = \u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y\n[PROOFSTEP]\nrw [smul_eq_mul, \u2190 mul_assoc, mul_div, mul_neg, div_mul_cancel, neg_neg, mul_pow _ (n : \u2102), \u2190 div_div, div_self]\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\nn : \u2124\n\u22a2 (2 * \u2191\u03c0 * I) ^ k \u2260 0\n[PROOFSTEP]\nrw [Ne.def, pow_eq_zero_iff', not_and_or]\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\nn : \u2124\n\u22a2 \u00ac2 * \u2191\u03c0 * I = 0 \u2228 \u00ack \u2260 0\n[PROOFSTEP]\nexact Or.inl two_pi_I_ne_zero\n[GOAL]\ncase h.e'_5.h.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\nn : \u2124\n\u22a2 \u2191k ! \u2260 0\n[PROOFSTEP]\nexact Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero _)\n[GOAL]\ncase h.e'_6.h.e'_6\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\n\u22a2 \u2191(bernoulliFun k y) = \u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y\n[PROOFSTEP]\nrw [ContinuousMap.coe_mk, Function.comp_apply, ofReal_inj, periodizedBernoulli,\n  AddCircle.liftIco_coe_apply (by rwa [zero_add])]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\ny : \u211d\nhy : y \u2208 Ico 0 1\nB : C(\ud835\udd4c, \u2102) := ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)\nstep1 : \u2200 (n : \u2124), fourierCoeff (\u2191B) n = -\u2191k ! / (2 * \u2191\u03c0 * I * \u2191n) ^ k\nstep2 :\n  HasSum (fun i => (-\u2191k ! / (2 * \u2191\u03c0 * I * \u2191i) ^ k) \u2022 \u2191(fourier i) \u2191y)\n    (\u2191(ContinuousMap.mk (ofReal' \u2218 periodizedBernoulli k)) \u2191y)\n\u22a2 y \u2208 Ico 0 (0 + 1)\n[PROOFSTEP]\nrwa [zero_add]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ k * (\u2191(fourier \u2191n) \u2191x + (-1) ^ k * \u2191(fourier (-\u2191n)) \u2191x))\n    (-(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x))\n[PROOFSTEP]\nconvert (hasSum_one_div_pow_mul_fourier_mul_bernoulliFun hk hx).sum_nat_of_sum_int using 1\n[GOAL]\ncase h.e'_5\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (fun n => 1 / \u2191n ^ k * (\u2191(fourier \u2191n) \u2191x + (-1) ^ k * \u2191(fourier (-\u2191n)) \u2191x)) = fun n =>\n    1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / \u2191(-\u2191n) ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 1 / \u2191n ^ k * (\u2191(fourier \u2191n) \u2191x + (-1) ^ k * \u2191(fourier (-\u2191n)) \u2191x) =\n    1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / \u2191(-\u2191n) ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\nrw [Int.cast_neg, mul_add, \u2190 mul_assoc]\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 1 / \u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / \u2191n ^ k * (-1) ^ k * \u2191(fourier (-\u2191n)) \u2191x =\n    1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / (-\u2191\u2191n) ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\nconv_rhs => rw [neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n| 1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / (-\u2191\u2191n) ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n| 1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / (-\u2191\u2191n) ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n| 1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / (-\u2191\u2191n) ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 1 / \u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / \u2191n ^ k * (-1) ^ k * \u2191(fourier (-\u2191n)) \u2191x =\n    1 / \u2191\u2191n ^ k * \u2191(fourier \u2191n) \u2191x + 1 / (-1) ^ k / \u2191\u2191n ^ k * \u2191(fourier (-\u2191n)) \u2191x\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.e'_5.h.e_a.e_a\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 1 / \u2191n ^ k * (-1) ^ k = 1 / (-1) ^ k / \u2191\u2191n ^ k\n[PROOFSTEP]\nrw [div_mul_eq_mul_div\u2080, one_mul]\n[GOAL]\ncase h.e'_5.h.e_a.e_a\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 (-1) ^ k / \u2191n ^ k = 1 / (-1) ^ k / \u2191\u2191n ^ k\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_5.h.e_a.e_a.e_a\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 (-1) ^ k = 1 / (-1) ^ k\n[PROOFSTEP]\nrw [eq_div_iff, \u2190 mul_pow, \u2190 neg_eq_neg_one_mul, neg_neg, one_pow]\n[GOAL]\ncase h.e'_5.h.e_a.e_a.e_a\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 (-1) ^ k \u2260 0\n[PROOFSTEP]\napply pow_ne_zero\n[GOAL]\ncase h.e'_5.h.e_a.e_a.e_a.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 -1 \u2260 0\n[PROOFSTEP]\nrw [neg_ne_zero]\n[GOAL]\ncase h.e'_5.h.e_a.e_a.e_a.h\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 -(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x) =\n    -(2 * \u2191\u03c0 * I) ^ k / \u2191k ! * \u2191(bernoulliFun k x) + 1 / \u21910 ^ k * \u2191(fourier 0) \u2191x\n[PROOFSTEP]\nrw [Int.cast_zero, zero_pow (by linarith : 0 < k), div_zero, zero_mul, add_zero]\n[GOAL]\nk : \u2115\nhk : 2 \u2264 k\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 0 < k\n[PROOFSTEP]\nlinarith\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))\n[PROOFSTEP]\nhave :\n  HasSum (fun n : \u2115 => 1 / (n : \u2102) ^ (2 * k) * (fourier n (x : \ud835\udd4c) + fourier (-n) (x : \ud835\udd4c)))\n    ((-1 : \u2102) ^ (k + 1) * (2 * (\u03c0 : \u2102)) ^ (2 * k) / (2 * k)! * bernoulliFun (2 * k) x) :=\n  by\n  convert hasSum_one_div_nat_pow_mul_fourier (by linarith [Nat.one_le_iff_ne_zero.mpr hk] : 2 \u2264 2 * k) hx using 3\n  \u00b7 rw [pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul]\n  \u00b7 rw [pow_add, pow_one]\n    conv_rhs =>\n      rw [mul_pow]\n      congr\n      congr\n      \u00b7skip\n      \u00b7rw [pow_mul, I_sq]\n    ring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n[PROOFSTEP]\nconvert hasSum_one_div_nat_pow_mul_fourier (by linarith [Nat.one_le_iff_ne_zero.mpr hk] : 2 \u2264 2 * k) hx using 3\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 2 \u2264 2 * k\n[PROOFSTEP]\nlinarith [Nat.one_le_iff_ne_zero.mpr hk]\n[GOAL]\ncase h.e'_5.h.h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nx\u271d : \u2115\n\u22a2 \u2191(fourier \u2191x\u271d) \u2191x + \u2191(fourier (-\u2191x\u271d)) \u2191x = \u2191(fourier \u2191x\u271d) \u2191x + (-1) ^ (2 * k) * \u2191(fourier (-\u2191x\u271d)) \u2191x\n[PROOFSTEP]\nrw [pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul]\n[GOAL]\ncase h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)\n[PROOFSTEP]\nrw [pow_add, pow_one]\n[GOAL]\ncase h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)\n[PROOFSTEP]\nconv_rhs =>\n  rw [mul_pow]\n  congr\n  congr\n  \u00b7skip\n  \u00b7rw [pow_mul, I_sq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -(2 * \u2191\u03c0 * I) ^ (2 * k)\n[PROOFSTEP]\n  rw [mul_pow]\n  congr\n  congr\n  \u00b7skip\n  \u00b7rw [pow_mul, I_sq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -(2 * \u2191\u03c0 * I) ^ (2 * k)\n[PROOFSTEP]\n  rw [mul_pow]\n  congr\n  congr\n  \u00b7skip\n  \u00b7rw [pow_mul, I_sq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -(2 * \u2191\u03c0 * I) ^ (2 * k)\n[PROOFSTEP]\nrw [mul_pow]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -((2 * \u2191\u03c0) ^ (2 * k) * I ^ (2 * k))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k) * I ^ (2 * k)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k)\ncase a.a k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 | I ^ (2 * k)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k)\n[PROOFSTEP]\n\u00b7rw [pow_mul, I_sq]\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k)\n[PROOFSTEP]\nrw [pow_mul, I_sq]\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k)\n[PROOFSTEP]\nrw [pow_mul, I_sq]\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k)\n[PROOFSTEP]\nrw [pow_mul, I_sq]\n[GOAL]\ncase h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -((2 * \u2191\u03c0) ^ (2 * k) * (-1) ^ k)\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))\n[PROOFSTEP]\nhave ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n\u22a2 \u21912 = 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))\n[PROOFSTEP]\nconvert ((hasSum_iff _ _).mp (this.div_const 2)).1 with n\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x) = (1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2).re\n[PROOFSTEP]\nconvert (ofReal_re _).symm\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n[PROOFSTEP]\nrw [ofReal_mul]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))\n[PROOFSTEP]\nrw [\u2190 mul_div]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2) = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = \u2191(1 / \u2191n ^ (2 * k))\n[PROOFSTEP]\nrw [ofReal_div, ofReal_one, ofReal_pow]\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191\u2191n ^ (2 * k)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(Real.cos (2 * \u03c0 * \u2191n * x))\n[PROOFSTEP]\nrw [ofReal_cos, ofReal_mul, fourier_coe_apply, fourier_coe_apply, cos, ofReal_one, div_one, div_one, ofReal_mul,\n  ofReal_mul, ofReal_two, Int.cast_neg, Int.cast_ofNat, ofReal_nat_cast]\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (exp (2 * \u2191\u03c0 * I * \u2191n * \u2191x) + exp (2 * \u2191\u03c0 * I * -\u2191n * \u2191x)) / 2 =\n    (exp (2 * \u2191\u03c0 * \u2191n * \u2191x * I) + exp (-(2 * \u2191\u03c0 * \u2191n * \u2191x) * I)) / 2\n[PROOFSTEP]\ncongr 3\n[GOAL]\ncase h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * \u2191n * \u2191x = 2 * \u2191\u03c0 * \u2191n * \u2191x * I\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * -\u2191n * \u2191x = -(2 * \u2191\u03c0 * \u2191n * \u2191x) * I\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))) =\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2).re\n[PROOFSTEP]\nconvert (ofReal_re _).symm\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n        Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))\n[PROOFSTEP]\nrw [ofReal_mul, ofReal_div, ofReal_div, ofReal_mul, ofReal_pow, ofReal_pow, ofReal_neg, ofReal_nat_cast, ofReal_mul,\n  ofReal_two, ofReal_one]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))\n[PROOFSTEP]\nrw [bernoulliFun]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! *\n        \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) /\n      2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * Real.sin (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nhave :\n  HasSum (fun n : \u2115 => 1 / (n : \u2102) ^ (2 * k + 1) * (fourier n (x : \ud835\udd4c) - fourier (-n) (x : \ud835\udd4c)))\n    ((-1 : \u2102) ^ (k + 1) * I * (2 * \u03c0 : \u2102) ^ (2 * k + 1) / (2 * k + 1)! * bernoulliFun (2 * k + 1) x) :=\n  by\n  convert hasSum_one_div_nat_pow_mul_fourier (by linarith [Nat.one_le_iff_ne_zero.mpr hk] : 2 \u2264 2 * k + 1) hx using 1\n  \u00b7 ext1 n\n    rw [pow_add (-1 : \u2102), pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul, pow_one, \u2190 neg_eq_neg_one_mul, \u2190\n      sub_eq_add_neg]\n  \u00b7 congr\n    rw [pow_add, pow_one]\n    conv_rhs =>\n      rw [mul_pow]\n      congr\n      congr\n      \u00b7skip\n      \u00b7rw [pow_add, pow_one, pow_mul, I_sq]\n    ring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\n[PROOFSTEP]\nconvert hasSum_one_div_nat_pow_mul_fourier (by linarith [Nat.one_le_iff_ne_zero.mpr hk] : 2 \u2264 2 * k + 1) hx using 1\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 2 \u2264 2 * k + 1\n[PROOFSTEP]\nlinarith [Nat.one_le_iff_ne_zero.mpr hk]\n[GOAL]\ncase h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x)) = fun n =>\n    1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x + (-1) ^ (2 * k + 1) * \u2191(fourier (-\u2191n)) \u2191x)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x) =\n    1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x + (-1) ^ (2 * k + 1) * \u2191(fourier (-\u2191n)) \u2191x)\n[PROOFSTEP]\nrw [pow_add (-1 : \u2102), pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul, pow_one, \u2190 neg_eq_neg_one_mul, \u2190 sub_eq_add_neg]\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x) =\n    -(2 * \u2191\u03c0 * I) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_6.e_a.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) = -(2 * \u2191\u03c0 * I) ^ (2 * k + 1)\n[PROOFSTEP]\nrw [pow_add, pow_one]\n[GOAL]\ncase h.e'_6.e_a.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * I * (2 * \u2191\u03c0) ^ (2 * k + 1) = -(2 * \u2191\u03c0 * I) ^ (2 * k + 1)\n[PROOFSTEP]\nconv_rhs =>\n  rw [mul_pow]\n  congr\n  congr\n  \u00b7skip\n  \u00b7rw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -(2 * \u2191\u03c0 * I) ^ (2 * k + 1)\n[PROOFSTEP]\n  rw [mul_pow]\n  congr\n  congr\n  \u00b7skip\n  \u00b7rw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -(2 * \u2191\u03c0 * I) ^ (2 * k + 1)\n[PROOFSTEP]\n  rw [mul_pow]\n  congr\n  congr\n  \u00b7skip\n  \u00b7rw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -(2 * \u2191\u03c0 * I) ^ (2 * k + 1)\n[PROOFSTEP]\nrw [mul_pow]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| -((2 * \u2191\u03c0) ^ (2 * k + 1) * I ^ (2 * k + 1))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k + 1) * I ^ (2 * k + 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k + 1)\ncase a.a k : \u2115 hk : k \u2260 0 x : \u211d hx : x \u2208 Icc 0 1 | I ^ (2 * k + 1)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| (2 * \u2191\u03c0) ^ (2 * k + 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k + 1)\n[PROOFSTEP]\n\u00b7rw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k + 1)\n[PROOFSTEP]\nrw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k + 1)\n[PROOFSTEP]\nrw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\ncase a.a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n| I ^ (2 * k + 1)\n[PROOFSTEP]\nrw [pow_add, pow_one, pow_mul, I_sq]\n[GOAL]\ncase h.e'_6.e_a.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * I * (2 * \u2191\u03c0) ^ (2 * k + 1) = -((2 * \u2191\u03c0) ^ (2 * k + 1) * ((-1) ^ k * I))\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * Real.sin (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nhave ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\n\u22a2 \u21912 = 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * Real.sin (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nconvert ((hasSum_iff _ _).mp (this.div_const (2 * I))).1\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 1 / \u2191x\u271d ^ (2 * k + 1) * Real.sin (2 * \u03c0 * \u2191x\u271d * x) =\n    (1 / \u2191x\u271d ^ (2 * k + 1) * (\u2191(fourier \u2191x\u271d) \u2191x - \u2191(fourier (-\u2191x\u271d)) \u2191x) / (2 * I)).re\n[PROOFSTEP]\nconvert (ofReal_re _).symm\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 1 / \u2191x\u271d ^ (2 * k + 1) * (\u2191(fourier \u2191x\u271d) \u2191x - \u2191(fourier (-\u2191x\u271d)) \u2191x) / (2 * I) =\n    \u2191(1 / \u2191x\u271d ^ (2 * k + 1) * Real.sin (2 * \u03c0 * \u2191x\u271d * x))\n[PROOFSTEP]\nrw [ofReal_mul]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 1 / \u2191x\u271d ^ (2 * k + 1) * (\u2191(fourier \u2191x\u271d) \u2191x - \u2191(fourier (-\u2191x\u271d)) \u2191x) / (2 * I) =\n    \u2191(1 / \u2191x\u271d ^ (2 * k + 1)) * \u2191(Real.sin (2 * \u03c0 * \u2191x\u271d * x))\n[PROOFSTEP]\nrw [\u2190 mul_div]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 1 / \u2191x\u271d ^ (2 * k + 1) * ((\u2191(fourier \u2191x\u271d) \u2191x - \u2191(fourier (-\u2191x\u271d)) \u2191x) / (2 * I)) =\n    \u2191(1 / \u2191x\u271d ^ (2 * k + 1)) * \u2191(Real.sin (2 * \u03c0 * \u2191x\u271d * x))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 1 / \u2191x\u271d ^ (2 * k + 1) = \u2191(1 / \u2191x\u271d ^ (2 * k + 1))\n[PROOFSTEP]\nrw [ofReal_div, ofReal_one, ofReal_pow]\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 1 / \u2191x\u271d ^ (2 * k + 1) = 1 / \u2191\u2191x\u271d ^ (2 * k + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 (\u2191(fourier \u2191x\u271d) \u2191x - \u2191(fourier (-\u2191x\u271d)) \u2191x) / (2 * I) = \u2191(Real.sin (2 * \u03c0 * \u2191x\u271d * x))\n[PROOFSTEP]\nrw [ofReal_sin, ofReal_mul, fourier_coe_apply, fourier_coe_apply, sin, ofReal_one, div_one, div_one, ofReal_mul,\n  ofReal_mul, ofReal_two, Int.cast_neg, Int.cast_ofNat, ofReal_nat_cast, \u2190 div_div, div_I, div_mul_eq_mul_div\u2080, \u2190\n  neg_div, \u2190 neg_mul, neg_sub]\n[GOAL]\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 (exp (2 * \u2191\u03c0 * I * -\u2191x\u271d * \u2191x) - exp (2 * \u2191\u03c0 * I * \u2191x\u271d * \u2191x)) * I / 2 =\n    (exp (-(2 * \u2191\u03c0 * \u2191x\u271d * \u2191x) * I) - exp (2 * \u2191\u03c0 * \u2191x\u271d * \u2191x * I)) * I / 2\n[PROOFSTEP]\ncongr 4\n[GOAL]\ncase h.e'_3.h.e'_1.e_a.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 2 * \u2191\u03c0 * I * -\u2191x\u271d * \u2191x = -(2 * \u2191\u03c0 * \u2191x\u271d * \u2191x) * I\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_3.h.e'_1.e_a.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nx\u271d : \u2115\n\u22a2 2 * \u2191\u03c0 * I * \u2191x\u271d * \u2191x = 2 * \u2191\u03c0 * \u2191x\u271d * \u2191x * I\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))) =\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x) / (2 * I)).re\n[PROOFSTEP]\nconvert (ofReal_re _).symm\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x) / (2 * I) =\n    \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n        Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nrw [ofReal_mul, ofReal_div, ofReal_div, ofReal_mul, ofReal_pow, ofReal_pow, ofReal_neg, ofReal_nat_cast, ofReal_mul,\n  ofReal_two, ofReal_one, \u2190 div_div, div_I, div_mul_eq_mul_div\u2080]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\n\u22a2 -((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x) * I / 2) =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nhave : \u2200 \u03b1 \u03b2 \u03b3 \u03b4 : \u2102, \u03b1 * I * \u03b2 / \u03b3 * \u03b4 * I = I ^ 2 * \u03b1 * \u03b2 / \u03b3 * \u03b4 := by intros; ring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\n\u22a2 \u2200 (\u03b1 \u03b2 \u03b3 \u03b4 : \u2102), \u03b1 * I * \u03b2 / \u03b3 * \u03b4 * I = I ^ 2 * \u03b1 * \u03b2 / \u03b3 * \u03b4\n[PROOFSTEP]\nintros\n[GOAL]\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\n\u03b1\u271d \u03b2\u271d \u03b3\u271d \u03b4\u271d : \u2102\n\u22a2 \u03b1\u271d * I * \u03b2\u271d / \u03b3\u271d * \u03b4\u271d * I = I ^ 2 * \u03b1\u271d * \u03b2\u271d / \u03b3\u271d * \u03b4\u271d\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis\u271d :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nthis : \u2200 (\u03b1 \u03b2 \u03b3 \u03b4 : \u2102), \u03b1 * I * \u03b2 / \u03b3 * \u03b4 * I = I ^ 2 * \u03b1 * \u03b2 / \u03b3 * \u03b4\n\u22a2 -((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x) * I / 2) =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nrw [this, I_sq]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis\u271d :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nthis : \u2200 (\u03b1 \u03b2 \u03b3 \u03b4 : \u2102), \u03b1 * I * \u03b2 / \u03b3 * \u03b4 * I = I ^ 2 * \u03b1 * \u03b2 / \u03b3 * \u03b4\n\u22a2 -(-1 * (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x) / 2) =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nrw [bernoulliFun]\n[GOAL]\ncase h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis\u271d :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k + 1) * (\u2191(fourier \u2191n) \u2191x - \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * I * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! * \u2191(bernoulliFun (2 * k + 1) x))\nofReal_two : \u21912 = 2\nthis : \u2200 (\u03b1 \u03b2 \u03b3 \u03b4 : \u2102), \u03b1 * I * \u03b2 / \u03b3 * \u03b4 * I = I ^ 2 * \u03b1 * \u03b2 / \u03b3 * \u03b4\n\u22a2 -(-1 * (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k + 1) / \u2191(2 * k + 1)! *\n          \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1)))) /\n        2) =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k + 1) / 2 / \u2191(2 * k + 1)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k + 1))))\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k)) ((-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)!)\n[PROOFSTEP]\nconvert hasSum_one_div_nat_pow_mul_cos hk (left_mem_Icc.mpr zero_le_one) using 1\n[GOAL]\ncase h.e'_5\nk : \u2115\nhk : k \u2260 0\n\u22a2 (fun n => 1 / \u2191n ^ (2 * k)) = fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * 0)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * 0)\n[PROOFSTEP]\nrw [mul_zero, Real.cos_zero, mul_one]\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\n\u22a2 (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)! =\n    (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval 0 (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))\n[PROOFSTEP]\nrw [Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\n\u22a2 (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)! =\n    (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * \u2191(bernoulli (2 * k))\n[PROOFSTEP]\nhave : (2 : \u211d) ^ (2 * k - 1) = (2 : \u211d) ^ (2 * k) / 2 :=\n  by\n  rw [eq_div_iff (two_ne_zero' \u211d)]\n  conv_lhs =>\n    congr\n    \u00b7skip\n    \u00b7rw [\u2190 pow_one (2 : \u211d)]\n  rw [\u2190 pow_add, Nat.sub_add_cancel]\n  linarith [Nat.one_le_iff_ne_zero.mpr hk]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 2 ^ (2 * k - 1) = 2 ^ (2 * k) / 2\n[PROOFSTEP]\nrw [eq_div_iff (two_ne_zero' \u211d)]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 2 ^ (2 * k - 1) * 2 = 2 ^ (2 * k)\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1) * 2\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1) * 2\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1) * 2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1)\ncase a k : \u2115 hk : k \u2260 0 | 2\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2 ^ (2 * k - 1)\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\n\u00b7rw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\nrw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\nrw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\ncase a\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\nrw [\u2190 pow_one (2 : \u211d)]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 2 ^ (2 * k - 1) * 2 ^ 1 = 2 ^ (2 * k)\n[PROOFSTEP]\nrw [\u2190 pow_add, Nat.sub_add_cancel]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 1 \u2264 2 * k\n[PROOFSTEP]\nlinarith [Nat.one_le_iff_ne_zero.mpr hk]\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nthis : 2 ^ (2 * k - 1) = 2 ^ (2 * k) / 2\n\u22a2 (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)! =\n    (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! * \u2191(bernoulli (2 * k))\n[PROOFSTEP]\nrw [this, mul_pow]\n[GOAL]\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nthis : 2 ^ (2 * k - 1) = 2 ^ (2 * k) / 2\n\u22a2 (-1) ^ (k + 1) * (2 ^ (2 * k) / 2) * \u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)! =\n    (-1) ^ (k + 1) * (2 ^ (2 * k) * \u03c0 ^ (2 * k)) / 2 / \u2191(2 * k)! * \u2191(bernoulli (2 * k))\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 HasSum (fun n => 1 / \u2191n ^ 2) (\u03c0 ^ 2 / 6)\n[PROOFSTEP]\nconvert hasSum_zeta_nat one_ne_zero using 1\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 2 / 6 = (-1) ^ (1 + 1) * 2 ^ (2 * 1 - 1) * \u03c0 ^ (2 * 1) * \u2191(bernoulli (2 * 1)) / \u2191(2 * 1)!\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 2 / 6 = (-1) ^ (1 + 1) * 2 ^ (2 - 1) * \u03c0 ^ 2 * \u2191(bernoulli 2) / \u21912!\n[PROOFSTEP]\nrw [bernoulli_eq_bernoulli'_of_ne_one (by decide : 2 \u2260 1), bernoulli'_two]\n[GOAL]\n\u22a2 2 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 2 / 6 = (-1) ^ (1 + 1) * 2 ^ (2 - 1) * \u03c0 ^ 2 * \u2191(1 / 6) / \u21912!\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 2 / 6 = 2 * \u03c0 ^ 2 * (1 / 6) / 2\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 2 * (6 * 2) = 2 * \u03c0 ^ 2 * 6\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 HasSum (fun n => 1 / \u2191n ^ 4) (\u03c0 ^ 4 / 90)\n[PROOFSTEP]\nconvert hasSum_zeta_nat two_ne_zero using 1\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 4 / 90 = (-1) ^ (2 + 1) * 2 ^ (2 * 2 - 1) * \u03c0 ^ (2 * 2) * \u2191(bernoulli (2 * 2)) / \u2191(2 * 2)!\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 4 / 90 = -(8 * \u03c0 ^ 4 * \u2191(bernoulli 4)) / 24\n[PROOFSTEP]\nrw [bernoulli_eq_bernoulli'_of_ne_one, bernoulli'_four]\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 4 / 90 = -(8 * \u03c0 ^ 4 * \u2191(-1 / 30)) / 24\ncase h.e'_6 \u22a2 4 \u2260 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 4 / 90 = 8 * \u03c0 ^ 4 * (1 / 30) / 24\ncase h.e'_6 \u22a2 4 \u2260 1\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h.e'_6\n\u22a2 \u03c0 ^ 4 * (30 * 24) = 8 * \u03c0 ^ 4 * 90\ncase h.e'_6 \u22a2 4 \u2260 1\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_6\n\u22a2 4 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 eval (1 / 4) (bernoulli 3) = 3 / 64\n[PROOFSTEP]\nsimp_rw [Polynomial.bernoulli, Finset.sum_range_succ, Polynomial.eval_add, Polynomial.eval_monomial]\n[GOAL]\n\u22a2 eval (1 / 4) (Finset.sum (Finset.range 0) fun x => \u2191(monomial (3 - x)) (_root_.bernoulli x * \u2191(Nat.choose 3 x))) +\n            _root_.bernoulli 0 * \u2191(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) +\n          _root_.bernoulli 1 * \u2191(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +\n        _root_.bernoulli 2 * \u2191(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +\n      _root_.bernoulli 3 * \u2191(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =\n    3 / 64\n[PROOFSTEP]\nrw [Finset.sum_range_zero, Polynomial.eval_zero, zero_add, bernoulli_one]\n[GOAL]\n\u22a2 _root_.bernoulli 0 * \u2191(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * \u2191(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +\n        _root_.bernoulli 2 * \u2191(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +\n      _root_.bernoulli 3 * \u2191(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =\n    3 / 64\n[PROOFSTEP]\nrw [bernoulli_eq_bernoulli'_of_ne_one zero_ne_one, bernoulli'_zero,\n  bernoulli_eq_bernoulli'_of_ne_one (by decide : 2 \u2260 1), bernoulli'_two,\n  bernoulli_eq_bernoulli'_of_ne_one (by decide : 3 \u2260 1), bernoulli'_three]\n[GOAL]\n\u22a2 2 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 3 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 1 * \u2191(Nat.choose 3 0) * (1 / 4) ^ (3 - 0) + -1 / 2 * \u2191(Nat.choose 3 1) * (1 / 4) ^ (3 - 1) +\n        1 / 6 * \u2191(Nat.choose 3 2) * (1 / 4) ^ (3 - 2) +\n      0 * \u2191(Nat.choose 3 3) * (1 / 4) ^ (3 - 3) =\n    3 / 64\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 HasSum (fun n => 1 / \u2191n ^ 3 * Real.sin (\u03c0 * \u2191n / 2)) (\u03c0 ^ 3 / 32)\n[PROOFSTEP]\napply (congr_arg\u2082 HasSum ?_ ?_).to_iff.mp <| hasSum_one_div_nat_pow_mul_sin one_ne_zero (?_ : 1 / 4 \u2208 Icc (0 : \u211d) 1)\n[GOAL]\n\u22a2 (fun n => 1 / \u2191n ^ (2 * 1 + 1) * Real.sin (2 * \u03c0 * \u2191n * (1 / 4))) = fun n => 1 / \u2191n ^ 3 * Real.sin (\u03c0 * \u2191n / 2)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * 1 + 1) * Real.sin (2 * \u03c0 * \u2191n * (1 / 4)) = 1 / \u2191n ^ 3 * Real.sin (\u03c0 * \u2191n / 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\nn : \u2115\n\u22a2 Real.sin (2 * \u03c0 * \u2191n * (1 / 4)) = Real.sin (\u03c0 * \u2191n / 2) \u2228 n = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase h.h\nn : \u2115\n\u22a2 Real.sin (2 * \u03c0 * \u2191n * (1 / 4)) = Real.sin (\u03c0 * \u2191n / 2)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.h.e_x\nn : \u2115\n\u22a2 2 * \u03c0 * \u2191n * (1 / 4) = \u03c0 * \u2191n / 2\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 (-1) ^ (1 + 1) * (2 * \u03c0) ^ (2 * 1 + 1) / 2 / \u2191(2 * 1 + 1)! *\n      Polynomial.eval (1 / 4) (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * 1 + 1))) =\n    \u03c0 ^ 3 / 32\n[PROOFSTEP]\nhave : (1 / 4 : \u211d) = (algebraMap \u211a \u211d) (1 / 4 : \u211a) := by norm_num\n[GOAL]\n\u22a2 1 / 4 = \u2191(algebraMap \u211a \u211d) (1 / 4)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis : 1 / 4 = \u2191(algebraMap \u211a \u211d) (1 / 4)\n\u22a2 (-1) ^ (1 + 1) * (2 * \u03c0) ^ (2 * 1 + 1) / 2 / \u2191(2 * 1 + 1)! *\n      Polynomial.eval (1 / 4) (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * 1 + 1))) =\n    \u03c0 ^ 3 / 32\n[PROOFSTEP]\nrw [this, mul_pow, Polynomial.eval_map, Polynomial.eval\u2082_at_apply, (by decide : 2 * 1 + 1 = 3),\n  Polynomial.bernoulli_three_eval_one_quarter]\n[GOAL]\nthis : 1 / 4 = \u2191(algebraMap \u211a \u211d) (1 / 4)\n\u22a2 2 * 1 + 1 = 3\n[PROOFSTEP]\ndecide\n[GOAL]\nthis : 1 / 4 = \u2191(algebraMap \u211a \u211d) (1 / 4)\n\u22a2 (-1) ^ (1 + 1) * (2 ^ 3 * \u03c0 ^ 3) / 2 / \u21913! * \u2191(algebraMap \u211a \u211d) (3 / 64) = \u03c0 ^ 3 / 32\n[PROOFSTEP]\nnorm_num\n[GOAL]\nthis : 1 / 4 = \u2191(algebraMap \u211a \u211d) (1 / 4)\n\u22a2 8 * \u03c0 ^ 3 / 2 / 6 * (3 / 64) = \u03c0 ^ 3 / 32\n[PROOFSTEP]\nfield_simp\n[GOAL]\nthis : 1 / 4 = \u2191(algebraMap \u211a \u211d) (1 / 4)\n\u22a2 8 * \u03c0 ^ 3 * 3 * 32 = \u03c0 ^ 3 * (2 * 6 * 64)\n[PROOFSTEP]\nring\n[GOAL]\n\u22a2 1 / 4 \u2208 Icc 0 1\n[PROOFSTEP]\nrw [mem_Icc]\n[GOAL]\n\u22a2 0 \u2264 1 / 4 \u2227 1 / 4 \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u22a2 0 \u2264 1 / 4\ncase right \u22a2 1 / 4 \u2264 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase right\n\u22a2 1 / 4 \u2264 1\n[PROOFSTEP]\nlinarith\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.ZetaValues", "llama_tokens": 36407, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527632, "lm_q2_score": 0.6825737344123242, "lm_q1q2_score": 0.5515654556108707}}
{"text": "[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : f n = true\n\u22a2 cantorFunctionAux c f n = c ^ n\n[PROOFSTEP]\nsimp [cantorFunctionAux, h]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : f n = false\n\u22a2 cantorFunctionAux c f n = 0\n[PROOFSTEP]\nsimp [cantorFunctionAux, h]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : 0 \u2264 c\n\u22a2 0 \u2264 cantorFunctionAux c f n\n[PROOFSTEP]\ncases h' : f n\n[GOAL]\ncase false\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : 0 \u2264 c\nh' : f n = false\n\u22a2 0 \u2264 cantorFunctionAux c f n\n[PROOFSTEP]\nsimp [h']\n[GOAL]\ncase true\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : 0 \u2264 c\nh' : f n = true\n\u22a2 0 \u2264 cantorFunctionAux c f n\n[PROOFSTEP]\nsimp [h']\n[GOAL]\ncase true\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : 0 \u2264 c\nh' : f n = true\n\u22a2 0 \u2264 c ^ n\n[PROOFSTEP]\napply pow_nonneg h\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh : f n = g n\n\u22a2 cantorFunctionAux c f n = cantorFunctionAux c g n\n[PROOFSTEP]\nsimp [cantorFunctionAux, h]\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\n\u22a2 cantorFunctionAux c f 0 = bif f 0 then 1 else 0\n[PROOFSTEP]\ncases h : f 0\n[GOAL]\ncase false\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh : f 0 = false\n\u22a2 cantorFunctionAux c f 0 = bif false then 1 else 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase true\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh : f 0 = true\n\u22a2 cantorFunctionAux c f 0 = bif true then 1 else 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\n\u22a2 (fun n => cantorFunctionAux c f (n + 1)) = fun n => c * cantorFunctionAux c (fun n => f (n + 1)) n\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn\u271d : \u2115\nf : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 cantorFunctionAux c f (n + 1) = c * cantorFunctionAux c (fun n => f (n + 1)) n\n[PROOFSTEP]\ncases h : f (n + 1)\n[GOAL]\ncase h.false\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn\u271d : \u2115\nf : \u2115 \u2192 Bool\nn : \u2115\nh : f (n + 1) = false\n\u22a2 cantorFunctionAux c f (n + 1) = c * cantorFunctionAux c (fun n => f (n + 1)) n\n[PROOFSTEP]\nsimp [h, _root_.pow_succ]\n[GOAL]\ncase h.true\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn\u271d : \u2115\nf : \u2115 \u2192 Bool\nn : \u2115\nh : f (n + 1) = true\n\u22a2 cantorFunctionAux c f (n + 1) = c * cantorFunctionAux c (fun n => f (n + 1)) n\n[PROOFSTEP]\nsimp [h, _root_.pow_succ]\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\n\u22a2 Summable (cantorFunctionAux c f)\n[PROOFSTEP]\napply (summable_geometric_of_lt_1 h1 h2).summable_of_eq_zero_or_self\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\n\u22a2 \u2200 (b : \u2115), cantorFunctionAux c f b = 0 \u2228 cantorFunctionAux c f b = c ^ b\n[PROOFSTEP]\nintro n\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn\u271d : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\nn : \u2115\n\u22a2 cantorFunctionAux c f n = 0 \u2228 cantorFunctionAux c f n = c ^ n\n[PROOFSTEP]\ncases h : f n\n[GOAL]\ncase false\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn\u271d : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\nn : \u2115\nh : f n = false\n\u22a2 cantorFunctionAux c f n = 0 \u2228 cantorFunctionAux c f n = c ^ n\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase true\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn\u271d : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\nn : \u2115\nh : f n = true\n\u22a2 cantorFunctionAux c f n = 0 \u2228 cantorFunctionAux c f n = c ^ n\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nh3 : \u2200 (n : \u2115), f n = true \u2192 g n = true\n\u22a2 cantorFunction c f \u2264 cantorFunction c g\n[PROOFSTEP]\napply tsum_le_tsum _ (summable_cantor_function f h1 h2) (summable_cantor_function g h1 h2)\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nh3 : \u2200 (n : \u2115), f n = true \u2192 g n = true\n\u22a2 \u2200 (i : \u2115), cantorFunctionAux c f i \u2264 cantorFunctionAux c g i\n[PROOFSTEP]\nintro n\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nh3 : \u2200 (n : \u2115), f n = true \u2192 g n = true\nn : \u2115\n\u22a2 cantorFunctionAux c f n \u2264 cantorFunctionAux c g n\n[PROOFSTEP]\ncases h : f n\n[GOAL]\ncase false\nc : \u211d\nf g : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nh3 : \u2200 (n : \u2115), f n = true \u2192 g n = true\nn : \u2115\nh : f n = false\n\u22a2 cantorFunctionAux c f n \u2264 cantorFunctionAux c g n\ncase true\nc : \u211d\nf g : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nh3 : \u2200 (n : \u2115), f n = true \u2192 g n = true\nn : \u2115\nh : f n = true\n\u22a2 cantorFunctionAux c f n \u2264 cantorFunctionAux c g n\n[PROOFSTEP]\nsimp [h, cantorFunctionAux_nonneg h1]\n[GOAL]\ncase true\nc : \u211d\nf g : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nh3 : \u2200 (n : \u2115), f n = true \u2192 g n = true\nn : \u2115\nh : f n = true\n\u22a2 cantorFunctionAux c f n \u2264 cantorFunctionAux c g n\n[PROOFSTEP]\nreplace h3 : g n = true := h3 n h\n[GOAL]\ncase true\nc : \u211d\nf g : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 \u2264 c\nh2 : c < 1\nn : \u2115\nh : f n = true\nh3 : g n = true\n\u22a2 cantorFunctionAux c f n \u2264 cantorFunctionAux c g n\n[PROOFSTEP]\nsimp [h, h3]\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\n\u22a2 cantorFunction c f = (bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)\n[PROOFSTEP]\nrw [cantorFunction, tsum_eq_zero_add (summable_cantor_function f h1 h2)]\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\n\u22a2 cantorFunctionAux c f 0 + \u2211' (b : \u2115), cantorFunctionAux c f (b + 1) =\n    (bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)\n[PROOFSTEP]\nrw [cantorFunctionAux_succ, tsum_mul_left, cantorFunctionAux, _root_.pow_zero]\n[GOAL]\nc : \u211d\nf\u271d g : \u2115 \u2192 Bool\nn : \u2115\nf : \u2115 \u2192 Bool\nh1 : 0 \u2264 c\nh2 : c < 1\n\u22a2 (bif f 0 then 1 else 0) + c * \u2211' (x : \u2115), cantorFunctionAux c (fun n => f (n + 1)) x =\n    (bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\ngn : g n = true\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nhave h3 : c < 1 := by\n  apply h2.trans\n  norm_num\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\ngn : g n = true\n\u22a2 c < 1\n[PROOFSTEP]\napply h2.trans\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\ngn : g n = true\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\ngn : g n = true\nh3 : c < 1\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\ninduction' n with n ih generalizing f g\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nlet f_max : \u2115 \u2192 Bool := fun n => Nat.rec false (fun _ _ => true) n\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nhave hf_max : \u2200 n, f n \u2192 f_max n := by\n  intro n hn\n  cases n\n  rw [fn] at hn \n  contradiction\n  apply rfl\n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\n\u22a2 \u2200 (n : \u2115), f n = true \u2192 f_max n = true\n[PROOFSTEP]\nintro n hn\n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nn : \u2115\nhn : f n = true\n\u22a2 f_max n = true\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhn : f zero = true\n\u22a2 f_max zero = true\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nn\u271d : \u2115\nhn : f (succ n\u271d) = true\n\u22a2 f_max (succ n\u271d) = true\n[PROOFSTEP]\nrw [fn] at hn \n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhn : false = true\n\u22a2 f_max zero = true\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nn\u271d : \u2115\nhn : f (succ n\u271d) = true\n\u22a2 f_max (succ n\u271d) = true\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nn\u271d : \u2115\nhn : f (succ n\u271d) = true\n\u22a2 f_max (succ n\u271d) = true\n[PROOFSTEP]\napply rfl\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nlet g_min : \u2115 \u2192 Bool := fun n => Nat.rec true (fun _ _ => false) n\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nhave hg_min : \u2200 n, g_min n \u2192 g n := by\n  intro n hn\n  cases n\n  rw [gn]\n  simp at hn \n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\n\u22a2 \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n[PROOFSTEP]\nintro n hn\n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nn : \u2115\nhn : g_min n = true\n\u22a2 g n = true\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhn : g_min zero = true\n\u22a2 g zero = true\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nn\u271d : \u2115\nhn : g_min (succ n\u271d) = true\n\u22a2 g (succ n\u271d) = true\n[PROOFSTEP]\nrw [gn]\n[GOAL]\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nn\u271d : \u2115\nhn : g_min (succ n\u271d) = true\n\u22a2 g (succ n\u271d) = true\n[PROOFSTEP]\nsimp at hn \n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\napply (cantorFunction_le (le_of_lt h1) h3 hf_max).trans_lt\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 (cantorFunction c fun n => f_max n) < cantorFunction c g\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (cantorFunction_le (le_of_lt h1) h3 hg_min)\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 (cantorFunction c fun n => f_max n) < cantorFunction c fun n => g_min n\n[PROOFSTEP]\nhave : c / (1 - c) < 1 := by\n  rw [div_lt_one, lt_sub_iff_add_lt]\n  \u00b7 convert _root_.add_lt_add h2 h2\n    norm_num\n  rwa [sub_pos]\n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 c / (1 - c) < 1\n[PROOFSTEP]\nrw [div_lt_one, lt_sub_iff_add_lt]\n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 c + c < 1\n[PROOFSTEP]\nconvert _root_.add_lt_add h2 h2\n[GOAL]\ncase h.e'_4\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 1 = 1 / 2 + 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\n\u22a2 0 < 1 - c\n[PROOFSTEP]\nrwa [sub_pos]\n[GOAL]\ncase zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\n\u22a2 (cantorFunction c fun n => f_max n) < cantorFunction c fun n => g_min n\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_3\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\n\u22a2 (cantorFunction c fun n => f_max n) = c / (1 - c)\n[PROOFSTEP]\nrw [cantorFunction_succ _ (le_of_lt h1) h3, div_eq_mul_inv, \u2190 tsum_geometric_of_lt_1 (le_of_lt h1) h3]\n[GOAL]\ncase h.e'_3\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\n\u22a2 ((bif f_max 0 then 1 else 0) + c * cantorFunction c fun n => f_max (n + 1)) = c * \u2211' (n : \u2115), c ^ n\n[PROOFSTEP]\napply zero_add\n[GOAL]\ncase h.e'_4\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\n\u22a2 (cantorFunction c fun n => g_min n) = 1\n[PROOFSTEP]\nrefine' (tsum_eq_single 0 _).trans _\n[GOAL]\ncase h.e'_4.refine'_1\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\n\u22a2 \u2200 (b' : \u2115), b' \u2260 0 \u2192 cantorFunctionAux c (fun n => g_min n) b' = 0\n[PROOFSTEP]\nintro n hn\n[GOAL]\ncase h.e'_4.refine'_1\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\nn : \u2115\nhn : n \u2260 0\n\u22a2 cantorFunctionAux c (fun n => g_min n) n = 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase h.e'_4.refine'_1.zero\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\nhn : zero \u2260 0\n\u22a2 cantorFunctionAux c (fun n => g_min n) zero = 0\ncase h.e'_4.refine'_1.succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\nn\u271d : \u2115\nhn : succ n\u271d \u2260 0\n\u22a2 cantorFunctionAux c (fun n => g_min n) (succ n\u271d) = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase h.e'_4.refine'_1.succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d\u00b9 : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\nn\u271d : \u2115\nhn : succ n\u271d \u2260 0\n\u22a2 cantorFunctionAux c (fun n => g_min n) (succ n\u271d) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_4.refine'_2\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n = false\ngn\u271d : g\u271d n = true\nh3 : c < 1\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < zero \u2192 f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : \u2115 \u2192 Bool := fun n => rec false (fun x x => true) n\nhf_max : \u2200 (n : \u2115), f n = true \u2192 f_max n = true\ng_min : \u2115 \u2192 Bool := fun n => rec true (fun x x => false) n\nhg_min : \u2200 (n : \u2115), g_min n = true \u2192 g n = true\nthis : c / (1 - c) < 1\n\u22a2 cantorFunctionAux c (fun n => g_min n) 0 = 1\n[PROOFSTEP]\nexact cantorFunctionAux_zero _\n[GOAL]\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nn : \u2115\nih :\n  \u2200 {f g : \u2115 \u2192 Bool},\n    (\u2200 (k : \u2115), k < n \u2192 f k = g k) \u2192 f n = false \u2192 g n = true \u2192 cantorFunction c f < cantorFunction c g\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < succ n \u2192 f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nrw [cantorFunction_succ f (le_of_lt h1) h3, cantorFunction_succ g (le_of_lt h1) h3]\n[GOAL]\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nn : \u2115\nih :\n  \u2200 {f g : \u2115 \u2192 Bool},\n    (\u2200 (k : \u2115), k < n \u2192 f k = g k) \u2192 f n = false \u2192 g n = true \u2192 cantorFunction c f < cantorFunction c g\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < succ n \u2192 f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n\u22a2 ((bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <\n    (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)\n[PROOFSTEP]\nrw [hn 0 <| zero_lt_succ n]\n[GOAL]\ncase succ\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nn : \u2115\nih :\n  \u2200 {f g : \u2115 \u2192 Bool},\n    (\u2200 (k : \u2115), k < n \u2192 f k = g k) \u2192 f n = false \u2192 g n = true \u2192 cantorFunction c f < cantorFunction c g\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < succ n \u2192 f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n\u22a2 ((bif g 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <\n    (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)\n[PROOFSTEP]\napply add_lt_add_left\n[GOAL]\ncase succ.bc\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nn : \u2115\nih :\n  \u2200 {f g : \u2115 \u2192 Bool},\n    (\u2200 (k : \u2115), k < n \u2192 f k = g k) \u2192 f n = false \u2192 g n = true \u2192 cantorFunction c f < cantorFunction c g\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < succ n \u2192 f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n\u22a2 (c * cantorFunction c fun n => f (n + 1)) < c * cantorFunction c fun n => g (n + 1)\n[PROOFSTEP]\nrw [mul_lt_mul_left h1]\n[GOAL]\ncase succ.bc\nc : \u211d\nf\u271d\u00b9 g\u271d\u00b9 : \u2115 \u2192 Bool\nn\u271d\u00b9 : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nn\u271d : \u2115\nf\u271d g\u271d : \u2115 \u2192 Bool\nhn\u271d : \u2200 (k : \u2115), k < n\u271d \u2192 f\u271d k = g\u271d k\nfn\u271d : f\u271d n\u271d = false\ngn\u271d : g\u271d n\u271d = true\nh3 : c < 1\nn : \u2115\nih :\n  \u2200 {f g : \u2115 \u2192 Bool},\n    (\u2200 (k : \u2115), k < n \u2192 f k = g k) \u2192 f n = false \u2192 g n = true \u2192 cantorFunction c f < cantorFunction c g\nf g : \u2115 \u2192 Bool\nhn : \u2200 (k : \u2115), k < succ n \u2192 f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n\u22a2 (cantorFunction c fun n => f (n + 1)) < cantorFunction c fun n => g (n + 1)\n[PROOFSTEP]\nexact ih (fun k hk => hn _ <| Nat.succ_lt_succ hk) fn gn\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\n\u22a2 Function.Injective (cantorFunction c)\n[PROOFSTEP]\nintro f g hfg\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nhfg : cantorFunction c f = cantorFunction c g\n\u22a2 f = g\n[PROOFSTEP]\nclassical\nby_contra h\nrevert hfg\nhave : \u2203 n, f n \u2260 g n := by\n  rw [\u2190 not_forall]\n  intro h'\n  apply h\n  ext\n  apply h'\nlet n := Nat.find this\nhave hn : \u2200 k : \u2115, k < n \u2192 f k = g k := by\n  intro k hk\n  apply of_not_not\n  exact Nat.find_min this hk\ncases fn : f n\n\u00b7 apply _root_.ne_of_lt\n  refine' increasing_cantorFunction h1 h2 hn fn _\n  apply Bool.eq_true_of_not_eq_false\n  rw [\u2190 fn]\n  apply Ne.symm\n  exact Nat.find_spec this\n\u00b7 apply _root_.ne_of_gt\n  refine' increasing_cantorFunction h1 h2 (fun k hk => (hn k hk).symm) _ fn\n  apply Bool.eq_false_of_not_eq_true\n  rw [\u2190 fn]\n  apply Ne.symm\n  exact Nat.find_spec this\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nhfg : cantorFunction c f = cantorFunction c g\n\u22a2 f = g\n[PROOFSTEP]\nby_contra h\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nhfg : cantorFunction c f = cantorFunction c g\nh : \u00acf = g\n\u22a2 False\n[PROOFSTEP]\nrevert hfg\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\n\u22a2 cantorFunction c f = cantorFunction c g \u2192 False\n[PROOFSTEP]\nhave : \u2203 n, f n \u2260 g n := by\n  rw [\u2190 not_forall]\n  intro h'\n  apply h\n  ext\n  apply h'\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\n\u22a2 \u2203 n, f n \u2260 g n\n[PROOFSTEP]\nrw [\u2190 not_forall]\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\n\u22a2 \u00ac\u2200 (x : \u2115), f x = g x\n[PROOFSTEP]\nintro h'\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nh' : \u2200 (x : \u2115), f x = g x\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nh' : \u2200 (x : \u2115), f x = g x\n\u22a2 f = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nh' : \u2200 (x : \u2115), f x = g x\nx\u271d : \u2115\n\u22a2 f x\u271d = g x\u271d\n[PROOFSTEP]\napply h'\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\n\u22a2 cantorFunction c f = cantorFunction c g \u2192 False\n[PROOFSTEP]\nlet n := Nat.find this\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\n\u22a2 cantorFunction c f = cantorFunction c g \u2192 False\n[PROOFSTEP]\nhave hn : \u2200 k : \u2115, k < n \u2192 f k = g k := by\n  intro k hk\n  apply of_not_not\n  exact Nat.find_min this hk\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\n\u22a2 \u2200 (k : \u2115), k < n \u2192 f k = g k\n[PROOFSTEP]\nintro k hk\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nk : \u2115\nhk : k < n\n\u22a2 f k = g k\n[PROOFSTEP]\napply of_not_not\n[GOAL]\ncase a\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nk : \u2115\nhk : k < n\n\u22a2 \u00ac\u00acf k = g k\n[PROOFSTEP]\nexact Nat.find_min this hk\n[GOAL]\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\n\u22a2 cantorFunction c f = cantorFunction c g \u2192 False\n[PROOFSTEP]\ncases fn : f n\n[GOAL]\ncase false\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\n\u22a2 cantorFunction c f = cantorFunction c g \u2192 False\n[PROOFSTEP]\napply _root_.ne_of_lt\n[GOAL]\ncase false.h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\n\u22a2 cantorFunction c f < cantorFunction c g\n[PROOFSTEP]\nrefine' increasing_cantorFunction h1 h2 hn fn _\n[GOAL]\ncase false.h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\n\u22a2 g n = true\n[PROOFSTEP]\napply Bool.eq_true_of_not_eq_false\n[GOAL]\ncase false.h.a\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\n\u22a2 \u00acg n = false\n[PROOFSTEP]\nrw [\u2190 fn]\n[GOAL]\ncase false.h.a\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\n\u22a2 \u00acg n = f n\n[PROOFSTEP]\napply Ne.symm\n[GOAL]\ncase false.h.a.h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = false\n\u22a2 f n \u2260 g n\n[PROOFSTEP]\nexact Nat.find_spec this\n[GOAL]\ncase true\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = true\n\u22a2 cantorFunction c f = cantorFunction c g \u2192 False\n[PROOFSTEP]\napply _root_.ne_of_gt\n[GOAL]\ncase true.h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = true\n\u22a2 cantorFunction c g < cantorFunction c f\n[PROOFSTEP]\nrefine' increasing_cantorFunction h1 h2 (fun k hk => (hn k hk).symm) _ fn\n[GOAL]\ncase true.h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = true\n\u22a2 g n = false\n[PROOFSTEP]\napply Bool.eq_false_of_not_eq_true\n[GOAL]\ncase true.h.a\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = true\n\u22a2 \u00acg n = true\n[PROOFSTEP]\nrw [\u2190 fn]\n[GOAL]\ncase true.h.a\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = true\n\u22a2 \u00acg n = f n\n[PROOFSTEP]\napply Ne.symm\n[GOAL]\ncase true.h.a.h\nc : \u211d\nf\u271d g\u271d : \u2115 \u2192 Bool\nn\u271d : \u2115\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : \u2115 \u2192 Bool\nh : \u00acf = g\nthis : \u2203 n, f n \u2260 g n\nn : \u2115 := Nat.find this\nhn : \u2200 (k : \u2115), k < n \u2192 f k = g k\nfn : f n = true\n\u22a2 f n \u2260 g n\n[PROOFSTEP]\nexact Nat.find_spec this\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 #\u211d = \ud835\udd20\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 #\u211d \u2264 \ud835\udd20\n[PROOFSTEP]\nrw [Real.equivCauchy.cardinal_eq]\n[GOAL]\ncase a\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 #(CauSeq.Completion.Cauchy abs) \u2264 \ud835\udd20\n[PROOFSTEP]\napply mk_quotient_le.trans\n[GOAL]\ncase a\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 #(CauSeq \u211a abs) \u2264 \ud835\udd20\n[PROOFSTEP]\napply (mk_subtype_le _).trans_eq\n[GOAL]\ncase a\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 #(\u2115 \u2192 \u211a) = \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 power_def, mk_nat, mkRat, aleph0_power_aleph0]\n[GOAL]\ncase a\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 \ud835\udd20 \u2264 #\u211d\n[PROOFSTEP]\nconvert mk_le_of_injective (cantorFunction_injective _ _)\n[GOAL]\ncase h.e'_3\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 \ud835\udd20 = #(\u2115 \u2192 Bool)\ncase a.convert_1\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 \u211d\ncase a.convert_2\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 0 < ?a.convert_1\u271d\ncase a.convert_3 c : \u211d f g : \u2115 \u2192 Bool n : \u2115 \u22a2 ?a.convert_1\u271d < 1 / 2\n[PROOFSTEP]\nrw [\u2190 power_def, mk_bool, mk_nat, two_power_aleph0]\n[GOAL]\ncase a.convert_1\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 \u211d\ncase a.convert_2\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 0 < ?a.convert_1\u271d\ncase a.convert_3 c : \u211d f g : \u2115 \u2192 Bool n : \u2115 \u22a2 ?a.convert_1\u271d < 1 / 2\n[PROOFSTEP]\nexact 1 / 3\n[GOAL]\ncase a.convert_2\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 0 < 1 / 3\ncase a.convert_3 c : \u211d f g : \u2115 \u2192 Bool n : \u2115 \u22a2 1 / 3 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase a.convert_3\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 1 / 3 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 #\u2191Set.univ = \ud835\udd20\n[PROOFSTEP]\nrw [mk_univ, mk_real]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 \u00acSet.Countable Set.univ\n[PROOFSTEP]\nrw [\u2190 le_aleph0_iff_set_countable, not_le, mk_univ_real]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\n\u22a2 \u2135\u2080 < \ud835\udd20\n[PROOFSTEP]\napply cantor\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\n\u22a2 #\u2191(Ioi a) = \ud835\udd20\n[PROOFSTEP]\nrefine' le_antisymm (mk_real \u25b8 mk_set_le _) _\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\n\u22a2 \ud835\udd20 \u2264 #\u2191(Ioi a)\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\n\u22a2 \u00ac#\u2191(Ioi a) < \ud835\udd20\n[PROOFSTEP]\nintro h\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\n\u22a2 False\n[PROOFSTEP]\nrefine' _root_.ne_of_lt _ mk_univ_real\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\n\u22a2 #\u2191Set.univ < \ud835\udd20\n[PROOFSTEP]\nhave hu : Iio a \u222a { a } \u222a Ioi a = Set.univ :=\n  by\n  convert @Iic_union_Ioi \u211d _ _\n  exact Iio_union_right\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\n\u22a2 Iio a \u222a {a} \u222a Ioi a = Set.univ\n[PROOFSTEP]\nconvert @Iic_union_Ioi \u211d _ _\n[GOAL]\ncase h.e'_2.h.e'_3\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\n\u22a2 Iio a \u222a {a} = Iic a\n[PROOFSTEP]\nexact Iio_union_right\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\n\u22a2 #\u2191Set.univ < \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 hu]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\n\u22a2 #\u2191(Iio a \u222a {a} \u222a Ioi a) < \ud835\udd20\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (mk_union_le _ _) _\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\n\u22a2 #\u2191(Iio a \u222a {a}) + #\u2191(Ioi a) < \ud835\udd20\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (add_le_add_right (mk_union_le _ _) _) _\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\n\u22a2 #\u2191(Iio a) + #\u2191{a} + #\u2191(Ioi a) < \ud835\udd20\n[PROOFSTEP]\nhave h2 : (fun x => a + a - x) '' Ioi a = Iio a :=\n  by\n  convert @image_const_sub_Ioi \u211d _ _ _\n  simp\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\n\u22a2 (fun x => a + a - x) '' Ioi a = Iio a\n[PROOFSTEP]\nconvert @image_const_sub_Ioi \u211d _ _ _\n[GOAL]\ncase h.e'_3.h.e'_3\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\n\u22a2 a = a + a - a\n[PROOFSTEP]\nsimp\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\nh2 : (fun x => a + a - x) '' Ioi a = Iio a\n\u22a2 #\u2191(Iio a) + #\u2191{a} + #\u2191(Ioi a) < \ud835\udd20\n[PROOFSTEP]\nrw [\u2190 h2]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\nh2 : (fun x => a + a - x) '' Ioi a = Iio a\n\u22a2 #\u2191((fun x => a + a - x) '' Ioi a) + #\u2191{a} + #\u2191(Ioi a) < \ud835\udd20\n[PROOFSTEP]\nrefine' add_lt_of_lt (cantor _).le _ h\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\nh2 : (fun x => a + a - x) '' Ioi a = Iio a\n\u22a2 #\u2191((fun x => a + a - x) '' Ioi a) + #\u2191{a} < \ud835\udd20\n[PROOFSTEP]\nrefine' add_lt_of_lt (cantor _).le (mk_image_le.trans_lt h) _\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\nh2 : (fun x => a + a - x) '' Ioi a = Iio a\n\u22a2 #\u2191{a} < \ud835\udd20\n[PROOFSTEP]\nrw [mk_singleton]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh : #\u2191(Ioi a) < \ud835\udd20\nhu : Iio a \u222a {a} \u222a Ioi a = Set.univ\nh2 : (fun x => a + a - x) '' Ioi a = Iio a\n\u22a2 1 < \ud835\udd20\n[PROOFSTEP]\nexact one_lt_aleph0.trans (cantor _)\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\n\u22a2 #\u2191(Iio a) = \ud835\udd20\n[PROOFSTEP]\nrefine' le_antisymm (mk_real \u25b8 mk_set_le _) _\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\n\u22a2 \ud835\udd20 \u2264 #\u2191(Iio a)\n[PROOFSTEP]\nhave h2 : (fun x => a + a - x) '' Iio a = Ioi a := by simp only [image_const_sub_Iio, add_sub_cancel]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\n\u22a2 (fun x => a + a - x) '' Iio a = Ioi a\n[PROOFSTEP]\nsimp only [image_const_sub_Iio, add_sub_cancel]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na : \u211d\nh2 : (fun x => a + a - x) '' Iio a = Ioi a\n\u22a2 \ud835\udd20 \u2264 #\u2191(Iio a)\n[PROOFSTEP]\nexact mk_Ioi_real a \u25b8 h2 \u25b8 mk_image_le\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\n\u22a2 #\u2191(Ioo a b) = \ud835\udd20\n[PROOFSTEP]\nrefine' le_antisymm (mk_real \u25b8 mk_set_le _) _\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\n\u22a2 \ud835\udd20 \u2264 #\u2191(Ioo a b)\n[PROOFSTEP]\nhave h1 : #((fun x => x - a) '' Ioo a b) \u2264 #(Ioo a b) := mk_image_le\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : #\u2191((fun x => x - a) '' Ioo a b) \u2264 #\u2191(Ioo a b)\n\u22a2 \ud835\udd20 \u2264 #\u2191(Ioo a b)\n[PROOFSTEP]\nrefine' le_trans _ h1\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : #\u2191((fun x => x - a) '' Ioo a b) \u2264 #\u2191(Ioo a b)\n\u22a2 \ud835\udd20 \u2264 #\u2191((fun x => x - a) '' Ioo a b)\n[PROOFSTEP]\nrw [image_sub_const_Ioo, sub_self]\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : #\u2191((fun x => x - a) '' Ioo a b) \u2264 #\u2191(Ioo a b)\n\u22a2 \ud835\udd20 \u2264 #\u2191(Ioo 0 (b - a))\n[PROOFSTEP]\nreplace h := sub_pos_of_lt h\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : #\u2191((fun x => x - a) '' Ioo a b) \u2264 #\u2191(Ioo a b)\nh : 0 < b - a\n\u22a2 \ud835\udd20 \u2264 #\u2191(Ioo 0 (b - a))\n[PROOFSTEP]\nhave h2 : #(Inv.inv '' Ioo 0 (b - a)) \u2264 #(Ioo 0 (b - a)) := mk_image_le\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : #\u2191((fun x => x - a) '' Ioo a b) \u2264 #\u2191(Ioo a b)\nh : 0 < b - a\nh2 : #\u2191(Inv.inv '' Ioo 0 (b - a)) \u2264 #\u2191(Ioo 0 (b - a))\n\u22a2 \ud835\udd20 \u2264 #\u2191(Ioo 0 (b - a))\n[PROOFSTEP]\nrefine' le_trans _ h2\n[GOAL]\nc : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : #\u2191((fun x => x - a) '' Ioo a b) \u2264 #\u2191(Ioo a b)\nh : 0 < b - a\nh2 : #\u2191(Inv.inv '' Ioo 0 (b - a)) \u2264 #\u2191(Ioo 0 (b - a))\n\u22a2 \ud835\udd20 \u2264 #\u2191(Inv.inv '' Ioo 0 (b - a))\n[PROOFSTEP]\nrw [image_inv, inv_Ioo_0_left h, mk_Ioi_real]\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.Cardinality", "llama_tokens": 22959, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.870597268408361, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5514452453397474}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nE : Type u_9\ninst\u271d\u2077 : AddCommMonoid E\ninst\u271d\u2076 : Module R E\nF : Type u_10\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\u2081\ninst\u271d\u00b3 : SMul R\u2081 R\ninst\u271d\u00b2 : IsScalarTower R\u2081 R R\ninst\u271d\u00b9 : SMul R\u2081 (s i)\ninst\u271d : IsScalarTower R\u2081 R (s i)\n\u22a2 tprodCoeff R z (update f i (r \u2022 f i)) = tprodCoeff R (r \u2022 z) f\n[PROOFSTEP]\nhave h\u2081 : r \u2022 z = r \u2022 (1 : R) * z := by rw [smul_mul_assoc, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nE : Type u_9\ninst\u271d\u2077 : AddCommMonoid E\ninst\u271d\u2076 : Module R E\nF : Type u_10\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\u2081\ninst\u271d\u00b3 : SMul R\u2081 R\ninst\u271d\u00b2 : IsScalarTower R\u2081 R R\ninst\u271d\u00b9 : SMul R\u2081 (s i)\ninst\u271d : IsScalarTower R\u2081 R (s i)\n\u22a2 r \u2022 z = r \u2022 1 * z\n[PROOFSTEP]\nrw [smul_mul_assoc, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nE : Type u_9\ninst\u271d\u2077 : AddCommMonoid E\ninst\u271d\u2076 : Module R E\nF : Type u_10\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\u2081\ninst\u271d\u00b3 : SMul R\u2081 R\ninst\u271d\u00b2 : IsScalarTower R\u2081 R R\ninst\u271d\u00b9 : SMul R\u2081 (s i)\ninst\u271d : IsScalarTower R\u2081 R (s i)\nh\u2081 : r \u2022 z = r \u2022 1 * z\n\u22a2 tprodCoeff R z (update f i (r \u2022 f i)) = tprodCoeff R (r \u2022 z) f\n[PROOFSTEP]\nhave h\u2082 : r \u2022 f i = (r \u2022 (1 : R)) \u2022 f i := (smul_one_smul _ _ _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nE : Type u_9\ninst\u271d\u2077 : AddCommMonoid E\ninst\u271d\u2076 : Module R E\nF : Type u_10\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\u2081\ninst\u271d\u00b3 : SMul R\u2081 R\ninst\u271d\u00b2 : IsScalarTower R\u2081 R R\ninst\u271d\u00b9 : SMul R\u2081 (s i)\ninst\u271d : IsScalarTower R\u2081 R (s i)\nh\u2081 : r \u2022 z = r \u2022 1 * z\nh\u2082 : r \u2022 f i = (r \u2022 1) \u2022 f i\n\u22a2 tprodCoeff R z (update f i (r \u2022 f i)) = tprodCoeff R (r \u2022 z) f\n[PROOFSTEP]\nrw [h\u2081, h\u2082]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b2 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nE : Type u_9\ninst\u271d\u2077 : AddCommMonoid E\ninst\u271d\u2076 : Module R E\nF : Type u_10\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\u2081\ninst\u271d\u00b3 : SMul R\u2081 R\ninst\u271d\u00b2 : IsScalarTower R\u2081 R R\ninst\u271d\u00b9 : SMul R\u2081 (s i)\ninst\u271d : IsScalarTower R\u2081 R (s i)\nh\u2081 : r \u2022 z = r \u2022 1 * z\nh\u2082 : r \u2022 f i = (r \u2022 1) \u2022 f i\n\u22a2 tprodCoeff R z (update f i ((r \u2022 1) \u2022 f i)) = tprodCoeff R (r \u2022 1 * z) f\n[PROOFSTEP]\nexact smul_tprodCoeff_aux z f i _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : R \u00d7 ((i : \u03b9) \u2192 s i) \u2192 F\nC0 : \u2200 (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9), f i = 0 \u2192 \u03c6 (r, f) = 0\nC0' : \u2200 (f : (i : \u03b9) \u2192 s i), \u03c6 (0, f) = 0\nC_add :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (m\u2081 m\u2082 : s i),\n    \u03c6 (r, update f i m\u2081) + \u03c6 (r, update f i m\u2082) = \u03c6 (r, update f i (m\u2081 + m\u2082))\nC_add_scalar : \u2200 (r r' : R) (f : (i : \u03b9) \u2192 s i), \u03c6 (r, f) + \u03c6 (r', f) = \u03c6 (r + r', f)\nC_smul :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (r' : R), \u03c6 (r, update f i (r' \u2022 f i)) = \u03c6 (r' * r, f)\nx y : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\nhxy : Eqv R s x y\nr' : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nhf : f i = 0\n\u22a2 \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (r', f)) = \u2191(\u2191FreeAddMonoid.lift \u03c6) 0\n[PROOFSTEP]\nsimp [FreeAddMonoid.lift_eval_of, C0 r' f i hf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : R \u00d7 ((i : \u03b9) \u2192 s i) \u2192 F\nC0 : \u2200 (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9), f i = 0 \u2192 \u03c6 (r, f) = 0\nC0' : \u2200 (f : (i : \u03b9) \u2192 s i), \u03c6 (0, f) = 0\nC_add :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (m\u2081 m\u2082 : s i),\n    \u03c6 (r, update f i m\u2081) + \u03c6 (r, update f i m\u2082) = \u03c6 (r, update f i (m\u2081 + m\u2082))\nC_add_scalar : \u2200 (r r' : R) (f : (i : \u03b9) \u2192 s i), \u03c6 (r, f) + \u03c6 (r', f) = \u03c6 (r + r', f)\nC_smul :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (r' : R), \u03c6 (r, update f i (r' \u2022 f i)) = \u03c6 (r' * r, f)\nx y : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\nhxy : Eqv R s x y\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (0, f)) = \u2191(\u2191FreeAddMonoid.lift \u03c6) 0\n[PROOFSTEP]\nsimp [FreeAddMonoid.lift_eval_of, C0']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : R \u00d7 ((i : \u03b9) \u2192 s i) \u2192 F\nC0 : \u2200 (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9), f i = 0 \u2192 \u03c6 (r, f) = 0\nC0' : \u2200 (f : (i : \u03b9) \u2192 s i), \u03c6 (0, f) = 0\nC_add :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (m\u2081 m\u2082 : s i),\n    \u03c6 (r, update f i m\u2081) + \u03c6 (r, update f i m\u2082) = \u03c6 (r, update f i (m\u2081 + m\u2082))\nC_add_scalar : \u2200 (r r' : R) (f : (i : \u03b9) \u2192 s i), \u03c6 (r, f) + \u03c6 (r', f) = \u03c6 (r + r', f)\nC_smul :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (r' : R), \u03c6 (r, update f i (r' \u2022 f i)) = \u03c6 (r' * r, f)\nx y : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\nhxy : Eqv R s x y\ninst : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nm\u2081 m\u2082 : s i\n\u22a2 \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (z, update f i m\u2081) + FreeAddMonoid.of (z, update f i m\u2082)) =\n    \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (z, update f i (m\u2081 + m\u2082)))\n[PROOFSTEP]\nsimp [FreeAddMonoid.lift_eval_of, @C_add inst]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : R \u00d7 ((i : \u03b9) \u2192 s i) \u2192 F\nC0 : \u2200 (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9), f i = 0 \u2192 \u03c6 (r, f) = 0\nC0' : \u2200 (f : (i : \u03b9) \u2192 s i), \u03c6 (0, f) = 0\nC_add :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (m\u2081 m\u2082 : s i),\n    \u03c6 (r, update f i m\u2081) + \u03c6 (r, update f i m\u2082) = \u03c6 (r, update f i (m\u2081 + m\u2082))\nC_add_scalar : \u2200 (r r' : R) (f : (i : \u03b9) \u2192 s i), \u03c6 (r, f) + \u03c6 (r', f) = \u03c6 (r + r', f)\nC_smul :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (r' : R), \u03c6 (r, update f i (r' \u2022 f i)) = \u03c6 (r' * r, f)\nx y : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\nhxy : Eqv R s x y\nz\u2081 z\u2082 : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (z\u2081, f) + FreeAddMonoid.of (z\u2082, f)) =\n    \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (z\u2081 + z\u2082, f))\n[PROOFSTEP]\nsimp [FreeAddMonoid.lift_eval_of, C_add_scalar]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : R \u00d7 ((i : \u03b9) \u2192 s i) \u2192 F\nC0 : \u2200 (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9), f i = 0 \u2192 \u03c6 (r, f) = 0\nC0' : \u2200 (f : (i : \u03b9) \u2192 s i), \u03c6 (0, f) = 0\nC_add :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (m\u2081 m\u2082 : s i),\n    \u03c6 (r, update f i m\u2081) + \u03c6 (r, update f i m\u2082) = \u03c6 (r, update f i (m\u2081 + m\u2082))\nC_add_scalar : \u2200 (r r' : R) (f : (i : \u03b9) \u2192 s i), \u03c6 (r, f) + \u03c6 (r', f) = \u03c6 (r + r', f)\nC_smul :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (r' : R), \u03c6 (r, update f i (r' \u2022 f i)) = \u03c6 (r' * r, f)\nx y : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\nhxy : Eqv R s x y\ninst : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr' : R\n\u22a2 \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (z, update f i (r' \u2022 f i))) =\n    \u2191(\u2191FreeAddMonoid.lift \u03c6) (FreeAddMonoid.of (r' * z, f))\n[PROOFSTEP]\nsimp [FreeAddMonoid.lift_eval_of, @C_smul inst]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : R \u00d7 ((i : \u03b9) \u2192 s i) \u2192 F\nC0 : \u2200 (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9), f i = 0 \u2192 \u03c6 (r, f) = 0\nC0' : \u2200 (f : (i : \u03b9) \u2192 s i), \u03c6 (0, f) = 0\nC_add :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (m\u2081 m\u2082 : s i),\n    \u03c6 (r, update f i m\u2081) + \u03c6 (r, update f i m\u2082) = \u03c6 (r, update f i (m\u2081 + m\u2082))\nC_add_scalar : \u2200 (r r' : R) (f : (i : \u03b9) \u2192 s i), \u03c6 (r, f) + \u03c6 (r', f) = \u03c6 (r + r', f)\nC_smul :\n  \u2200 [inst : DecidableEq \u03b9] (r : R) (f : (i : \u03b9) \u2192 s i) (i : \u03b9) (r' : R), \u03c6 (r, update f i (r' \u2022 f i)) = \u03c6 (r' * r, f)\nx\u271d y\u271d : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\nhxy : Eqv R s x\u271d y\u271d\nx y : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 s i))\n\u22a2 \u2191(\u2191FreeAddMonoid.lift \u03c6) (x + y) = \u2191(\u2191FreeAddMonoid.lift \u03c6) (y + x)\n[PROOFSTEP]\nsimp_rw [AddMonoidHom.map_add, add_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\n\u22a2 C z\n[PROOFSTEP]\nhave C0 : C 0 := by\n  have h\u2081 := @C1 0 0\n  rwa [zero_tprodCoeff] at h\u2081 \n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\n\u22a2 C 0\n[PROOFSTEP]\nhave h\u2081 := @C1 0 0\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\nh\u2081 : C (tprodCoeff R 0 0)\n\u22a2 C 0\n[PROOFSTEP]\nrwa [zero_tprodCoeff] at h\u2081 \n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\nC0 : C 0\n\u22a2 C z\n[PROOFSTEP]\nrefine' AddCon.induction_on z fun x \u21a6 FreeAddMonoid.recOn x C0 _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\nC0 : C 0\nx : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i))\n\u22a2 \u2200 (x : R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i)) (xs : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i))),\n    C \u2191xs \u2192 C \u2191(FreeAddMonoid.of x + xs)\n[PROOFSTEP]\nsimp_rw [AddCon.coe_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\nC0 : C 0\nx : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i))\n\u22a2 \u2200 (x : R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i)) (xs : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i))),\n    C \u2191xs \u2192 C (\u2191(FreeAddMonoid.of x) + \u2191xs)\n[PROOFSTEP]\nrefine' fun f y ih \u21a6 Cp _ ih\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\nC0 : C 0\nx : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i))\nf : R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i)\ny : FreeAddMonoid (R \u00d7 ((i : \u03b9) \u2192 (fun i => s i) i))\nih : C \u2191y\n\u22a2 C \u2191(FreeAddMonoid.of f)\n[PROOFSTEP]\nconvert @C1 f.1 f.2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nr : R\u2081\nr' : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nhf : f i = 0\n\u22a2 (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r', f) = 0\n[PROOFSTEP]\nsimp_rw [zero_tprodCoeff' _ f i hf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nr : R\u2081\nf : (i : \u03b9) \u2192 s i\n\u22a2 (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (0, f) = 0\n[PROOFSTEP]\nsimp [zero_tprodCoeff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2074 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nE : Type u_9\ninst\u271d\u2079 : AddCommMonoid E\ninst\u271d\u2078 : Module R E\nF : Type u_10\ninst\u271d\u2077 : AddCommMonoid F\ninst\u271d\u2076 : Monoid R\u2081\ninst\u271d\u2075 : DistribMulAction R\u2081 R\ninst\u271d\u2074 : SMulCommClass R\u2081 R R\ninst\u271d\u00b3 : Monoid R\u2082\ninst\u271d\u00b2 : DistribMulAction R\u2082 R\ninst\u271d\u00b9 : SMulCommClass R\u2082 R R\nr : R\u2081\ninst\u271d : DecidableEq \u03b9\nr' : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nm\u2081 m\u2082 : s i\n\u22a2 (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r', update f i m\u2081) +\n      (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r', update f i m\u2082) =\n    (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r', update f i (m\u2081 + m\u2082))\n[PROOFSTEP]\nsimp [add_tprodCoeff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nr : R\u2081\nr' r'' : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r', f) + (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r'', f) =\n    (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r' + r'', f)\n[PROOFSTEP]\nsimp [add_tprodCoeff', mul_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2074 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nE : Type u_9\ninst\u271d\u2079 : AddCommMonoid E\ninst\u271d\u2078 : Module R E\nF : Type u_10\ninst\u271d\u2077 : AddCommMonoid F\ninst\u271d\u2076 : Monoid R\u2081\ninst\u271d\u2075 : DistribMulAction R\u2081 R\ninst\u271d\u2074 : SMulCommClass R\u2081 R R\ninst\u271d\u00b3 : Monoid R\u2082\ninst\u271d\u00b2 : DistribMulAction R\u2082 R\ninst\u271d\u00b9 : SMulCommClass R\u2082 R R\nr : R\u2081\ninst\u271d : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr' : R\n\u22a2 (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (z, update f i (r' \u2022 f i)) =\n    (fun f => tprodCoeff R (r \u2022 f.fst) f.snd) (r' * z, f)\n[PROOFSTEP]\nsimp [smul_tprodCoeff, mul_smul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nx : \u2a02[R] (i : \u03b9), s i\nr : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 1 \u2022 tprodCoeff R r f = tprodCoeff R r f\n[PROOFSTEP]\nrw [smul_tprodCoeff', one_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nx z y : \u2a02[R] (i : \u03b9), s i\nihz : 1 \u2022 z = z\nihy : 1 \u2022 y = y\n\u22a2 1 \u2022 (z + y) = z + y\n[PROOFSTEP]\nsimp_rw [PiTensorProduct.smul_add, ihz, ihy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nr r' : R\u2081\nx : \u2a02[R] (i : \u03b9), s i\nr'' : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 (r * r') \u2022 tprodCoeff R r'' f = r \u2022 r' \u2022 tprodCoeff R r'' f\n[PROOFSTEP]\nsimp [smul_tprodCoeff', smul_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u00b3 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\nE : Type u_9\ninst\u271d\u2078 : AddCommMonoid E\ninst\u271d\u2077 : Module R E\nF : Type u_10\ninst\u271d\u2076 : AddCommMonoid F\ninst\u271d\u2075 : Monoid R\u2081\ninst\u271d\u2074 : DistribMulAction R\u2081 R\ninst\u271d\u00b3 : SMulCommClass R\u2081 R R\ninst\u271d\u00b2 : Monoid R\u2082\ninst\u271d\u00b9 : DistribMulAction R\u2082 R\ninst\u271d : SMulCommClass R\u2082 R R\nr r' : R\u2081\nx\u271d x y : \u2a02[R] (i : \u03b9), s i\nihx : (r * r') \u2022 x = r \u2022 r' \u2022 x\nihy : (r * r') \u2022 y = r \u2022 r' \u2022 y\n\u22a2 (r * r') \u2022 (x + y) = r \u2022 r' \u2022 (x + y)\n[PROOFSTEP]\nsimp_rw [PiTensorProduct.smul_add, ihx, ihy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2074 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nE : Type u_9\ninst\u271d\u2079 : AddCommMonoid E\ninst\u271d\u2078 : Module R E\nF : Type u_10\ninst\u271d\u2077 : AddCommMonoid F\ninst\u271d\u2076 : Monoid R\u2081\ninst\u271d\u2075 : DistribMulAction R\u2081 R\ninst\u271d\u2074 : SMulCommClass R\u2081 R R\ninst\u271d\u00b3 : Monoid R\u2082\ninst\u271d\u00b2 : DistribMulAction R\u2082 R\ninst\u271d\u00b9 : SMulCommClass R\u2082 R R\ninst\u271d : SMulCommClass R\u2081 R\u2082 R\nr' : R\u2081\nr'' : R\u2082\nx : \u2a02[R] (i : \u03b9), s i\nxr : R\nxf : (i : \u03b9) \u2192 s i\n\u22a2 r' \u2022 r'' \u2022 tprodCoeff R xr xf = r'' \u2022 r' \u2022 tprodCoeff R xr xf\n[PROOFSTEP]\nsimp only [smul_tprodCoeff', smul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2074 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nE : Type u_9\ninst\u271d\u2079 : AddCommMonoid E\ninst\u271d\u2078 : Module R E\nF : Type u_10\ninst\u271d\u2077 : AddCommMonoid F\ninst\u271d\u2076 : Monoid R\u2081\ninst\u271d\u2075 : DistribMulAction R\u2081 R\ninst\u271d\u2074 : SMulCommClass R\u2081 R R\ninst\u271d\u00b3 : Monoid R\u2082\ninst\u271d\u00b2 : DistribMulAction R\u2082 R\ninst\u271d\u00b9 : SMulCommClass R\u2082 R R\ninst\u271d : SMulCommClass R\u2081 R\u2082 R\nr' : R\u2081\nr'' : R\u2082\nx z y : \u2a02[R] (i : \u03b9), s i\nihz : r' \u2022 r'' \u2022 z = r'' \u2022 r' \u2022 z\nihy : r' \u2022 r'' \u2022 y = r'' \u2022 r' \u2022 y\n\u22a2 r' \u2022 r'' \u2022 (z + y) = r'' \u2022 r' \u2022 (z + y)\n[PROOFSTEP]\nsimp_rw [PiTensorProduct.smul_add, ihz, ihy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2075 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : Module R M\nE : Type u_9\ninst\u271d\u00b9\u2070 : AddCommMonoid E\ninst\u271d\u2079 : Module R E\nF : Type u_10\ninst\u271d\u2078 : AddCommMonoid F\ninst\u271d\u2077 : Monoid R\u2081\ninst\u271d\u2076 : DistribMulAction R\u2081 R\ninst\u271d\u2075 : SMulCommClass R\u2081 R R\ninst\u271d\u2074 : Monoid R\u2082\ninst\u271d\u00b3 : DistribMulAction R\u2082 R\ninst\u271d\u00b2 : SMulCommClass R\u2082 R R\ninst\u271d\u00b9 : SMul R\u2081 R\u2082\ninst\u271d : IsScalarTower R\u2081 R\u2082 R\nr' : R\u2081\nr'' : R\u2082\nx : \u2a02[R] (i : \u03b9), s i\nxr : R\nxf : (i : \u03b9) \u2192 s i\n\u22a2 (r' \u2022 r'') \u2022 tprodCoeff R xr xf = r' \u2022 r'' \u2022 tprodCoeff R xr xf\n[PROOFSTEP]\nsimp only [smul_tprodCoeff', smul_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2075 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : Module R M\nE : Type u_9\ninst\u271d\u00b9\u2070 : AddCommMonoid E\ninst\u271d\u2079 : Module R E\nF : Type u_10\ninst\u271d\u2078 : AddCommMonoid F\ninst\u271d\u2077 : Monoid R\u2081\ninst\u271d\u2076 : DistribMulAction R\u2081 R\ninst\u271d\u2075 : SMulCommClass R\u2081 R R\ninst\u271d\u2074 : Monoid R\u2082\ninst\u271d\u00b3 : DistribMulAction R\u2082 R\ninst\u271d\u00b2 : SMulCommClass R\u2082 R R\ninst\u271d\u00b9 : SMul R\u2081 R\u2082\ninst\u271d : IsScalarTower R\u2081 R\u2082 R\nr' : R\u2081\nr'' : R\u2082\nx z y : \u2a02[R] (i : \u03b9), s i\nihz : (r' \u2022 r'') \u2022 z = r' \u2022 r'' \u2022 z\nihy : (r' \u2022 r'') \u2022 y = r' \u2022 r'' \u2022 y\n\u22a2 (r' \u2022 r'') \u2022 (z + y) = r' \u2022 r'' \u2022 (z + y)\n[PROOFSTEP]\nsimp_rw [PiTensorProduct.smul_add, ihz, ihy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nE : Type u_9\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : Module R E\nF : Type u_10\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Module R\u2081 R\ninst\u271d : SMulCommClass R\u2081 R R\nsrc\u271d : DistribMulAction R\u2081 (\u2a02[R] (i : \u03b9), s i) := distribMulAction'\nr\u271d r' : R\u2081\nx : \u2a02[R] (i : \u03b9), s i\nr : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 (r\u271d + r') \u2022 tprodCoeff R r f = r\u271d \u2022 tprodCoeff R r f + r' \u2022 tprodCoeff R r f\n[PROOFSTEP]\nsimp_rw [smul_tprodCoeff', add_smul, add_tprodCoeff']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nE : Type u_9\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : Module R E\nF : Type u_10\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Module R\u2081 R\ninst\u271d : SMulCommClass R\u2081 R R\nsrc\u271d : DistribMulAction R\u2081 (\u2a02[R] (i : \u03b9), s i) := distribMulAction'\nr r' : R\u2081\nx\u271d x y : \u2a02[R] (i : \u03b9), s i\nihx : (r + r') \u2022 x = r \u2022 x + r' \u2022 x\nihy : (r + r') \u2022 y = r \u2022 y + r' \u2022 y\n\u22a2 (r + r') \u2022 (x + y) = r \u2022 (x + y) + r' \u2022 (x + y)\n[PROOFSTEP]\nsimp_rw [PiTensorProduct.smul_add, ihx, ihy, add_add_add_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nE : Type u_9\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : Module R E\nF : Type u_10\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Module R\u2081 R\ninst\u271d : SMulCommClass R\u2081 R R\nsrc\u271d : DistribMulAction R\u2081 (\u2a02[R] (i : \u03b9), s i) := distribMulAction'\nx : \u2a02[R] (i : \u03b9), s i\nr : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 0 \u2022 tprodCoeff R r f = 0\n[PROOFSTEP]\nsimp_rw [smul_tprodCoeff', zero_smul, zero_tprodCoeff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u00b9\u2070 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nE : Type u_9\ninst\u271d\u2075 : AddCommMonoid E\ninst\u271d\u2074 : Module R E\nF : Type u_10\ninst\u271d\u00b3 : AddCommMonoid F\ninst\u271d\u00b2 : Semiring R\u2081\ninst\u271d\u00b9 : Module R\u2081 R\ninst\u271d : SMulCommClass R\u2081 R R\nsrc\u271d : DistribMulAction R\u2081 (\u2a02[R] (i : \u03b9), s i) := distribMulAction'\nx\u271d x y : \u2a02[R] (i : \u03b9), s i\nihx : 0 \u2022 x = 0\nihy : 0 \u2022 y = 0\n\u22a2 0 \u2022 (x + y) = 0\n[PROOFSTEP]\nsimp_rw [PiTensorProduct.smul_add, ihx, ihy, add_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nx\u271d : DecidableEq \u03b9\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\nx : s i\n\u22a2 tprodCoeff R 1 (update f i (r \u2022 x)) = r \u2022 tprodCoeff R 1 (update f i x)\n[PROOFSTEP]\nrw [smul_tprodCoeff', \u2190 smul_tprodCoeff (1 : R) _ i, update_idem, update_same]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nz : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 tprodCoeff R z f = z \u2022 \u2191(tprod R) f\n[PROOFSTEP]\nhave : z = z \u2022 (1 : R) := by simp only [mul_one, Algebra.id.smul_eq_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nz : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 z = z \u2022 1\n[PROOFSTEP]\nsimp only [mul_one, Algebra.id.smul_eq_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nz : R\nf : (i : \u03b9) \u2192 s i\nthis : z = z \u2022 1\n\u22a2 tprodCoeff R z f = z \u2022 \u2191(tprod R) f\n[PROOFSTEP]\nconv_lhs => rw [this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nz : R\nf : (i : \u03b9) \u2192 s i\nthis : z = z \u2022 1\n| tprodCoeff R z f\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nz : R\nf : (i : \u03b9) \u2192 s i\nthis : z = z \u2022 1\n| tprodCoeff R z f\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nz : R\nf : (i : \u03b9) \u2192 s i\nthis : z = z \u2022 1\n| tprodCoeff R z f\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (r \u2022 \u2191(tprod R) f)\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\n\u22a2 C z\n[PROOFSTEP]\nsimp_rw [\u2190 tprodCoeff_eq_smul_tprod] at C1 \n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nC : (\u2a02[R] (i : \u03b9), s i) \u2192 Prop\nz : \u2a02[R] (i : \u03b9), s i\nCp : \u2200 {x y : \u2a02[R] (i : \u03b9), s i}, C x \u2192 C y \u2192 C (x + y)\nC1 : \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, C (tprodCoeff R r f)\n\u22a2 C z\n[PROOFSTEP]\nexact PiTensorProduct.induction_on' z @C1 @Cp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\n\u22a2 \u03c6\u2081 = \u03c6\u2082\n[PROOFSTEP]\nrefine' LinearMap.ext _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\n\u22a2 \u2200 (x : \u2a02[R] (i : \u03b9), s i), \u2191\u03c6\u2081 x = \u2191\u03c6\u2082 x\n[PROOFSTEP]\nrefine' fun z \u21a6 PiTensorProduct.induction_on' z _ fun {x y} hx hy \u21a6 by rw [\u03c6\u2081.map_add, \u03c6\u2082.map_add, hx, hy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\nz x y : \u2a02[R] (i : \u03b9), s i\nhx : \u2191\u03c6\u2081 x = \u2191\u03c6\u2082 x\nhy : \u2191\u03c6\u2081 y = \u2191\u03c6\u2082 y\n\u22a2 \u2191\u03c6\u2081 (x + y) = \u2191\u03c6\u2082 (x + y)\n[PROOFSTEP]\nrw [\u03c6\u2081.map_add, \u03c6\u2082.map_add, hx, hy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\nz : \u2a02[R] (i : \u03b9), s i\n\u22a2 \u2200 {r : R} {f : (i : \u03b9) \u2192 s i}, \u2191\u03c6\u2081 (tprodCoeff R r f) = \u2191\u03c6\u2082 (tprodCoeff R r f)\n[PROOFSTEP]\nintro r f\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\nz : \u2a02[R] (i : \u03b9), s i\nr : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191\u03c6\u2081 (tprodCoeff R r f) = \u2191\u03c6\u2082 (tprodCoeff R r f)\n[PROOFSTEP]\nrw [tprodCoeff_eq_smul_tprod, \u03c6\u2081.map_smul, \u03c6\u2082.map_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\nz : \u2a02[R] (i : \u03b9), s i\nr : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 r \u2022 \u2191\u03c6\u2081 (\u2191(tprod R) f) = r \u2022 \u2191\u03c6\u2082 (\u2191(tprod R) f)\n[PROOFSTEP]\napply _root_.congr_arg\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nH : LinearMap.compMultilinearMap \u03c6\u2081 (tprod R) = LinearMap.compMultilinearMap \u03c6\u2082 (tprod R)\nz : \u2a02[R] (i : \u03b9), s i\nr : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191\u03c6\u2081 (\u2191(tprod R) f) = \u2191\u03c6\u2082 (\u2191(tprod R) f)\n[PROOFSTEP]\nexact MultilinearMap.congr_fun H f\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nhf : f i = 0\n\u22a2 (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z, f) = 0\n[PROOFSTEP]\nsimp_rw [map_coord_zero \u03c6 i hf, smul_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (0, f) = 0\n[PROOFSTEP]\nsimp_rw [zero_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nm\u2081 m\u2082 : s i\n\u22a2 (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z, update f i m\u2081) + (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z, update f i m\u2082) =\n    (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z, update f i (m\u2081 + m\u2082))\n[PROOFSTEP]\nsimp_rw [\u2190 smul_add, \u03c6.map_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nz\u2081 z\u2082 : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z\u2081, f) + (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z\u2082, f) = (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z\u2081 + z\u2082, f)\n[PROOFSTEP]\nrw [\u2190 add_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : DecidableEq \u03b9\nz : R\nf : (i : \u03b9) \u2192 s i\ni : \u03b9\nr : R\n\u22a2 (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (z, update f i (r \u2022 f i)) = (fun p => p.fst \u2022 \u2191\u03c6 p.snd) (r * z, f)\n[PROOFSTEP]\nsimp [\u03c6.map_smul, smul_smul, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(liftAux \u03c6) (\u2191(tprod R) f) = \u2191\u03c6 f\n[PROOFSTEP]\nsimp only [liftAux, liftAddHom, tprod_eq_tprodCoeff_one, tprodCoeff, AddCon.coe_mk']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(AddCon.lift (addConGen (Eqv R fun i => s i)) (\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd)\n          (_ : addConGen (Eqv R fun i => s i) \u2264 AddCon.ker (\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd)))\n      \u2191(FreeAddMonoid.of (1, f)) =\n    \u2191\u03c6 f\n[PROOFSTEP]\nrw [FreeAddMonoid.of, FreeAddMonoid.ofList, Equiv.refl_apply, AddCon.lift_coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd) [(1, f)] = \u2191\u03c6 f\n[PROOFSTEP]\ndsimp [FreeAddMonoid.lift, FreeAddMonoid.sumAux]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.sumAux.match_1 (fun x => E) [1 \u2022 \u2191\u03c6 f] (fun _ => 0) fun x xs =>\n      List.foldl (fun x x_1 => x + x_1) x xs) =\n    \u2191\u03c6 f\n[PROOFSTEP]\nshow _ \u2022 _ = _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 1 \u2022 \u2191\u03c6 f = \u2191\u03c6 f\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nr : R\nx : \u2a02[R] (i : \u03b9), s i\n\u22a2 \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x\n[PROOFSTEP]\nrefine' PiTensorProduct.induction_on' x _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nr : R\nx : \u2a02[R] (i : \u03b9), s i\n\u22a2 \u2200 {r_1 : R} {f : (i : \u03b9) \u2192 s i}, \u2191(liftAux \u03c6) (r \u2022 tprodCoeff R r_1 f) = r \u2022 \u2191(liftAux \u03c6) (tprodCoeff R r_1 f)\n[PROOFSTEP]\nintro z f\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nr : R\nx : \u2a02[R] (i : \u03b9), s i\nz : R\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(liftAux \u03c6) (r \u2022 tprodCoeff R z f) = r \u2022 \u2191(liftAux \u03c6) (tprodCoeff R z f)\n[PROOFSTEP]\nrw [smul_tprodCoeff' r z f, liftAux_tprodCoeff, liftAux_tprodCoeff, smul_assoc]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nr : R\nx : \u2a02[R] (i : \u03b9), s i\n\u22a2 \u2200 {x y : \u2a02[R] (i : \u03b9), s i},\n    \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x \u2192\n      \u2191(liftAux \u03c6) (r \u2022 y) = r \u2022 \u2191(liftAux \u03c6) y \u2192 \u2191(liftAux \u03c6) (r \u2022 (x + y)) = r \u2022 \u2191(liftAux \u03c6) (x + y)\n[PROOFSTEP]\nintro z y ihz ihy\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nr : R\nx z y : \u2a02[R] (i : \u03b9), s i\nihz : \u2191(liftAux \u03c6) (r \u2022 z) = r \u2022 \u2191(liftAux \u03c6) z\nihy : \u2191(liftAux \u03c6) (r \u2022 y) = r \u2022 \u2191(liftAux \u03c6) y\n\u22a2 \u2191(liftAux \u03c6) (r \u2022 (z + y)) = r \u2022 \u2191(liftAux \u03c6) (z + y)\n[PROOFSTEP]\nrw [smul_add, (liftAux \u03c6).map_add, ihz, ihy, (liftAux \u03c6).map_add, smul_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E\n\u22a2 (fun \u03c6 =>\n        let src := liftAux \u03c6;\n        {\n          toAddHom :=\n            { toFun := src.toFun,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                    ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n          map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n      (\u03c6\u2081 + \u03c6\u2082) =\n    (fun \u03c6 =>\n          let src := liftAux \u03c6;\n          {\n            toAddHom :=\n              { toFun := src.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n            map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n        \u03c6\u2081 +\n      (fun \u03c6 =>\n          let src := liftAux \u03c6;\n          {\n            toAddHom :=\n              { toFun := src.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n            map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n        \u03c6\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E\nx\u271d : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          ((fun \u03c6 =>\n              let src := liftAux \u03c6;\n              {\n                toAddHom :=\n                  { toFun := src.toFun,\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                          ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n            (\u03c6\u2081 + \u03c6\u2082))\n          (tprod R))\n      x\u271d =\n    \u2191(LinearMap.compMultilinearMap\n          ((fun \u03c6 =>\n                let src := liftAux \u03c6;\n                {\n                  toAddHom :=\n                    { toFun := src.toFun,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                            ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                  map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n              \u03c6\u2081 +\n            (fun \u03c6 =>\n                let src := liftAux \u03c6;\n                {\n                  toAddHom :=\n                    { toFun := src.toFun,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                            ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                  map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n              \u03c6\u2082)\n          (tprod R))\n      x\u271d\n[PROOFSTEP]\nsimp [liftAux_tprod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nr : R\n\u03c6\u2082 : MultilinearMap R s E\n\u22a2 AddHom.toFun\n      {\n        toFun := fun \u03c6 =>\n          let src := liftAux \u03c6;\n          {\n            toAddHom :=\n              { toFun := src.toFun,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n            map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n        map_add' :=\n          (_ :\n            \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n              (fun \u03c6 =>\n                    let src := liftAux \u03c6;\n                    {\n                      toAddHom :=\n                        { toFun := src.toFun,\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                      map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                  (\u03c6\u2081 + \u03c6\u2082) =\n                (fun \u03c6 =>\n                      let src := liftAux \u03c6;\n                      {\n                        toAddHom :=\n                          { toFun := src.toFun,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                  ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                        map_smul' :=\n                          (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                    \u03c6\u2081 +\n                  (fun \u03c6 =>\n                      let src := liftAux \u03c6;\n                      {\n                        toAddHom :=\n                          { toFun := src.toFun,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                  ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                        map_smul' :=\n                          (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                    \u03c6\u2082) }\n      (r \u2022 \u03c6\u2082) =\n    \u2191(RingHom.id R) r \u2022\n      AddHom.toFun\n        {\n          toFun := fun \u03c6 =>\n            let src := liftAux \u03c6;\n            {\n              toAddHom :=\n                { toFun := src.toFun,\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                        ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n              map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n          map_add' :=\n            (_ :\n              \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                (fun \u03c6 =>\n                      let src := liftAux \u03c6;\n                      {\n                        toAddHom :=\n                          { toFun := src.toFun,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                  ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                        map_smul' :=\n                          (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                    (\u03c6\u2081 + \u03c6\u2082) =\n                  (fun \u03c6 =>\n                        let src := liftAux \u03c6;\n                        {\n                          toAddHom :=\n                            { toFun := src.toFun,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                    ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                          map_smul' :=\n                            (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                      \u03c6\u2081 +\n                    (fun \u03c6 =>\n                        let src := liftAux \u03c6;\n                        {\n                          toAddHom :=\n                            { toFun := src.toFun,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                    ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                          map_smul' :=\n                            (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                      \u03c6\u2082) }\n        \u03c6\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\nr : R\n\u03c6\u2082 : MultilinearMap R s E\nx\u271d : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (AddHom.toFun\n            {\n              toFun := fun \u03c6 =>\n                let src := liftAux \u03c6;\n                {\n                  toAddHom :=\n                    { toFun := src.toFun,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                            ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                  map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n              map_add' :=\n                (_ :\n                  \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                    (fun \u03c6 =>\n                          let src := liftAux \u03c6;\n                          {\n                            toAddHom :=\n                              { toFun := src.toFun,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                            map_smul' :=\n                              (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                        (\u03c6\u2081 + \u03c6\u2082) =\n                      (fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                          \u03c6\u2081 +\n                        (fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                          \u03c6\u2082) }\n            (r \u2022 \u03c6\u2082))\n          (tprod R))\n      x\u271d =\n    \u2191(LinearMap.compMultilinearMap\n          (\u2191(RingHom.id R) r \u2022\n            AddHom.toFun\n              {\n                toFun := fun \u03c6 =>\n                  let src := liftAux \u03c6;\n                  {\n                    toAddHom :=\n                      { toFun := src.toFun,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                              ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                    map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                map_add' :=\n                  (_ :\n                    \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                      (fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                          (\u03c6\u2081 + \u03c6\u2082) =\n                        (fun \u03c6 =>\n                              let src := liftAux \u03c6;\n                              {\n                                toAddHom :=\n                                  { toFun := src.toFun,\n                                    map_add' :=\n                                      (_ :\n                                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                          ZeroHom.toFun (\u2191src) (x + y) =\n                                            ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                map_smul' :=\n                                  (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                            \u03c6\u2081 +\n                          (fun \u03c6 =>\n                              let src := liftAux \u03c6;\n                              {\n                                toAddHom :=\n                                  { toFun := src.toFun,\n                                    map_add' :=\n                                      (_ :\n                                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                          ZeroHom.toFun (\u2191src) (x + y) =\n                                            ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                map_smul' :=\n                                  (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                            \u03c6\u2082) }\n              \u03c6\u2082)\n          (tprod R))\n      x\u271d\n[PROOFSTEP]\nsimp [liftAux_tprod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\n\u22a2 (fun \u03c6' => LinearMap.compMultilinearMap \u03c6' (tprod R))\n      (AddHom.toFun\n        {\n            toAddHom :=\n              {\n                toFun := fun \u03c6 =>\n                  let src := liftAux \u03c6;\n                  {\n                    toAddHom :=\n                      { toFun := src.toFun,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                              ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                    map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                map_add' :=\n                  (_ :\n                    \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                      (fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                          (\u03c6\u2081 + \u03c6\u2082) =\n                        (fun \u03c6 =>\n                              let src := liftAux \u03c6;\n                              {\n                                toAddHom :=\n                                  { toFun := src.toFun,\n                                    map_add' :=\n                                      (_ :\n                                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                          ZeroHom.toFun (\u2191src) (x + y) =\n                                            ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                map_smul' :=\n                                  (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                            \u03c6\u2081 +\n                          (fun \u03c6 =>\n                              let src := liftAux \u03c6;\n                              {\n                                toAddHom :=\n                                  { toFun := src.toFun,\n                                    map_add' :=\n                                      (_ :\n                                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                          ZeroHom.toFun (\u2191src) (x + y) =\n                                            ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                map_smul' :=\n                                  (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                            \u03c6\u2082) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (\u03c6\u2082 : MultilinearMap R s E),\n                  AddHom.toFun\n                      {\n                        toFun := fun \u03c6 =>\n                          let src := liftAux \u03c6;\n                          {\n                            toAddHom :=\n                              { toFun := src.toFun,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                            map_smul' :=\n                              (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                              (fun \u03c6 =>\n                                    let src := liftAux \u03c6;\n                                    {\n                                      toAddHom :=\n                                        { toFun := src.toFun,\n                                          map_add' :=\n                                            (_ :\n                                              \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                ZeroHom.toFun (\u2191src) (x + y) =\n                                                  ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                      map_smul' :=\n                                        (_ :\n                                          \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                            \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                  (\u03c6\u2081 + \u03c6\u2082) =\n                                (fun \u03c6 =>\n                                      let src := liftAux \u03c6;\n                                      {\n                                        toAddHom :=\n                                          { toFun := src.toFun,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                  ZeroHom.toFun (\u2191src) (x + y) =\n                                                    ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                        map_smul' :=\n                                          (_ :\n                                            \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                              \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                    \u03c6\u2081 +\n                                  (fun \u03c6 =>\n                                      let src := liftAux \u03c6;\n                                      {\n                                        toAddHom :=\n                                          { toFun := src.toFun,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                  ZeroHom.toFun (\u2191src) (x + y) =\n                                                    ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                        map_smul' :=\n                                          (_ :\n                                            \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                              \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                    \u03c6\u2082) }\n                      (r \u2022 \u03c6\u2082) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        {\n                          toFun := fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                          map_add' :=\n                            (_ :\n                              \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                                (fun \u03c6 =>\n                                      let src := liftAux \u03c6;\n                                      {\n                                        toAddHom :=\n                                          { toFun := src.toFun,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                  ZeroHom.toFun (\u2191src) (x + y) =\n                                                    ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                        map_smul' :=\n                                          (_ :\n                                            \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                              \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                    (\u03c6\u2081 + \u03c6\u2082) =\n                                  (fun \u03c6 =>\n                                        let src := liftAux \u03c6;\n                                        {\n                                          toAddHom :=\n                                            { toFun := src.toFun,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                    ZeroHom.toFun (\u2191src) (x + y) =\n                                                      ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                          map_smul' :=\n                                            (_ :\n                                              \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                      \u03c6\u2081 +\n                                    (fun \u03c6 =>\n                                        let src := liftAux \u03c6;\n                                        {\n                                          toAddHom :=\n                                            { toFun := src.toFun,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                    ZeroHom.toFun (\u2191src) (x + y) =\n                                                      ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                          map_smul' :=\n                                            (_ :\n                                              \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                      \u03c6\u2082) }\n                        \u03c6\u2082) }.toAddHom\n        \u03c6) =\n    \u03c6\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx\u271d : (i : \u03b9) \u2192 s i\n\u22a2 \u2191((fun \u03c6' => LinearMap.compMultilinearMap \u03c6' (tprod R))\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  {\n                    toFun := fun \u03c6 =>\n                      let src := liftAux \u03c6;\n                      {\n                        toAddHom :=\n                          { toFun := src.toFun,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                  ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                        map_smul' :=\n                          (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                    map_add' :=\n                      (_ :\n                        \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                          (fun \u03c6 =>\n                                let src := liftAux \u03c6;\n                                {\n                                  toAddHom :=\n                                    { toFun := src.toFun,\n                                      map_add' :=\n                                        (_ :\n                                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                            ZeroHom.toFun (\u2191src) (x + y) =\n                                              ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                  map_smul' :=\n                                    (_ :\n                                      \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                              (\u03c6\u2081 + \u03c6\u2082) =\n                            (fun \u03c6 =>\n                                  let src := liftAux \u03c6;\n                                  {\n                                    toAddHom :=\n                                      { toFun := src.toFun,\n                                        map_add' :=\n                                          (_ :\n                                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                              ZeroHom.toFun (\u2191src) (x + y) =\n                                                ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                    map_smul' :=\n                                      (_ :\n                                        \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                \u03c6\u2081 +\n                              (fun \u03c6 =>\n                                  let src := liftAux \u03c6;\n                                  {\n                                    toAddHom :=\n                                      { toFun := src.toFun,\n                                        map_add' :=\n                                          (_ :\n                                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                              ZeroHom.toFun (\u2191src) (x + y) =\n                                                ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                    map_smul' :=\n                                      (_ :\n                                        \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                \u03c6\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (\u03c6\u2082 : MultilinearMap R s E),\n                      AddHom.toFun\n                          {\n                            toFun := fun \u03c6 =>\n                              let src := liftAux \u03c6;\n                              {\n                                toAddHom :=\n                                  { toFun := src.toFun,\n                                    map_add' :=\n                                      (_ :\n                                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                          ZeroHom.toFun (\u2191src) (x + y) =\n                                            ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                map_smul' :=\n                                  (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                            map_add' :=\n                              (_ :\n                                \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                                  (fun \u03c6 =>\n                                        let src := liftAux \u03c6;\n                                        {\n                                          toAddHom :=\n                                            { toFun := src.toFun,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                    ZeroHom.toFun (\u2191src) (x + y) =\n                                                      ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                          map_smul' :=\n                                            (_ :\n                                              \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                      (\u03c6\u2081 + \u03c6\u2082) =\n                                    (fun \u03c6 =>\n                                          let src := liftAux \u03c6;\n                                          {\n                                            toAddHom :=\n                                              { toFun := src.toFun,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                      ZeroHom.toFun (\u2191src) (x + y) =\n                                                        ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                            map_smul' :=\n                                              (_ :\n                                                \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                  \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                        \u03c6\u2081 +\n                                      (fun \u03c6 =>\n                                          let src := liftAux \u03c6;\n                                          {\n                                            toAddHom :=\n                                              { toFun := src.toFun,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                      ZeroHom.toFun (\u2191src) (x + y) =\n                                                        ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                            map_smul' :=\n                                              (_ :\n                                                \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                  \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                        \u03c6\u2082) }\n                          (r \u2022 \u03c6\u2082) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            {\n                              toFun := fun \u03c6 =>\n                                let src := liftAux \u03c6;\n                                {\n                                  toAddHom :=\n                                    { toFun := src.toFun,\n                                      map_add' :=\n                                        (_ :\n                                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                            ZeroHom.toFun (\u2191src) (x + y) =\n                                              ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                  map_smul' :=\n                                    (_ :\n                                      \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                              map_add' :=\n                                (_ :\n                                  \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                                    (fun \u03c6 =>\n                                          let src := liftAux \u03c6;\n                                          {\n                                            toAddHom :=\n                                              { toFun := src.toFun,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                      ZeroHom.toFun (\u2191src) (x + y) =\n                                                        ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                            map_smul' :=\n                                              (_ :\n                                                \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                  \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                        (\u03c6\u2081 + \u03c6\u2082) =\n                                      (fun \u03c6 =>\n                                            let src := liftAux \u03c6;\n                                            {\n                                              toAddHom :=\n                                                { toFun := src.toFun,\n                                                  map_add' :=\n                                                    (_ :\n                                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                              map_smul' :=\n                                                (_ :\n                                                  \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                    \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                          \u03c6\u2081 +\n                                        (fun \u03c6 =>\n                                            let src := liftAux \u03c6;\n                                            {\n                                              toAddHom :=\n                                                { toFun := src.toFun,\n                                                  map_add' :=\n                                                    (_ :\n                                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                              map_smul' :=\n                                                (_ :\n                                                  \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                    \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                          \u03c6\u2082) }\n                            \u03c6\u2082) }.toAddHom\n            \u03c6))\n      x\u271d =\n    \u2191\u03c6 x\u271d\n[PROOFSTEP]\nsimp [liftAux_tprod, LinearMap.compMultilinearMap]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            {\n              toFun := fun \u03c6 =>\n                let src := liftAux \u03c6;\n                {\n                  toAddHom :=\n                    { toFun := src.toFun,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                            ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                  map_smul' := (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n              map_add' :=\n                (_ :\n                  \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                    (fun \u03c6 =>\n                          let src := liftAux \u03c6;\n                          {\n                            toAddHom :=\n                              { toFun := src.toFun,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                            map_smul' :=\n                              (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                        (\u03c6\u2081 + \u03c6\u2082) =\n                      (fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                          \u03c6\u2081 +\n                        (fun \u03c6 =>\n                            let src := liftAux \u03c6;\n                            {\n                              toAddHom :=\n                                { toFun := src.toFun,\n                                  map_add' :=\n                                    (_ :\n                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                              map_smul' :=\n                                (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                          \u03c6\u2082) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (\u03c6\u2082 : MultilinearMap R s E),\n                AddHom.toFun\n                    {\n                      toFun := fun \u03c6 =>\n                        let src := liftAux \u03c6;\n                        {\n                          toAddHom :=\n                            { toFun := src.toFun,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                    ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                          map_smul' :=\n                            (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                      map_add' :=\n                        (_ :\n                          \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                            (fun \u03c6 =>\n                                  let src := liftAux \u03c6;\n                                  {\n                                    toAddHom :=\n                                      { toFun := src.toFun,\n                                        map_add' :=\n                                          (_ :\n                                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                              ZeroHom.toFun (\u2191src) (x + y) =\n                                                ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                    map_smul' :=\n                                      (_ :\n                                        \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                (\u03c6\u2081 + \u03c6\u2082) =\n                              (fun \u03c6 =>\n                                    let src := liftAux \u03c6;\n                                    {\n                                      toAddHom :=\n                                        { toFun := src.toFun,\n                                          map_add' :=\n                                            (_ :\n                                              \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                ZeroHom.toFun (\u2191src) (x + y) =\n                                                  ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                      map_smul' :=\n                                        (_ :\n                                          \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                            \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                  \u03c6\u2081 +\n                                (fun \u03c6 =>\n                                    let src := liftAux \u03c6;\n                                    {\n                                      toAddHom :=\n                                        { toFun := src.toFun,\n                                          map_add' :=\n                                            (_ :\n                                              \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                ZeroHom.toFun (\u2191src) (x + y) =\n                                                  ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                      map_smul' :=\n                                        (_ :\n                                          \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                            \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                  \u03c6\u2082) }\n                    (r \u2022 \u03c6\u2082) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      {\n                        toFun := fun \u03c6 =>\n                          let src := liftAux \u03c6;\n                          {\n                            toAddHom :=\n                              { toFun := src.toFun,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                      ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                            map_smul' :=\n                              (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                              (fun \u03c6 =>\n                                    let src := liftAux \u03c6;\n                                    {\n                                      toAddHom :=\n                                        { toFun := src.toFun,\n                                          map_add' :=\n                                            (_ :\n                                              \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                ZeroHom.toFun (\u2191src) (x + y) =\n                                                  ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                      map_smul' :=\n                                        (_ :\n                                          \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                            \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                  (\u03c6\u2081 + \u03c6\u2082) =\n                                (fun \u03c6 =>\n                                      let src := liftAux \u03c6;\n                                      {\n                                        toAddHom :=\n                                          { toFun := src.toFun,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                  ZeroHom.toFun (\u2191src) (x + y) =\n                                                    ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                        map_smul' :=\n                                          (_ :\n                                            \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                              \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                    \u03c6\u2081 +\n                                  (fun \u03c6 =>\n                                      let src := liftAux \u03c6;\n                                      {\n                                        toAddHom :=\n                                          { toFun := src.toFun,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                  ZeroHom.toFun (\u2191src) (x + y) =\n                                                    ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                        map_smul' :=\n                                          (_ :\n                                            \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                              \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                    \u03c6\u2082) }\n                      \u03c6\u2082) }.toAddHom\n      ((fun \u03c6' => LinearMap.compMultilinearMap \u03c6' (tprod R)) \u03c6) =\n    \u03c6\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : (\u2a02[R] (i : \u03b9), s i) \u2192\u2097[R] E\nx\u271d : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  {\n                    toFun := fun \u03c6 =>\n                      let src := liftAux \u03c6;\n                      {\n                        toAddHom :=\n                          { toFun := src.toFun,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                  ZeroHom.toFun (\u2191src) (x + y) = ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                        map_smul' :=\n                          (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                    map_add' :=\n                      (_ :\n                        \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                          (fun \u03c6 =>\n                                let src := liftAux \u03c6;\n                                {\n                                  toAddHom :=\n                                    { toFun := src.toFun,\n                                      map_add' :=\n                                        (_ :\n                                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                            ZeroHom.toFun (\u2191src) (x + y) =\n                                              ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                  map_smul' :=\n                                    (_ :\n                                      \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                              (\u03c6\u2081 + \u03c6\u2082) =\n                            (fun \u03c6 =>\n                                  let src := liftAux \u03c6;\n                                  {\n                                    toAddHom :=\n                                      { toFun := src.toFun,\n                                        map_add' :=\n                                          (_ :\n                                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                              ZeroHom.toFun (\u2191src) (x + y) =\n                                                ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                    map_smul' :=\n                                      (_ :\n                                        \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                \u03c6\u2081 +\n                              (fun \u03c6 =>\n                                  let src := liftAux \u03c6;\n                                  {\n                                    toAddHom :=\n                                      { toFun := src.toFun,\n                                        map_add' :=\n                                          (_ :\n                                            \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                              ZeroHom.toFun (\u2191src) (x + y) =\n                                                ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                    map_smul' :=\n                                      (_ :\n                                        \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                \u03c6\u2082) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (\u03c6\u2082 : MultilinearMap R s E),\n                      AddHom.toFun\n                          {\n                            toFun := fun \u03c6 =>\n                              let src := liftAux \u03c6;\n                              {\n                                toAddHom :=\n                                  { toFun := src.toFun,\n                                    map_add' :=\n                                      (_ :\n                                        \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                          ZeroHom.toFun (\u2191src) (x + y) =\n                                            ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                map_smul' :=\n                                  (_ : \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                            map_add' :=\n                              (_ :\n                                \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                                  (fun \u03c6 =>\n                                        let src := liftAux \u03c6;\n                                        {\n                                          toAddHom :=\n                                            { toFun := src.toFun,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                    ZeroHom.toFun (\u2191src) (x + y) =\n                                                      ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                          map_smul' :=\n                                            (_ :\n                                              \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                      (\u03c6\u2081 + \u03c6\u2082) =\n                                    (fun \u03c6 =>\n                                          let src := liftAux \u03c6;\n                                          {\n                                            toAddHom :=\n                                              { toFun := src.toFun,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                      ZeroHom.toFun (\u2191src) (x + y) =\n                                                        ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                            map_smul' :=\n                                              (_ :\n                                                \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                  \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                        \u03c6\u2081 +\n                                      (fun \u03c6 =>\n                                          let src := liftAux \u03c6;\n                                          {\n                                            toAddHom :=\n                                              { toFun := src.toFun,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                      ZeroHom.toFun (\u2191src) (x + y) =\n                                                        ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                            map_smul' :=\n                                              (_ :\n                                                \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                  \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                        \u03c6\u2082) }\n                          (r \u2022 \u03c6\u2082) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            {\n                              toFun := fun \u03c6 =>\n                                let src := liftAux \u03c6;\n                                {\n                                  toAddHom :=\n                                    { toFun := src.toFun,\n                                      map_add' :=\n                                        (_ :\n                                          \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                            ZeroHom.toFun (\u2191src) (x + y) =\n                                              ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                  map_smul' :=\n                                    (_ :\n                                      \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i), \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) },\n                              map_add' :=\n                                (_ :\n                                  \u2200 (\u03c6\u2081 \u03c6\u2082 : MultilinearMap R s E),\n                                    (fun \u03c6 =>\n                                          let src := liftAux \u03c6;\n                                          {\n                                            toAddHom :=\n                                              { toFun := src.toFun,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                      ZeroHom.toFun (\u2191src) (x + y) =\n                                                        ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                            map_smul' :=\n                                              (_ :\n                                                \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                  \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                        (\u03c6\u2081 + \u03c6\u2082) =\n                                      (fun \u03c6 =>\n                                            let src := liftAux \u03c6;\n                                            {\n                                              toAddHom :=\n                                                { toFun := src.toFun,\n                                                  map_add' :=\n                                                    (_ :\n                                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                              map_smul' :=\n                                                (_ :\n                                                  \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                    \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                          \u03c6\u2081 +\n                                        (fun \u03c6 =>\n                                            let src := liftAux \u03c6;\n                                            {\n                                              toAddHom :=\n                                                { toFun := src.toFun,\n                                                  map_add' :=\n                                                    (_ :\n                                                      \u2200 (x y : \u2a02[R] (i : \u03b9), s i),\n                                                        ZeroHom.toFun (\u2191src) (x + y) =\n                                                          ZeroHom.toFun (\u2191src) x + ZeroHom.toFun (\u2191src) y) },\n                                              map_smul' :=\n                                                (_ :\n                                                  \u2200 (r : R) (x : \u2a02[R] (i : \u03b9), s i),\n                                                    \u2191(liftAux \u03c6) (r \u2022 x) = r \u2022 \u2191(liftAux \u03c6) x) })\n                                          \u03c6\u2082) }\n                            \u03c6\u2082) }.toAddHom\n            ((fun \u03c6' => LinearMap.compMultilinearMap \u03c6' (tprod R)) \u03c6))\n          (tprod R))\n      x\u271d =\n    \u2191(LinearMap.compMultilinearMap \u03c6 (tprod R)) x\u271d\n[PROOFSTEP]\nsimp [liftAux_tprod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\n\u22a2 LinearMap.comp (\u2191lift (domDomCongr e.symm (tprod R))) (\u2191lift (domDomCongr e (tprod R))) = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d : \u03b9\u2082 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (LinearMap.comp (\u2191lift (domDomCongr e.symm (tprod R))) (\u2191lift (domDomCongr e (tprod R)))) (tprod R))\n      x\u271d =\n    \u2191(LinearMap.compMultilinearMap LinearMap.id (tprod R)) x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, LinearMap.id_apply, lift_tprod, LinearMap.compMultilinearMap_apply, lift.tprod,\n  domDomCongr_apply]\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d : \u03b9\u2082 \u2192 M\n\u22a2 (\u2a02\u209c[R] (i : \u03b9\u2082), x\u271d (\u2191e (\u2191e.symm i))) = \u2191(tprod R) x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase H.H.h.e_6.h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d : \u03b9\u2082 \u2192 M\n\u22a2 (fun i => x\u271d (\u2191e (\u2191e.symm i))) = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase H.H.h.e_6.h.h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d\u00b9 : \u03b9\u2082 \u2192 M\nx\u271d : \u03b9\u2082\n\u22a2 x\u271d\u00b9 (\u2191e (\u2191e.symm x\u271d)) = x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrw [e.apply_symm_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\n\u22a2 LinearMap.comp (\u2191lift (domDomCongr e (tprod R))) (\u2191lift (domDomCongr e.symm (tprod R))) = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (LinearMap.comp (\u2191lift (domDomCongr e (tprod R))) (\u2191lift (domDomCongr e.symm (tprod R)))) (tprod R))\n      x\u271d =\n    \u2191(LinearMap.compMultilinearMap LinearMap.id (tprod R)) x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, LinearMap.id_apply, lift_tprod, LinearMap.compMultilinearMap_apply, lift.tprod,\n  domDomCongr_apply]\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d : \u03b9 \u2192 M\n\u22a2 (\u2a02\u209c[R] (i : \u03b9), x\u271d (\u2191e.symm (\u2191e i))) = \u2191(tprod R) x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase H.H.h.e_6.h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d : \u03b9 \u2192 M\n\u22a2 (fun i => x\u271d (\u2191e.symm (\u2191e i))) = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase H.H.h.e_6.h.h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nx\u271d\u00b9 : \u03b9 \u2192 M\nx\u271d : \u03b9\n\u22a2 x\u271d\u00b9 (\u2191e.symm (\u2191e x\u271d)) = x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrw [e.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nf : \u03b9 \u2192 M\n\u22a2 \u2191(reindex R M e) (\u2191(tprod R) f) = \u2a02\u209c[R] (i : \u03b9\u2082), f (\u2191e.symm i)\n[PROOFSTEP]\ndsimp [reindex]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u2191lift (domDomCongr e.symm (tprod R))) (\u2191(tprod R) f) = \u2a02\u209c[R] (i : \u03b9\u2082), f (\u2191e.symm i)\n[PROOFSTEP]\nexact liftAux_tprod _ f\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u271d : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\n\u03c6 : MultilinearMap R (fun x => M) E\n\u22a2 LinearMap.comp (\u2191lift \u03c6) \u2191(reindex R M e) = \u2191lift (domDomCongr e.symm \u03c6)\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6\u271d : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\n\u03c6 : MultilinearMap R (fun x => M) E\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap (LinearMap.comp (\u2191lift \u03c6) \u2191(reindex R M e)) (tprod R)) x\u271d =\n    \u2191(LinearMap.compMultilinearMap (\u2191lift (domDomCongr e.symm \u03c6)) (tprod R)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\ne' : \u03b9\u2082 \u2243 \u03b9\u2083\n\u22a2 LinearEquiv.trans (reindex R M e) (reindex R M e') = reindex R M (e.trans e')\n[PROOFSTEP]\napply LinearEquiv.toLinearMap_injective\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\ne' : \u03b9\u2082 \u2243 \u03b9\u2083\n\u22a2 \u2191(LinearEquiv.trans (reindex R M e) (reindex R M e')) = \u2191(reindex R M (e.trans e'))\n[PROOFSTEP]\next f\n[GOAL]\ncase a.H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\ne' : \u03b9\u2082 \u2243 \u03b9\u2083\nf : \u03b9 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap (\u2191(LinearEquiv.trans (reindex R M e) (reindex R M e'))) (tprod R)) f =\n    \u2191(LinearMap.compMultilinearMap (\u2191(reindex R M (e.trans e'))) (tprod R)) f\n[PROOFSTEP]\nsimp only [LinearEquiv.trans_apply, LinearEquiv.coe_coe, reindex_tprod, LinearMap.coe_compMultilinearMap,\n  Function.comp_apply, MultilinearMap.domDomCongr_apply, reindex_comp_tprod]\n[GOAL]\ncase a.H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\ne' : \u03b9\u2082 \u2243 \u03b9\u2083\nf : \u03b9 \u2192 M\n\u22a2 (\u2a02\u209c[R] (i : \u03b9\u2083), f (\u2191e.symm (\u2191e'.symm i))) = \u2a02\u209c[R] (i : \u03b9\u2083), f (\u2191(e.trans e').symm i)\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\n\u22a2 reindex R M (Equiv.refl \u03b9) = LinearEquiv.refl R (\u2a02[R] (x : \u03b9), M)\n[PROOFSTEP]\napply LinearEquiv.toLinearMap_injective\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\n\u22a2 \u2191(reindex R M (Equiv.refl \u03b9)) = \u2191(LinearEquiv.refl R (\u2a02[R] (x : \u03b9), M))\n[PROOFSTEP]\next\n[GOAL]\ncase a.H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap (\u2191(reindex R M (Equiv.refl \u03b9))) (tprod R)) x\u271d =\n    \u2191(LinearMap.compMultilinearMap (\u2191(LinearEquiv.refl R (\u2a02[R] (x : \u03b9), M))) (tprod R)) x\u271d\n[PROOFSTEP]\nrw [reindex_comp_tprod, LinearEquiv.refl_toLinearMap, Equiv.refl_symm]\n[GOAL]\ncase a.H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx\u271d : \u03b9 \u2192 M\n\u22a2 \u2191(domDomCongr (Equiv.refl \u03b9) (tprod R)) x\u271d = \u2191(LinearMap.compMultilinearMap LinearMap.id (tprod R)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nr : R\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 AddHom.toFun\n      { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n        map_add' :=\n          (_ :\n            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x + \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n      (r \u2022 x) =\n    \u2191(RingHom.id R) r \u2022\n      AddHom.toFun\n        { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n          map_add' :=\n            (_ :\n              \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x + \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n        x\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nr : R\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (r \u2022 x) = \u2191(RingHom.id R) r \u2022 \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x\n[PROOFSTEP]\nexact LinearMap.map_smul _ r x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 (fun r => r \u2022 \u2191(tprod R) isEmptyElim)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x + \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                  AddHom.toFun\n                      { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                      (r \u2022 x) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                        x) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\nrefine x.induction_on ?_ ?_\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 \u2200 {r : R} {f : \u03b9 \u2192 M},\n    (fun r => r \u2022 \u2191(tprod R) isEmptyElim)\n        (AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                          \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                            \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                    AddHom.toFun\n                        { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                        (r \u2022 x) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                          x) }.toAddHom\n          (r \u2022 \u2191(tprod R) f)) =\n      r \u2022 \u2191(tprod R) f\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx\u271d : \u2a02[R] (x : \u03b9), M\nx : R\ny : \u03b9 \u2192 M\n\u22a2 (fun r => r \u2022 \u2191(tprod R) isEmptyElim)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x + \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                  AddHom.toFun\n                      { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                      (r \u2022 x) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                        x) }.toAddHom\n        (x \u2022 \u2191(tprod R) y)) =\n    x \u2022 \u2191(tprod R) y\n[PROOFSTEP]\nsimp only [map_smul\u209b\u2097, RingHom.id_apply, lift.tprod, constOfIsEmpty_apply, const_apply, smul_eq_mul, mul_one]\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx\u271d : \u2a02[R] (x : \u03b9), M\nx : R\ny : \u03b9 \u2192 M\n\u22a2 x \u2022 \u2191(tprod R) isEmptyElim = x \u2022 \u2191(tprod R) y\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine_1.e_a.h.e_6.h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx\u271d : \u2a02[R] (x : \u03b9), M\nx : R\ny : \u03b9 \u2192 M\n\u22a2 isEmptyElim = y\n[PROOFSTEP]\naesop\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 \u2200 {x y : \u2a02[R] (i : \u03b9), M},\n    (fun r => r \u2022 \u2191(tprod R) isEmptyElim)\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                          \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                            \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                      AddHom.toFun\n                          { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                          (r \u2022 x) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                            x) }.toAddHom\n            x) =\n        x \u2192\n      (fun r => r \u2022 \u2191(tprod R) isEmptyElim)\n            (AddHom.toFun\n              {\n                  toAddHom :=\n                    { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                            \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                        AddHom.toFun\n                            { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                            (r \u2022 x) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                          \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                              x) }.toAddHom\n              y) =\n          y \u2192\n        (fun r => r \u2022 \u2191(tprod R) isEmptyElim)\n            (AddHom.toFun\n              {\n                  toAddHom :=\n                    { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                            \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                        AddHom.toFun\n                            { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                            (r \u2022 x) =\n                          \u2191(RingHom.id R) r \u2022\n                            AddHom.toFun\n                              { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                          \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                              x) }.toAddHom\n              (x + y)) =\n          x + y\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 \u2200 {x y : \u2a02[R] (i : \u03b9), M},\n    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x \u2022 \u2191(tprod R) isEmptyElim = x \u2192\n      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y \u2022 \u2191(tprod R) isEmptyElim = y \u2192\n        \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) \u2022 \u2191(tprod R) isEmptyElim = x + y\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nx\u271d x y : \u2a02[R] (i : \u03b9), M\nhx : \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x \u2022 \u2191(tprod R) isEmptyElim = x\nhy : \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y \u2022 \u2191(tprod R) isEmptyElim = y\n\u22a2 \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) \u2022 \u2191(tprod R) isEmptyElim = x + y\n[PROOFSTEP]\nrw [map_add, add_smul, hx, hy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : IsEmpty \u03b9\nt : R\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                    \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                      \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x + \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                AddHom.toFun\n                    { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                            \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                              \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) (x + y) =\n                                \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) x +\n                                  \u2191(\u2191lift (constOfIsEmpty R (fun i => M) 1)) y) }\n                      x) }.toAddHom\n      ((fun r => r \u2022 \u2191(tprod R) isEmptyElim) t) =\n    t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nr : R\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 AddHom.toFun\n      { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n        map_add' :=\n          (_ :\n            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n              \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) }\n      (r \u2022 x) =\n    \u2191(RingHom.id R) r \u2022\n      AddHom.toFun\n        { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n          map_add' :=\n            (_ :\n              \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                  \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) }\n        x\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nr : R\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (r \u2022 x) = \u2191(RingHom.id R) r \u2022 \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x\n[PROOFSTEP]\nexact LinearMap.map_smul _ r x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 (fun m => \u2a02\u209c[R] (x : \u03b9), m)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                      \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                        \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                  AddHom.toFun\n                      { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                              \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                                \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) }\n                      (r \u2022 x) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                                \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                                  \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) }\n                        x) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 (\u2a02\u209c[R] (x_1 : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x) = x\n[PROOFSTEP]\nhave : \u2200 (f : \u03b9 \u2192 M) (z : M), (fun _ : \u03b9 \u21a6 z) = update f i\u2080 z :=\n  by\n  intro f z\n  ext i\n  rw [Subsingleton.elim i i\u2080, Function.update_same]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\n\u22a2 \u2200 (f : \u03b9 \u2192 M) (z : M), (fun x => z) = update f i\u2080 z\n[PROOFSTEP]\nintro f z\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\nf : \u03b9 \u2192 M\nz : M\n\u22a2 (fun x => z) = update f i\u2080 z\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\nf : \u03b9 \u2192 M\nz : M\ni : \u03b9\n\u22a2 z = update f i\u2080 z i\n[PROOFSTEP]\nrw [Subsingleton.elim i i\u2080, Function.update_same]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\nthis : \u2200 (f : \u03b9 \u2192 M) (z : M), (fun x => z) = update f i\u2080 z\n\u22a2 (\u2a02\u209c[R] (x_1 : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x) = x\n[PROOFSTEP]\nrefine x.induction_on ?_ ?_\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\nthis : \u2200 (f : \u03b9 \u2192 M) (z : M), (fun x => z) = update f i\u2080 z\n\u22a2 \u2200 {r : R} {f : \u03b9 \u2192 M}, (\u2a02\u209c[R] (x : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (r \u2022 \u2191(tprod R) f)) = r \u2022 \u2191(tprod R) f\n[PROOFSTEP]\nintro r f\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\nthis : \u2200 (f : \u03b9 \u2192 M) (z : M), (fun x => z) = update f i\u2080 z\nr : R\nf : \u03b9 \u2192 M\n\u22a2 (\u2a02\u209c[R] (x : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (r \u2022 \u2191(tprod R) f)) = r \u2022 \u2191(tprod R) f\n[PROOFSTEP]\nsimp only [LinearMap.map_smul, lift.tprod, ofSubsingleton_apply, Function.eval, this f, MultilinearMap.map_smul,\n  update_eq_self]\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx : \u2a02[R] (x : \u03b9), M\nthis : \u2200 (f : \u03b9 \u2192 M) (z : M), (fun x => z) = update f i\u2080 z\n\u22a2 \u2200 {x y : \u2a02[R] (i : \u03b9), M},\n    (\u2a02\u209c[R] (x_1 : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x) = x \u2192\n      (\u2a02\u209c[R] (x : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) = y \u2192\n        (\u2a02\u209c[R] (x_1 : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y)) = x + y\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nx\u271d : \u2a02[R] (x : \u03b9), M\nthis : \u2200 (f : \u03b9 \u2192 M) (z : M), (fun x => z) = update f i\u2080 z\nx y : \u2a02[R] (i : \u03b9), M\nhx : (\u2a02\u209c[R] (x_1 : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x) = x\nhy : (\u2a02\u209c[R] (x : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) = y\n\u22a2 (\u2a02\u209c[R] (x_1 : \u03b9), \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y)) = x + y\n[PROOFSTEP]\nrw [LinearMap.map_add, this 0 (_ + _), MultilinearMap.map_add, \u2190 this 0 (lift _ _), hx, \u2190 this 0 (lift _ _), hy]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2078 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nE : Type u_9\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module R E\nF : Type u_10\ninst\u271d\u00b9 : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ninst\u271d : Subsingleton \u03b9\ni\u2080 : \u03b9\nt : M\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                    \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                      \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : \u2a02[R] (x : \u03b9), M),\n                AddHom.toFun\n                    { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                            \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                              \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := \u2191(\u2191lift (ofSubsingleton R M i\u2080)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : \u2a02[R] (x : \u03b9), M),\n                              \u2191(\u2191lift (ofSubsingleton R M i\u2080)) (x + y) =\n                                \u2191(\u2191lift (ofSubsingleton R M i\u2080)) x + \u2191(\u2191lift (ofSubsingleton R M i\u2080)) y) }\n                      x) }.toAddHom\n      ((fun m => \u2a02\u209c[R] (x : \u03b9), m) t) =\n    t\n[PROOFSTEP]\nsimp only [ofSubsingleton_apply, lift.tprod, Function.eval_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\na b : \u2a02[R] (x : \u03b9), M\n\u22a2 (fun a => \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a)) (a + b) =\n    (fun a => \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a)) a +\n      (fun a => \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a)) b\n[PROOFSTEP]\nsimp only [LinearEquiv.map_add, LinearMap.map_add]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nr : R\na : \u2a02[R] (x : \u03b9), M\n\u22a2 AddHom.toFun\n      { toFun := fun a => \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a),\n        map_add' :=\n          (_ :\n            \u2200 (a b : \u2a02[R] (x : \u03b9), M),\n              \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) (a + b)) =\n                \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a) +\n                  \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) b)) }\n      (r \u2022 a) =\n    \u2191(RingHom.id R) r \u2022\n      AddHom.toFun\n        { toFun := fun a => \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a),\n          map_add' :=\n            (_ :\n              \u2200 (a b : \u2a02[R] (x : \u03b9), M),\n                \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) (a + b)) =\n                  \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) a) +\n                    \u2191lift (\u2191(\u2191lift (\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R))) b)) }\n        a\n[PROOFSTEP]\nsimp only [LinearEquiv.map_smul, LinearMap.map_smul, RingHom.id_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\na : \u03b9 \u2192 M\nb : \u03b9\u2082 \u2192 M\n\u22a2 \u2191PiTensorProduct.tmul ((\u2a02\u209c[R] (i : \u03b9), a i) \u2297\u209c[R] \u2a02\u209c[R] (i : \u03b9\u2082), b i) = \u2a02\u209c[R] (i : \u03b9 \u2295 \u03b9\u2082), Sum.elim a b i\n[PROOFSTEP]\nerw [TensorProduct.lift.tmul, PiTensorProduct.lift.tprod, PiTensorProduct.lift.tprod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\na : \u03b9 \u2192 M\nb : \u03b9\u2082 \u2192 M\n\u22a2 (\u2191(\u2191(\u2191(currySumEquiv R \u03b9 (\u2a02[R] (i : \u03b9 \u2295 \u03b9\u2082), M) M \u03b9\u2082) (tprod R)) fun i => a i) fun i => b i) =\n    \u2a02\u209c[R] (i : \u03b9 \u2295 \u03b9\u2082), Sum.elim a b i\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\n\u22a2 LinearMap.comp PiTensorProduct.tmul PiTensorProduct.tmulSymm = LinearMap.id\n[PROOFSTEP]\next x\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx : \u03b9 \u2295 \u03b9\u2082 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap (LinearMap.comp PiTensorProduct.tmul PiTensorProduct.tmulSymm) (tprod R)) x =\n    \u2191(LinearMap.compMultilinearMap LinearMap.id (tprod R)) x\n[PROOFSTEP]\nshow tmul (tmulSymm (tprod R x)) = tprod R x\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx : \u03b9 \u2295 \u03b9\u2082 \u2192 M\n\u22a2 \u2191PiTensorProduct.tmul (\u2191PiTensorProduct.tmulSymm (\u2191(tprod R) x)) = \u2191(tprod R) x\n[PROOFSTEP]\nsimp only [tmulSymm_apply, tmul_apply]\n  -- Porting note (https://github.com/leanprover-community/mathlib4/issues/5026):\n        -- was part of `simp only` above\n[GOAL]\ncase H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx : \u03b9 \u2295 \u03b9\u2082 \u2192 M\n\u22a2 (\u2a02\u209c[R] (i : \u03b9 \u2295 \u03b9\u2082), Sum.elim (fun i => x (Sum.inl i)) (fun i => x (Sum.inr i)) i) = \u2191(tprod R) x\n[PROOFSTEP]\nerw [Sum.elim_comp_inl_inr]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\n\u22a2 LinearMap.comp PiTensorProduct.tmulSymm PiTensorProduct.tmul = LinearMap.id\n[PROOFSTEP]\next x y\n[GOAL]\ncase H.H.H.H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx : \u03b9 \u2192 M\ny : \u03b9\u2082 \u2192 M\n\u22a2 \u2191(LinearMap.compMultilinearMap\n          (\u2191(LinearMap.compMultilinearMap\n                (LinearMap.compr\u2082 (TensorProduct.mk R (\u2a02[R] (x : \u03b9), M) (\u2a02[R] (x : \u03b9\u2082), M))\n                  (LinearMap.comp PiTensorProduct.tmulSymm PiTensorProduct.tmul))\n                (tprod R))\n            x)\n          (tprod R))\n      y =\n    \u2191(LinearMap.compMultilinearMap\n          (\u2191(LinearMap.compMultilinearMap\n                (LinearMap.compr\u2082 (TensorProduct.mk R (\u2a02[R] (x : \u03b9), M) (\u2a02[R] (x : \u03b9\u2082), M)) LinearMap.id) (tprod R))\n            x)\n          (tprod R))\n      y\n[PROOFSTEP]\nshow tmulSymm (tmul (tprod R x \u2297\u209c[R] tprod R y)) = tprod R x \u2297\u209c[R] tprod R y\n[GOAL]\ncase H.H.H.H.H\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nx : \u03b9 \u2192 M\ny : \u03b9\u2082 \u2192 M\n\u22a2 \u2191PiTensorProduct.tmulSymm (\u2191PiTensorProduct.tmul (\u2191(tprod R) x \u2297\u209c[R] \u2191(tprod R) y)) = \u2191(tprod R) x \u2297\u209c[R] \u2191(tprod R) y\n[PROOFSTEP]\nsimp only [tmul_apply, tmulSymm_apply, Sum.elim_inl, Sum.elim_inr]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.PiTensorProduct", "llama_tokens": 66964, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8652240721511739, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5511743900104797}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\nx y z : M\nx\u271d\u00b9 : y \u2208 powers x\nx\u271d : z \u2208 powers x\nn\u2081 : \u2115\nh\u2081 : x ^ n\u2081 = y\nn\u2082 : \u2115\nh\u2082 : x ^ n\u2082 = z\n\u22a2 x ^ (n\u2081 + n\u2082) = y * z\n[PROOFSTEP]\nsimp only [pow_add, *]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\n\u22a2 IsSubmonoid Set.univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase one_mem\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\n\u22a2 1 \u2208 Set.univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mul_mem\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\n\u22a2 \u2200 {a b : M}, a \u2208 Set.univ \u2192 b \u2208 Set.univ \u2192 a * b \u2208 Set.univ\n[PROOFSTEP]\nsimp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nN : Type u_3\ninst\u271d : Monoid N\nf : M \u2192 N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\n\u22a2 f 1 \u2208 s\n[PROOFSTEP]\nrw [IsMonoidHom.map_one hf]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nN : Type u_3\ninst\u271d : Monoid N\nf : M \u2192 N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\n\u22a2 1 \u2208 s\n[PROOFSTEP]\nexact hs.one_mem\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nN : Type u_3\ninst\u271d : Monoid N\nf : M \u2192 N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\na b : M\nha : f a \u2208 s\nhb : f b \u2208 s\n\u22a2 f (a * b) \u2208 s\n[PROOFSTEP]\nrw [IsMonoidHom.map_mul' hf]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nN : Type u_3\ninst\u271d : Monoid N\nf : M \u2192 N\nhf : IsMonoidHom f\ns : Set N\nhs : IsSubmonoid s\na b : M\nha : f a \u2208 s\nhb : f b \u2208 s\n\u22a2 f a * f b \u2208 s\n[PROOFSTEP]\nexact hs.mul_mem ha hb\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\n\u03b3 : Type u_3\ninst\u271d : Monoid \u03b3\nf : M \u2192 \u03b3\nhf : IsMonoidHom f\ns : Set M\nhs : IsSubmonoid s\na b : \u03b3\nx\u271d\u00b9 : a \u2208 f '' s\nx\u271d : b \u2208 f '' s\nx : M\nhx : x \u2208 s \u2227 f x = a\ny : M\nhy : y \u2208 s \u2227 f y = b\n\u22a2 f (x * y) = a * b\n[PROOFSTEP]\nrw [hf.map_mul, hx.2, hy.2]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\n\u03b3 : Type u_3\ninst\u271d : Monoid \u03b3\nf : M \u2192 \u03b3\nhf : IsMonoidHom f\n\u22a2 IsSubmonoid (Set.range f)\n[PROOFSTEP]\nrw [\u2190 Set.image_univ]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\n\u03b3 : Type u_3\ninst\u271d : Monoid \u03b3\nf : M \u2192 \u03b3\nhf : IsMonoidHom f\n\u22a2 IsSubmonoid (f '' Set.univ)\n[PROOFSTEP]\nexact Univ.isSubmonoid.image hf\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\na : M\nhs : IsSubmonoid s\nh : a \u2208 s\n\u22a2 a ^ 0 \u2208 s\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\na : M\nhs : IsSubmonoid s\nh : a \u2208 s\n\u22a2 1 \u2208 s\n[PROOFSTEP]\nexact hs.one_mem\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\na : M\nhs : IsSubmonoid s\nh : a \u2208 s\nn : \u2115\n\u22a2 a ^ (n + 1) \u2208 s\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\na : M\nhs : IsSubmonoid s\nh : a \u2208 s\nn : \u2115\n\u22a2 a * a ^ n \u2208 s\n[PROOFSTEP]\nexact hs.mul_mem h (IsSubmonoid.pow_mem hs h)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\nhs : IsSubmonoid s\na : M\nl : List M\nh : \u2200 (x : M), x \u2208 a :: l \u2192 x \u2208 s\n\u22a2 a \u2208 s \u2227 \u2200 (x : M), x \u2208 l \u2192 x \u2208 s\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\nhs : IsSubmonoid s\na : M\nl : List M\nh : \u2200 (x : M), x \u2208 a :: l \u2192 x \u2208 s\nthis : a * List.prod l \u2208 s\n\u22a2 List.prod (a :: l) \u2208 s\n[PROOFSTEP]\nsimpa\n[GOAL]\nM\u271d : Type u_1\ninst\u271d\u00b2 : Monoid M\u271d\ns\u271d : Set M\u271d\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nM : Type u_3\ninst\u271d : CommMonoid M\ns : Set M\nhs : IsSubmonoid s\nm : Multiset M\n\u22a2 (\u2200 (a : M), a \u2208 m \u2192 a \u2208 s) \u2192 Multiset.prod m \u2208 s\n[PROOFSTEP]\nrefine' Quotient.inductionOn m fun l hl => _\n[GOAL]\nM\u271d : Type u_1\ninst\u271d\u00b2 : Monoid M\u271d\ns\u271d : Set M\u271d\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nM : Type u_3\ninst\u271d : CommMonoid M\ns : Set M\nhs : IsSubmonoid s\nm : Multiset M\nl : List M\nhl : \u2200 (a : M), a \u2208 Quotient.mk (List.isSetoid M) l \u2192 a \u2208 s\n\u22a2 Multiset.prod (Quotient.mk (List.isSetoid M) l) \u2208 s\n[PROOFSTEP]\nrw [Multiset.quot_mk_to_coe, Multiset.coe_prod]\n[GOAL]\nM\u271d : Type u_1\ninst\u271d\u00b2 : Monoid M\u271d\ns\u271d : Set M\u271d\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt : Set A\nM : Type u_3\ninst\u271d : CommMonoid M\ns : Set M\nhs : IsSubmonoid s\nm : Multiset M\nl : List M\nhl : \u2200 (a : M), a \u2208 Quotient.mk (List.isSetoid M) l \u2192 a \u2208 s\n\u22a2 List.prod l \u2208 s\n[PROOFSTEP]\nexact list_prod_mem hs hl\n[GOAL]\nM\u271d : Type u_1\ninst\u271d\u00b2 : Monoid M\u271d\ns\u271d : Set M\u271d\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nM : Type u_3\nA : Type u_4\ninst\u271d : CommMonoid M\ns : Set M\nhs : IsSubmonoid s\nf : A \u2192 M\nm : Multiset A\nhm : Multiset.Nodup m\nx\u271d : \u2200 (b : A), b \u2208 { val := m, nodup := hm } \u2192 f b \u2208 s\n\u22a2 \u2200 (a : M), a \u2208 Multiset.map (fun b => f b) { val := m, nodup := hm }.val \u2192 a \u2208 s\n[PROOFSTEP]\nsimpa\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt\u271d : Set A\ns t : Set M\nht : IsSubmonoid t\nh : s \u2286 t\na : M\nha : a \u2208 Closure s\n\u22a2 a \u2208 t\n[PROOFSTEP]\ninduction ha\n[GOAL]\ncase basic\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt\u271d : Set A\ns t : Set M\nht : IsSubmonoid t\nh : s \u2286 t\na a\u271d\u00b9 : M\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 a\u271d\u00b9 \u2208 t\n[PROOFSTEP]\nsimp [h _, *, IsSubmonoid.one_mem, IsSubmonoid.mul_mem]\n[GOAL]\ncase one\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt\u271d : Set A\ns t : Set M\nht : IsSubmonoid t\nh : s \u2286 t\na : M\n\u22a2 1 \u2208 t\n[PROOFSTEP]\nsimp [h _, *, IsSubmonoid.one_mem, IsSubmonoid.mul_mem]\n[GOAL]\ncase mul\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt\u271d : Set A\ns t : Set M\nht : IsSubmonoid t\nh : s \u2286 t\na a\u271d\u00b2 b\u271d : M\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 _, *, IsSubmonoid.one_mem, IsSubmonoid.mul_mem]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\n\u22a2 f '' Closure s \u2264 Closure (f '' s)\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\nx : M\nhx : x \u2208 Closure s\n\u22a2 f x \u2208 Closure (f '' s)\n[PROOFSTEP]\ninduction' hx with z hz\n[GOAL]\ncase intro.intro.basic\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\nx z : M\nhz : z \u2208 s\n\u22a2 f z \u2208 Closure (f '' s)\n[PROOFSTEP]\nsolve_by_elim [subset_closure, Set.mem_image_of_mem]\n[GOAL]\ncase intro.intro.one\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\nx : M\n\u22a2 f 1 \u2208 Closure (f '' s)\n[PROOFSTEP]\nrw [hf.map_one]\n[GOAL]\ncase intro.intro.one\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\nx : M\n\u22a2 1 \u2208 Closure (f '' s)\n[PROOFSTEP]\napply IsSubmonoid.one_mem (closure.isSubmonoid (f '' s))\n[GOAL]\ncase intro.intro.mul\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\nx a\u271d\u00b2 b\u271d : M\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 Closure (f '' s)\na_ih\u271d : 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]\ncase intro.intro.mul\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\ns\u271d : Set M\nA\u271d : Type u_2\ninst\u271d\u00b9 : AddMonoid A\u271d\nt : Set A\u271d\nA : Type u_3\ninst\u271d : Monoid A\nf : M \u2192 A\nhf : IsMonoidHom f\ns : Set M\nx a\u271d\u00b2 b\u271d : M\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\na_ih\u271d\u00b9 : f a\u271d\u00b2 \u2208 Closure (f '' s)\na_ih\u271d : f b\u271d \u2208 Closure (f '' s)\n\u22a2 f a\u271d\u00b2 * f b\u271d \u2208 Closure (f '' s)\n[PROOFSTEP]\nsolve_by_elim [(closure.isSubmonoid _).mul_mem]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na : M\nh : a \u2208 Closure s\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase basic\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na a\u271d\u00b9 : M\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b9\ncase one\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na : M\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = 1\ncase mul\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na a\u271d\u00b2 b\u271d : M\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\na_ih\u271d\u00b9 : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b2\na_ih\u271d : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = b\u271d\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b2 * b\u271d\n[PROOFSTEP]\ncase basic a ha => exists [a]; simp [ha]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d a : M\nha : a \u2208 s\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\n[PROOFSTEP]\ncase basic a ha => exists [a]; simp [ha]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d a : M\nha : a \u2208 s\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\n[PROOFSTEP]\nexists [a]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d a : M\nha : a \u2208 s\n\u22a2 (\u2200 (x : M), x \u2208 [a] \u2192 x \u2208 s) \u2227 List.prod [a] = a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase one\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na : M\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = 1\ncase mul\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na a\u271d\u00b2 b\u271d : M\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\na_ih\u271d\u00b9 : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b2\na_ih\u271d : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = b\u271d\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b2 * b\u271d\n[PROOFSTEP]\ncase one => exists []; simp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na : M\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = 1\n[PROOFSTEP]\ncase one => exists []; simp\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na : M\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = 1\n[PROOFSTEP]\nexists []\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na : M\n\u22a2 (\u2200 (x : M), x \u2208 [] \u2192 x \u2208 s) \u2227 List.prod [] = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mul\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na a\u271d\u00b2 b\u271d : M\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\na_ih\u271d\u00b9 : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b2\na_ih\u271d : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = b\u271d\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\u271d\u00b2 * b\u271d\n[PROOFSTEP]\ncase mul a b _ _ ha hb =>\n  rcases ha with \u27e8la, ha, eqa\u27e9\n  rcases hb with \u27e8lb, hb, eqb\u27e9\n  exists la ++ lb\n  simp [eqa.symm, eqb.symm, or_imp]\n  exact fun a => \u27e8ha a, hb a\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d\u00b2 a b : M\na\u271d\u00b9 : InClosure s a\na\u271d : InClosure s b\nha : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\nhb : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = b\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a * b\n[PROOFSTEP]\ncase mul a b _ _ ha hb =>\n  rcases ha with \u27e8la, ha, eqa\u27e9\n  rcases hb with \u27e8lb, hb, eqb\u27e9\n  exists la ++ lb\n  simp [eqa.symm, eqb.symm, or_imp]\n  exact fun a => \u27e8ha a, hb a\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d\u00b2 a b : M\na\u271d\u00b9 : InClosure s a\na\u271d : InClosure s b\nha : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a\nhb : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = b\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a * b\n[PROOFSTEP]\nrcases ha with \u27e8la, ha, eqa\u27e9\n[GOAL]\ncase intro.intro\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d\u00b2 a b : M\na\u271d\u00b9 : InClosure s a\na\u271d : InClosure s b\nhb : \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = b\nla : List M\nha : \u2200 (x : M), x \u2208 la \u2192 x \u2208 s\neqa : List.prod la = a\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a * b\n[PROOFSTEP]\nrcases hb with \u27e8lb, hb, eqb\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d\u00b2 a b : M\na\u271d\u00b9 : InClosure s a\na\u271d : InClosure s b\nla : List M\nha : \u2200 (x : M), x \u2208 la \u2192 x \u2208 s\neqa : List.prod la = a\nlb : List M\nhb : \u2200 (x : M), x \u2208 lb \u2192 x \u2208 s\neqb : List.prod lb = b\n\u22a2 \u2203 l, (\u2200 (x : M), x \u2208 l \u2192 x \u2208 s) \u2227 List.prod l = a * b\n[PROOFSTEP]\nexists la ++ lb\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d\u00b2 a b : M\na\u271d\u00b9 : InClosure s a\na\u271d : InClosure s b\nla : List M\nha : \u2200 (x : M), x \u2208 la \u2192 x \u2208 s\neqa : List.prod la = a\nlb : List M\nhb : \u2200 (x : M), x \u2208 lb \u2192 x \u2208 s\neqb : List.prod lb = b\n\u22a2 (\u2200 (x : M), x \u2208 la ++ lb \u2192 x \u2208 s) \u2227 List.prod (la ++ lb) = a * b\n[PROOFSTEP]\nsimp [eqa.symm, eqb.symm, or_imp]\n[GOAL]\ncase intro.intro.intro.intro\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns\u271d : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\ns : Set M\na\u271d\u00b2 a b : M\na\u271d\u00b9 : InClosure s a\na\u271d : InClosure s b\nla : List M\nha : \u2200 (x : M), x \u2208 la \u2192 x \u2208 s\neqa : List.prod la = a\nlb : List M\nhb : \u2200 (x : M), x \u2208 lb \u2192 x \u2208 s\neqb : List.prod lb = b\n\u22a2 \u2200 (x : M), (x \u2208 la \u2192 x \u2208 s) \u2227 (x \u2208 lb \u2192 x \u2208 s)\n[PROOFSTEP]\nexact fun a => \u27e8ha a, hb a\u27e9\n[GOAL]\nM\u271d : Type u_1\ninst\u271d\u00b2 : Monoid M\u271d\ns\u271d : Set M\u271d\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt\u271d : Set A\nM : Type u_3\ninst\u271d : CommMonoid M\ns t : Set M\nx : M\nhx : x \u2208 Closure (s \u222a t)\nL : List M\nHL1\u271d : \u2200 (x : M), x \u2208 L \u2192 x \u2208 s \u222a t\nHL2 : List.prod L = x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 x \u2208 s \u222a t) \u2192 \u2203 y, y \u2208 Closure s \u2227 \u2203 z, z \u2208 Closure t \u2227 y * z = List.prod tl\nHL1 : \u2200 (x : M), x \u2208 hd :: tl \u2192 x \u2208 s \u222a t\ny : M\nhy : y \u2208 Closure s\nz : M\nhz : z \u2208 Closure t\nhyzx : y * z = List.prod tl\nhs : hd \u2208 s\n\u22a2 hd * y * z = List.prod (hd :: tl)\n[PROOFSTEP]\nrw [mul_assoc, List.prod_cons, \u2190 hyzx]\n[GOAL]\nM\u271d : Type u_1\ninst\u271d\u00b2 : Monoid M\u271d\ns\u271d : Set M\u271d\nA : Type u_2\ninst\u271d\u00b9 : AddMonoid A\nt\u271d : Set A\nM : Type u_3\ninst\u271d : CommMonoid M\ns t : Set M\nx : M\nhx : x \u2208 Closure (s \u222a t)\nL : List M\nHL1\u271d : \u2200 (x : M), x \u2208 L \u2192 x \u2208 s \u222a t\nHL2 : List.prod L = x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 x \u2208 s \u222a t) \u2192 \u2203 y, y \u2208 Closure s \u2227 \u2203 z, z \u2208 Closure t \u2227 y * z = List.prod tl\nHL1 : \u2200 (x : M), x \u2208 hd :: tl \u2192 x \u2208 s \u222a t\ny : M\nhy : y \u2208 Closure s\nz : M\nhz : z \u2208 Closure t\nhyzx : y * z = List.prod tl\nht : hd \u2208 t\n\u22a2 y * (z * hd) = List.prod (hd :: tl)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, List.prod_cons, \u2190 hyzx, mul_comm hd]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Monoid M\ns : Set M\nA : Type u_2\ninst\u271d : AddMonoid A\nt : Set A\nS : Submonoid M\n\u22a2 IsSubmonoid \u2191S\n[PROOFSTEP]\nrefine' \u27e8S.2, S.1.2\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Deprecated.Submonoid", "llama_tokens": 8685, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.7090191460821871, "lm_q1q2_score": 0.5511204838198821}}
{"text": "[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\n\u22a2 ConformalAt f x \u2194 IsConformalMap (fderiv \u211d f x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\n\u22a2 ConformalAt f x \u2192 IsConformalMap (fderiv \u211d f x)\n[PROOFSTEP]\nrintro \u27e8f', hf, hf'\u27e9\n[GOAL]\ncase mp.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\nf' : X \u2192L[\u211d] Y\nhf : HasFDerivAt f f' x\nhf' : IsConformalMap f'\n\u22a2 IsConformalMap (fderiv \u211d f x)\n[PROOFSTEP]\nrwa [hf.fderiv]\n[GOAL]\ncase mpr\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\n\u22a2 IsConformalMap (fderiv \u211d f x) \u2192 ConformalAt f x\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\nH : IsConformalMap (fderiv \u211d f x)\n\u22a2 ConformalAt f x\n[PROOFSTEP]\nby_cases h : DifferentiableAt \u211d f x\n[GOAL]\ncase pos\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\nH : IsConformalMap (fderiv \u211d f x)\nh : DifferentiableAt \u211d f x\n\u22a2 ConformalAt f x\n[PROOFSTEP]\nexact \u27e8fderiv \u211d f x, h.hasFDerivAt, H\u27e9\n[GOAL]\ncase neg\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\nH : IsConformalMap (fderiv \u211d f x)\nh : \u00acDifferentiableAt \u211d f x\n\u22a2 ConformalAt f x\n[PROOFSTEP]\nnontriviality X\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\nx : X\nH : IsConformalMap (fderiv \u211d f x)\nh : \u00acDifferentiableAt \u211d f x\n\u271d : Nontrivial X\n\u22a2 ConformalAt f x\n[PROOFSTEP]\nexact absurd (fderiv_zero_of_not_differentiableAt h) H.ne_zero\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\ng : Y \u2192 Z\nx : X\nhg : ConformalAt g (f x)\nhf : ConformalAt f x\n\u22a2 ConformalAt (g \u2218 f) x\n[PROOFSTEP]\nrcases hf with \u27e8f', hf\u2081, cf\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\ng : Y \u2192 Z\nx : X\nhg : ConformalAt g (f x)\nf' : X \u2192L[\u211d] Y\nhf\u2081 : HasFDerivAt f f' x\ncf : IsConformalMap f'\n\u22a2 ConformalAt (g \u2218 f) x\n[PROOFSTEP]\nrcases hg with \u27e8g', hg\u2081, cg\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedAddCommGroup Y\ninst\u271d\u00b3 : NormedAddCommGroup Z\ninst\u271d\u00b2 : NormedSpace \u211d X\ninst\u271d\u00b9 : NormedSpace \u211d Y\ninst\u271d : NormedSpace \u211d Z\nf : X \u2192 Y\ng : Y \u2192 Z\nx : X\nf' : X \u2192L[\u211d] Y\nhf\u2081 : HasFDerivAt f f' x\ncf : IsConformalMap f'\ng' : Y \u2192L[\u211d] Z\nhg\u2081 : HasFDerivAt g g' (f x)\ncg : IsConformalMap g'\n\u22a2 ConformalAt (g \u2218 f) x\n[PROOFSTEP]\nexact \u27e8g'.comp f', hg\u2081.comp x hf\u2081, cg.comp cf\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "llama_tokens": 1994, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031738057795402, "lm_q2_score": 0.685949467848392, "lm_q1q2_score": 0.5509366446642433}}
{"text": "[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\nx y z : \u2124_[p]\nhz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)\n\u22a2 \u2016Polynomial.eval x F - Polynomial.eval y F\u2016 = \u2016z\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nsimp [hz]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\nx y z : \u2124_[p]\nhz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)\n\u22a2 \u2016z\u2016 * \u2016x - y\u2016 \u2264 1 * \u2016x - y\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\nx y z : \u2124_[p]\nhz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)\n\u22a2 \u2016z\u2016 \u2264 1\n[PROOFSTEP]\napply PadicInt.norm_le_one\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\nx y z : \u2124_[p]\nhz : Polynomial.eval x F - Polynomial.eval y F = z * (x - y)\n\u22a2 1 * \u2016x - y\u2016 = \u2016x - y\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nncs : CauSeq \u2124_[p] norm\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nncs_der_val :\n  \u2200 (n : \u2115), \u2016Polynomial.eval (\u2191ncs n) (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n\u22a2 Tendsto (fun i => \u2016Polynomial.eval (\u2191ncs i) (\u2191Polynomial.derivative F)\u2016) atTop\n    (\ud835\udcdd \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016)\n[PROOFSTEP]\nconvert @tendsto_const_nhds \u211d \u2115 _ _ _\n[GOAL]\ncase h.e'_3.h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nncs : CauSeq \u2124_[p] norm\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nncs_der_val :\n  \u2200 (n : \u2115), \u2016Polynomial.eval (\u2191ncs n) (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nx\u271d : \u2115\n\u22a2 \u2016Polynomial.eval (\u2191ncs x\u271d) (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [ncs_der_val]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nncs : CauSeq \u2124_[p] norm\nF : Polynomial \u2124_[p]\nhnorm : Tendsto (fun i => \u2016Polynomial.eval (\u2191ncs i) F\u2016) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun e => \u2016Polynomial.eval (\u2191ncs e) F - 0\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using hnorm\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nh : \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 = 0\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 = 0\n[PROOFSTEP]\nsimp [*, sq]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 T_gen p F a = \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nsimp [T_gen, \u2190 PadicInt.norm_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 T_gen p F a < 1\n[PROOFSTEP]\nhave h := (div_lt_one (deriv_sq_norm_pos hnorm)).2 hnorm\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nh : \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 < 1\n\u22a2 T_gen p F a < 1\n[PROOFSTEP]\nrw [T_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nh : \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 < 1\n\u22a2 \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 < 1\n[PROOFSTEP]\nexact h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2016Polynomial.eval a F\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ 0\n[PROOFSTEP]\nsimp [T_def, mul_div_cancel' _ (ne_of_gt (deriv_sq_norm_pos hnorm))]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\n\u22a2 \u2016\u2191(Polynomial.eval z F)\u2016 / \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 =\n    \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nsimp [hz.1]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\n\u22a2 \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 \u2264\n    \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n /\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\n\u22a2 \u2016Polynomial.eval z F\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\napply hz.2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nhz1 : \u2016z1\u2016 = \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nn : \u2115\nhz : ih_gen n z\n\u22a2 \u2016z' - z\u2016 = \u2016z1\u2016\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, add_assoc, hz', add_add_neg_cancel'_right, norm_neg]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nhz1 : \u2016z1\u2016 = \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nn : \u2115\nhz : ih_gen n z\n\u22a2 \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 \u2264\n    \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n /\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nhz1 : \u2016z1\u2016 = \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nn : \u2115\nhz : ih_gen n z\n\u22a2 \u2016Polynomial.eval z F\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\napply hz.2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 { q // Polynomial.eval z' F = q * z1 ^ 2 }\n[PROOFSTEP]\nhave hdzne : F.derivative.eval z \u2260 0 := mt norm_eq_zero.2 (by rw [hz.1]; apply deriv_norm_ne_zero; assumption)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u00ac\u2016Polynomial.eval z (\u2191Polynomial.derivative F)\u2016 = 0\n[PROOFSTEP]\nrw [hz.1]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u00ac\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 = 0\n[PROOFSTEP]\napply deriv_norm_ne_zero\n[GOAL]\ncase hnorm\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nassumption\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\n\u22a2 { q // Polynomial.eval z' F = q * z1 ^ 2 }\n[PROOFSTEP]\nhave hdzne' : (\u2191(F.derivative.eval z) : \u211a_[p]) \u2260 0 := fun h => hdzne (Subtype.ext_iff_val.2 h)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\n\u22a2 { q // Polynomial.eval z' F = q * z1 ^ 2 }\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := F.binomExpansion z (-z1)\n[GOAL]\ncase mk\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\n\u22a2 { q // Polynomial.eval z' F = q * z1 ^ 2 }\n[PROOFSTEP]\nhave : \u2016(\u2191(F.derivative.eval z) * (\u2191(F.eval z) / \u2191(F.derivative.eval z)) : \u211a_[p])\u2016 \u2264 1 :=\n  by\n  rw [padicNormE.mul]\n  exact mul_le_one (PadicInt.norm_le_one _) (norm_nonneg _) h1\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\n\u22a2 \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\n[PROOFSTEP]\nrw [padicNormE.mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\n\u22a2 \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 *\n      \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264\n    1\n[PROOFSTEP]\nexact mul_le_one (PadicInt.norm_le_one _) (norm_nonneg _) h1\n[GOAL]\ncase mk\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\n\u22a2 { q // Polynomial.eval z' F = q * z1 ^ 2 }\n[PROOFSTEP]\nhave : F.derivative.eval z * -z1 = -F.eval z := by\n  calc\n    F.derivative.eval z * -z1 = F.derivative.eval z * -\u27e8\u2191(F.eval z) / \u2191(F.derivative.eval z), h1\u27e9 := by rw [hzeq]\n    _ = -(F.derivative.eval z * \u27e8\u2191(F.eval z) / \u2191(F.derivative.eval z), h1\u27e9) := (mul_neg _ _)\n    _ = -\u27e8F.derivative.eval z * (F.eval z / (F.derivative.eval z : \u2124_[p]) : \u211a_[p]), this\u27e9 :=\n      (Subtype.ext <| by simp only [PadicInt.coe_neg, PadicInt.coe_mul, Subtype.coe_mk])\n    _ = -F.eval z := by simp only [mul_div_cancel' _ hdzne', Subtype.coe_eta]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\n\u22a2 Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 = -Polynomial.eval z F\n[PROOFSTEP]\ncalc\n  F.derivative.eval z * -z1 = F.derivative.eval z * -\u27e8\u2191(F.eval z) / \u2191(F.derivative.eval z), h1\u27e9 := by rw [hzeq]\n  _ = -(F.derivative.eval z * \u27e8\u2191(F.eval z) / \u2191(F.derivative.eval z), h1\u27e9) := (mul_neg _ _)\n  _ = -\u27e8F.derivative.eval z * (F.eval z / (F.derivative.eval z : \u2124_[p]) : \u211a_[p]), this\u27e9 :=\n    (Subtype.ext <| by simp only [PadicInt.coe_neg, PadicInt.coe_mul, Subtype.coe_mk])\n  _ = -F.eval z := by simp only [mul_div_cancel' _ hdzne', Subtype.coe_eta]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\n\u22a2 Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 =\n    Polynomial.eval z (\u2191Polynomial.derivative F) *\n      -{ val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n[PROOFSTEP]\nrw [hzeq]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\n\u22a2 \u2191(-(Polynomial.eval z (\u2191Polynomial.derivative F) *\n          { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 })) =\n    \u2191(-{\n          val :=\n            \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n              (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))),\n          property := this })\n[PROOFSTEP]\nsimp only [PadicInt.coe_neg, PadicInt.coe_mul, Subtype.coe_mk]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\n\u22a2 -{\n        val :=\n          \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n            (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))),\n        property := this } =\n    -Polynomial.eval z F\n[PROOFSTEP]\nsimp only [mul_div_cancel' _ hdzne', Subtype.coe_eta]\n[GOAL]\ncase mk\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis\u271d :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\nthis : Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 = -Polynomial.eval z F\n\u22a2 { q // Polynomial.eval z' F = q * z1 ^ 2 }\n[PROOFSTEP]\nexact \u27e8q, by simpa only [sub_eq_add_neg, this, hz', add_right_neg, neg_sq, zero_add] using hq\u27e9\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nhz' : z' = z - z1\nn : \u2115\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nhdzne : Polynomial.eval z (\u2191Polynomial.derivative F) \u2260 0\nhdzne' : \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) \u2260 0\nq : \u2124_[p]\nhq :\n  Polynomial.eval (z + -z1) F = Polynomial.eval z F + Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 + q * (-z1) ^ 2\nthis\u271d :\n  \u2016\u2191(Polynomial.eval z (\u2191Polynomial.derivative F)) *\n        (\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)))\u2016 \u2264\n    1\nthis : Polynomial.eval z (\u2191Polynomial.derivative F) * -z1 = -Polynomial.eval z F\n\u22a2 Polynomial.eval z' F = q * z1 ^ 2\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg, this, hz', add_right_neg, neg_sq, zero_add] using hq\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016Polynomial.eval z' F\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ (n + 1)\n[PROOFSTEP]\ncalc\n  \u2016F.eval z'\u2016 = \u2016q\u2016 * \u2016z1\u2016 ^ 2 := by simp [heq]\n  _ \u2264 1 * \u2016z1\u2016 ^ 2 := by gcongr; apply PadicInt.norm_le_one\n  _ = \u2016F.eval z\u2016 ^ 2 / \u2016F.derivative.eval a\u2016 ^ 2 := by simp [hzeq, hz.1, div_pow]\n  _ \u2264 (\u2016F.derivative.eval a\u2016 ^ 2 * T ^ 2 ^ n) ^ 2 / \u2016F.derivative.eval a\u2016 ^ 2 :=\n    by\n    gcongr\n    exact hz.2\n  _ = (\u2016F.derivative.eval a\u2016 ^ 2) ^ 2 * (T ^ 2 ^ n) ^ 2 / \u2016F.derivative.eval a\u2016 ^ 2 := by simp only [mul_pow]\n  _ = \u2016F.derivative.eval a\u2016 ^ 2 * (T ^ 2 ^ n) ^ 2 := (div_sq_cancel _ _)\n  _ = \u2016F.derivative.eval a\u2016 ^ 2 * T ^ 2 ^ (n + 1) := by\n    rw [\u2190 pow_mul, pow_succ' 2]\n      -- Porting note: unsupported option eqn_compiler.zeta\n      -- set_option eqn_compiler.zeta true\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016Polynomial.eval z' F\u2016 = \u2016q\u2016 * \u2016z1\u2016 ^ 2\n[PROOFSTEP]\nsimp [heq]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016q\u2016 * \u2016z1\u2016 ^ 2 \u2264 1 * \u2016z1\u2016 ^ 2\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016q\u2016 \u2264 1\n[PROOFSTEP]\napply PadicInt.norm_le_one\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 1 * \u2016z1\u2016 ^ 2 = \u2016Polynomial.eval z F\u2016 ^ 2 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nsimp [hzeq, hz.1, div_pow]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016Polynomial.eval z F\u2016 ^ 2 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 \u2264\n    (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n) ^ 2 /\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.hab\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016Polynomial.eval z F\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\nexact hz.2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n) ^ 2 /\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 =\n    (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2) ^ 2 * (T_gen p F a ^ 2 ^ n) ^ 2 /\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nsimp only [mul_pow]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz z' z1 : \u2124_[p]\nn : \u2115\nhz : ih_gen n z\nq : \u2124_[p]\nheq : Polynomial.eval z' F = q * z1 ^ 2\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nhzeq : z1 = { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * (T_gen p F a ^ 2 ^ n) ^ 2 =\n    \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ (n + 1)\n[PROOFSTEP]\nrw [\u2190 pow_mul, pow_succ' 2]\n  -- Porting note: unsupported option eqn_compiler.zeta\n  -- set_option eqn_compiler.zeta true\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nz1 : \u2124_[p] := { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nz' : \u2124_[p] := z - z1\n\u22a2 \u2016z1\u2016 = \u2016Polynomial.eval z F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nsimp [hz.1]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nz1 : \u2124_[p] := { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nz' : \u2124_[p] := z - z1\nhdist :\n  \u2016Polynomial.eval z' (\u2191Polynomial.derivative F) - Polynomial.eval z (\u2191Polynomial.derivative F)\u2016 <\n    \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n\u22a2 \u2016Polynomial.eval z' (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 hz.1, \u2190 norm_neg (F.derivative.eval z)] at hdist \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nz1 : \u2124_[p] := { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nz' : \u2124_[p] := z - z1\nhdist :\n  \u2016Polynomial.eval z' (\u2191Polynomial.derivative F) + -Polynomial.eval z (\u2191Polynomial.derivative F)\u2016 <\n    \u2016-Polynomial.eval z (\u2191Polynomial.derivative F)\u2016\n\u22a2 \u2016Polynomial.eval z' (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nhave := PadicInt.norm_eq_of_norm_add_lt_right hdist\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nz : \u2124_[p]\nhz : ih_gen n z\nh1 : \u2016\u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F))\u2016 \u2264 1\nz1 : \u2124_[p] := { val := \u2191(Polynomial.eval z F) / \u2191(Polynomial.eval z (\u2191Polynomial.derivative F)), property := h1 }\nz' : \u2124_[p] := z - z1\nhdist :\n  \u2016Polynomial.eval z' (\u2191Polynomial.derivative F) + -Polynomial.eval z (\u2191Polynomial.derivative F)\u2016 <\n    \u2016-Polynomial.eval z (\u2191Polynomial.derivative F)\u2016\nthis : \u2016Polynomial.eval z' (\u2191Polynomial.derivative F)\u2016 = \u2016-Polynomial.eval z (\u2191Polynomial.derivative F)\u2016\n\u22a2 \u2016Polynomial.eval z' (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrwa [norm_neg, hz.1] at this \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016newton_seq (n + 1) - newton_seq n\u2016 =\n    \u2016Polynomial.eval (newton_seq n) F\u2016 / \u2016Polynomial.eval (newton_seq n) (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [newton_seq_gen, newton_seq_gen, newton_seq_aux, ih_n]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016\u2191(let_fun h1 :=\n            (_ :\n              \u2016\u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F) /\n                    \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F))\u2016 \u2264\n                1);\n          let z1 :=\n            {\n              val :=\n                \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F) /\n                  \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F)),\n              property := h1 };\n          let z' := \u2191(newton_seq_aux hnorm n) - z1;\n          { val := z', property := (_ : ih_gen (n + 1) z') }) -\n        \u2191(newton_seq_aux hnorm n)\u2016 =\n    \u2016Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F\u2016 /\n      \u2016Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nsimp [sub_eq_add_neg, add_comm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016Polynomial.eval (newton_seq n) F\u2016 / \u2016Polynomial.eval (newton_seq n) (\u2191Polynomial.derivative F)\u2016 =\n    \u2016Polynomial.eval (newton_seq n) F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [newton_seq_deriv_norm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 T_gen p F a > 0\n[PROOFSTEP]\nrw [T_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 > 0\n[PROOFSTEP]\nexact div_pos (norm_pos_iff.2 hnsol) (deriv_sq_norm_pos hnorm)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 2 \u2264 2 ^ (n + 1)\n[PROOFSTEP]\nhave := pow_le_pow (by norm_num : 1 \u2264 2) (Nat.le_add_left _ _ : 1 \u2264 n + 1)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 1 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nthis : 2 ^ 1 \u2264 2 ^ (n + 1)\n\u22a2 2 \u2264 2 ^ (n + 1)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nthis : 2 \u2264 2 ^ (n + 1)\n\u22a2 1 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nthis : 2 \u2264 2 ^ (n + 1)\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 1 =\n    \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [T_gen, sq, pow_one, norm_div, \u2190 mul_div_assoc, PadicInt.padic_norm_e_of_padicInt, PadicInt.coe_mul, padicNormE.mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nthis : 2 \u2264 2 ^ (n + 1)\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * \u2016Polynomial.eval a F\u2016 /\n      (\u2016\u2191(Polynomial.eval a (\u2191Polynomial.derivative F))\u2016 * \u2016\u2191(Polynomial.eval a (\u2191Polynomial.derivative F))\u2016) =\n    \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\napply mul_div_mul_left\n[GOAL]\ncase hc\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nthis : 2 \u2264 2 ^ (n + 1)\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 \u2260 0\n[PROOFSTEP]\napply deriv_norm_ne_zero\n[GOAL]\ncase hc.hnorm\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\nthis : 2 \u2264 2 ^ (n + 1)\n\u22a2 \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nassumption\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016newton_seq (n + 0) - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\nsimp [T_pow_nonneg, mul_nonneg]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\n\u22a2 2 ^ n \u2264 2 ^ (n + k)\n[PROOFSTEP]\napply pow_le_pow\n[GOAL]\ncase ha\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\n\u22a2 1 \u2264 2\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\n\u22a2 n \u2264 n + k\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\n\u22a2 n \u2264 n + k\n[PROOFSTEP]\napply Nat.le_add_right\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nthis : 2 ^ n \u2264 2 ^ (n + k)\n\u22a2 \u2016newton_seq (n + (k + 1)) - newton_seq n\u2016 = \u2016newton_seq (n + k + 1) - newton_seq n\u2016\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nthis : 2 ^ n \u2264 2 ^ (n + k)\n\u22a2 \u2016newton_seq (n + k + 1) - newton_seq n\u2016 =\n    \u2016newton_seq (n + k + 1) - newton_seq (n + k) + (newton_seq (n + k) - newton_seq n)\u2016\n[PROOFSTEP]\nrw [\u2190 sub_add_sub_cancel]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\nhave hex : \u2203 m, k = n + m := exists_eq_add_of_le hnk\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\nlet \u27e8_, hex'\u27e9 := hex\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\nw\u271d : \u2115\nhex' : k = n + w\u271d\n\u22a2 \u2016newton_seq k - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\nrw [hex']\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn k : \u2115\nhnk : n \u2264 k\nhex : \u2203 m, k = n + m\nw\u271d : \u2115\nhex' : k = n + w\u271d\n\u22a2 \u2016newton_seq (n + w\u271d) - newton_seq n\u2016 \u2264 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n\n[PROOFSTEP]\napply newton_seq_dist_aux\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n_h : 0 < 1\n\u22a2 \u2016newton_seq 1 - a\u2016 = \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nsimp [sub_eq_add_neg, add_assoc, newton_seq_gen, newton_seq_aux, ih_n]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nk : \u2115\n_h : 0 < k + 2\n\u22a2 \u2016newton_seq (k + 2) - newton_seq (k + 1)\u2016 < \u2016newton_seq (k + 1) - a\u2016\n[PROOFSTEP]\nrw [newton_seq_dist_to_a (k + 1) (succ_pos _)]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nk : \u2115\n_h : 0 < k + 2\n\u22a2 \u2016newton_seq (k + 2) - newton_seq (k + 1)\u2016 < \u2016Polynomial.eval a F\u2016 / \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\napply newton_seq_succ_dist_weak\n[GOAL]\ncase hnsol\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nk : \u2115\n_h : 0 < k + 2\n\u22a2 Polynomial.eval a F \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nk : \u2115\n_h : 0 < k + 2\nhlt : \u2016newton_seq (k + 2) - newton_seq (k + 1)\u2016 < \u2016newton_seq (k + 1) - a\u2016\nhne' : \u2016newton_seq (k + 2) - newton_seq (k + 1)\u2016 \u2260 \u2016newton_seq (k + 1) - a\u2016\n\u22a2 \u2016newton_seq (k + 2) - a\u2016 = \u2016newton_seq (k + 2) - newton_seq (k + 1) + (newton_seq (k + 1) - a)\u2016\n[PROOFSTEP]\nrw [\u2190 sub_add_sub_cancel]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto (fun n => \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 mul_zero \u2016F.derivative.eval a\u2016]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto (fun n => \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n) atTop\n    (\ud835\udcdd (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * 0))\n[PROOFSTEP]\nexact\n  tendsto_const_nhds.mul\n    (Tendsto.comp (tendsto_pow_atTop_nhds_0_of_lt_1 (norm_nonneg _) (T_lt_one hnorm))\n      (Nat.tendsto_pow_atTop_atTop_of_one_lt (by norm_num)))\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 1 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2200 {\u03b5 : \u211d}, \u03b5 > 0 \u2192 \u2203 N, \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\nhave := bound' hnorm\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nthis : Tendsto (fun n => \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 {\u03b5 : \u211d}, \u03b5 > 0 \u2192 \u2203 N, \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\nsimp [Tendsto, nhds] at this \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nthis :\n  \u2200 (i : Set \u211d),\n    0 \u2208 i \u2192\n      IsOpen i \u2192 \u2203 a_3, \u2200 (b : \u2115), a_3 \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 i\n\u22a2 \u2200 {\u03b5 : \u211d}, \u03b5 > 0 \u2192 \u2203 N, \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nthis :\n  \u2200 (i : Set \u211d),\n    0 \u2208 i \u2192\n      IsOpen i \u2192 \u2203 a_3, \u2200 (b : \u2115), a_3 \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 i\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\ncases' this (ball 0 \u03b5) (mem_ball_self h\u03b5) isOpen_ball with N hN\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nthis :\n  \u2200 (i : Set \u211d),\n    0 \u2208 i \u2192\n      IsOpen i \u2192 \u2203 a_3, \u2200 (b : \u2115), a_3 \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 i\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 ball 0 \u03b5\n\u22a2 \u2203 N, \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\nexists N\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nthis :\n  \u2200 (i : Set \u211d),\n    0 \u2208 i \u2192\n      IsOpen i \u2192 \u2203 a_3, \u2200 (b : \u2115), a_3 \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 i\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 ball 0 \u03b5\n\u22a2 \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\nintro n hn\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nthis :\n  \u2200 (i : Set \u211d),\n    0 \u2208 i \u2192\n      IsOpen i \u2192 \u2203 a_3, \u2200 (b : \u2115), a_3 \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 i\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (b : \u2115), N \u2264 b \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ b \u2208 ball 0 \u03b5\nn : \u2115\nhn : n \u2265 N\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n[PROOFSTEP]\nsimpa [abs_of_nonneg T_nonneg] using hN _ hn\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto (fun n => \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2 * T_gen p F a ^ 2 ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 mul_zero \u2016F.derivative.eval a\u2016, sq]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto\n    (fun n =>\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 *\n        T_gen p F a ^ 2 ^ n)\n    atTop (\ud835\udcdd (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * 0))\n[PROOFSTEP]\nsimp only [mul_assoc]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto\n    (fun n =>\n      \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 *\n        (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n))\n    atTop (\ud835\udcdd (\u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * 0))\n[PROOFSTEP]\napply Tendsto.mul\n[GOAL]\ncase hf\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto (fun x => \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016) atTop\n    (\ud835\udcdd \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016)\n[PROOFSTEP]\napply tendsto_const_nhds\n[GOAL]\ncase hg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Tendsto (fun x => \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply bound'\n[GOAL]\ncase hg.hnorm\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nassumption\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 IsCauSeq norm newton_seq\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016newton_seq j - newton_seq i\u2016 < \u03b5\n[PROOFSTEP]\ncases' bound hnorm h\u03b5 with N hN\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016newton_seq j - newton_seq i\u2016 < \u03b5\n[PROOFSTEP]\nexists N\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\n\u22a2 \u2200 (j : \u2115), j \u2265 N \u2192 \u2016newton_seq j - newton_seq N\u2016 < \u03b5\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\nj : \u2115\nhj : j \u2265 N\n\u22a2 \u2016newton_seq j - newton_seq N\u2016 < \u03b5\n[PROOFSTEP]\napply lt_of_le_of_lt\n[GOAL]\ncase intro.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\nj : \u2115\nhj : j \u2265 N\n\u22a2 \u2016newton_seq j - newton_seq N\u2016 \u2264 ?intro.b\n[PROOFSTEP]\napply newton_seq_dist hnorm hj\n[GOAL]\ncase intro.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\nj : \u2115\nhj : j \u2265 N\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ N < \u03b5\n[PROOFSTEP]\napply hN\n[GOAL]\ncase intro.a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 {n : \u2115}, n \u2265 N \u2192 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * T_gen p F a ^ 2 ^ n < \u03b5\nj : \u2115\nhj : j \u2265 N\n\u22a2 N \u2265 N\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2016soln - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [soln_dist_to_a, div_lt_iff]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2016Polynomial.eval a F\u2016 <\n    \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 * \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrwa [sq] at hnorm \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 0 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\napply deriv_norm_pos\n[GOAL]\ncase hnorm\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hnsol\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\n\u22a2 Polynomial.eval a F \u2260 0\n[PROOFSTEP]\nexact hnsol\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n\u22a2 \u2016z - soln\u2016 = \u2016z - a + (a - soln)\u2016\n[PROOFSTEP]\nrw [sub_add_sub_cancel]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 0 = Polynomial.eval (soln + h) F\n[PROOFSTEP]\nsimp [hev]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 Polynomial.eval (soln + h) F = Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\n[PROOFSTEP]\nrw [hq, eval_soln, zero_add]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2 =\n    (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h\n[PROOFSTEP]\nrw [sq, right_distrib, mul_assoc]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\n\u22a2 Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n[PROOFSTEP]\nsimpa using eq_neg_of_add_eq_zero_left this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016Polynomial.eval soln (\u2191Polynomial.derivative F)\u2016 = \u2016-q * h\u2016\n[PROOFSTEP]\nrw [this]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016-q * h\u2016 \u2264 1 * \u2016h\u2016\n[PROOFSTEP]\nrw [PadicInt.norm_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016-q\u2016 * \u2016h\u2016 \u2264 1 * \u2016h\u2016\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right (PadicInt.norm_le_one _) (norm_nonneg _)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n\u22a2 1 * \u2016h\u2016 = \u2016z - soln\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016z - soln\u2016 < \u2016Polynomial.eval soln (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nrw [soln_deriv_norm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval soln (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\napply soln_dist\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nz : \u2124_[p]\nhev : Polynomial.eval z F = 0\nhnlt : \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nsoln_dist : \u2016z - soln\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z - soln\nq : \u2124_[p]\nhq :\n  Polynomial.eval (soln + h) F =\n    Polynomial.eval soln F + Polynomial.eval soln (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d : (Polynomial.eval soln (\u2191Polynomial.derivative F) + q * h) * h = 0\nthis : h = 0\n\u22a2 z - soln = 0\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 0 = Polynomial.eval (a + (z' - a)) F\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 0 = Polynomial.eval (z' - a + a) F\n[PROOFSTEP]\nsimp [hz']\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 Polynomial.eval (a + h) F = Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\n[PROOFSTEP]\nrw [hq, ha, zero_add]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\n\u22a2 Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2 =\n    (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h\n[PROOFSTEP]\nrw [sq, right_distrib, mul_assoc]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d : (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis : Polynomial.eval a (\u2191Polynomial.derivative F) + q * h = 0\n\u22a2 Polynomial.eval a (\u2191Polynomial.derivative F) = -q * h\n[PROOFSTEP]\nsimpa using eq_neg_of_add_eq_zero_left this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval a (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 = \u2016q\u2016 * \u2016h\u2016\n[PROOFSTEP]\nsimp [this]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval a (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016q\u2016 * \u2016h\u2016 \u2264 1 * \u2016h\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval a (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (\u2191Polynomial.derivative F) = -q * h\n\u22a2 \u2016q\u2016 \u2264 1\n[PROOFSTEP]\napply PadicInt.norm_le_one\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d\u00b9 : (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h = 0\nhne : \u00ach = 0\nthis\u271d : Polynomial.eval a (\u2191Polynomial.derivative F) + q * h = 0\nthis : Polynomial.eval a (\u2191Polynomial.derivative F) = -q * h\n\u22a2 1 * \u2016h\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nsimpa\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nha : Polynomial.eval a F = 0\nz' : \u2124_[p]\nhz' : Polynomial.eval z' F = 0\nhnormz' : \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\nh : \u2124_[p] := z' - a\nq : \u2124_[p]\nhq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (\u2191Polynomial.derivative F) * h + q * h ^ 2\nthis\u271d : (Polynomial.eval a (\u2191Polynomial.derivative F) + q * h) * h = 0\nthis : h = 0\n\u22a2 z' - a = 0\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nha : Polynomial.eval a F = 0\n\u22a2 \u2016a - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016\n[PROOFSTEP]\nsimp [deriv_ne_zero hnorm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nha : \u00acPolynomial.eval a F = 0\n\u22a2 \u2203 z,\n    Polynomial.eval z F = 0 \u2227\n      \u2016z - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 \u2227\n        \u2016Polynomial.eval z (\u2191Polynomial.derivative F)\u2016 = \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 \u2227\n          \u2200 (z' : \u2124_[p]), Polynomial.eval z' F = 0 \u2192 \u2016z' - a\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 \u2192 z' = z\n[PROOFSTEP]\nexact\n  \u27e8soln_gen hnorm, eval_soln hnorm, soln_dist_to_a_lt_deriv hnorm ha, soln_deriv_norm hnorm, fun z =>\n    soln_unique hnorm ha z\u27e9\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Padics.Hensel", "llama_tokens": 33000, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8688267864276108, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5503238038814835}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 hermite (n + 1) = X * hermite n - \u2191derivative (hermite n)\n[PROOFSTEP]\nrw [hermite]\n[GOAL]\nn : \u2115\n\u22a2 hermite n = (fun p => X * p - \u2191derivative p)^[n] 1\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u22a2 hermite Nat.zero = (fun p => X * p - \u2191derivative p)^[Nat.zero] 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn : \u2115\nih : hermite n = (fun p => X * p - \u2191derivative p)^[n] 1\n\u22a2 hermite (Nat.succ n) = (fun p => X * p - \u2191derivative p)^[Nat.succ n] 1\n[PROOFSTEP]\nrw [Function.iterate_succ_apply', \u2190 ih, hermite_succ]\n[GOAL]\n\u22a2 hermite 1 = X\n[PROOFSTEP]\nrw [hermite_succ, hermite_zero]\n[GOAL]\n\u22a2 X * \u2191C 1 - \u2191derivative (\u2191C 1) = X\n[PROOFSTEP]\nsimp only [map_one, mul_one, derivative_one, sub_zero]\n[GOAL]\nn : \u2115\n\u22a2 coeff (hermite (n + 1)) 0 = -coeff (hermite n) 1\n[PROOFSTEP]\nsimp [coeff_derivative]\n[GOAL]\nn k : \u2115\n\u22a2 coeff (hermite (n + 1)) (k + 1) = coeff (hermite n) k - (\u2191k + 2) * coeff (hermite n) (k + 2)\n[PROOFSTEP]\nrw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm]\n[GOAL]\nn k : \u2115\n\u22a2 coeff (hermite n) k - (\u2191(k + 1) + 1) * coeff (hermite n) (k + 1 + 1) =\n    coeff (hermite n) k - (\u2191k + 2) * coeff (hermite n) (k + 2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn k : \u2115\nhnk : n < k\n\u22a2 coeff (hermite n) k = 0\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_lt hnk\n[GOAL]\ncase intro\nn k : \u2115\nhnk : n < n + k + 1\n\u22a2 coeff (hermite n) (n + k + 1) = 0\n[PROOFSTEP]\nclear hnk\n[GOAL]\ncase intro\nn k : \u2115\n\u22a2 coeff (hermite n) (n + k + 1) = 0\n[PROOFSTEP]\ninduction' n with n ih generalizing k\n[GOAL]\ncase intro.zero\nk\u271d k : \u2115\n\u22a2 coeff (hermite Nat.zero) (Nat.zero + k + 1) = 0\n[PROOFSTEP]\napply coeff_C\n[GOAL]\ncase intro.succ\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (hermite n) (n + k + 1) = 0\nk : \u2115\n\u22a2 coeff (hermite (Nat.succ n)) (Nat.succ n + k + 1) = 0\n[PROOFSTEP]\nhave : n + k + 1 + 2 = n + (k + 2) + 1 := by ring\n[GOAL]\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (hermite n) (n + k + 1) = 0\nk : \u2115\n\u22a2 n + k + 1 + 2 = n + (k + 2) + 1\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.succ\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (hermite n) (n + k + 1) = 0\nk : \u2115\nthis : n + k + 1 + 2 = n + (k + 2) + 1\n\u22a2 coeff (hermite (Nat.succ n)) (Nat.succ n + k + 1) = 0\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, coeff_hermite_succ_succ, add_right_comm, this, ih k, ih (k + 2), mul_zero, sub_zero]\n[GOAL]\nn : \u2115\n\u22a2 coeff (hermite n) n = 1\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u22a2 coeff (hermite Nat.zero) Nat.zero = 1\n[PROOFSTEP]\napply coeff_C\n[GOAL]\ncase succ\nn : \u2115\nih : coeff (hermite n) n = 1\n\u22a2 coeff (hermite (Nat.succ n)) (Nat.succ n) = 1\n[PROOFSTEP]\nrw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero]\n[GOAL]\ncase succ\nn : \u2115\nih : coeff (hermite n) n = 1\n\u22a2 n < n + 2\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 degree (hermite n) = \u2191n\n[PROOFSTEP]\nrw [degree_eq_of_le_of_coeff_ne_zero]\n[GOAL]\ncase pn\nn : \u2115\n\u22a2 degree (hermite n) \u2264 \u2191n\ncase p1 n : \u2115 \u22a2 coeff (hermite n) n \u2260 0\n[PROOFSTEP]\nsimp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]\n[GOAL]\ncase pn\nn : \u2115\n\u22a2 \u2200 (m : \u2115), n < m \u2192 coeff (hermite n) m = 0\n[PROOFSTEP]\nrintro m hnm\n[GOAL]\ncase pn\nn m : \u2115\nhnm : n < m\n\u22a2 coeff (hermite n) m = 0\n[PROOFSTEP]\nexact coeff_hermite_of_lt hnm\n[GOAL]\ncase p1\nn : \u2115\n\u22a2 coeff (hermite n) n \u2260 0\n[PROOFSTEP]\nsimp [coeff_hermite_self n]\n[GOAL]\nn : \u2115\n\u22a2 leadingCoeff (hermite n) = 1\n[PROOFSTEP]\nrw [\u2190 coeff_natDegree, natDegree_hermite, coeff_hermite_self]\n[GOAL]\nn k : \u2115\nhnk : Odd (n + k)\n\u22a2 coeff (hermite n) k = 0\n[PROOFSTEP]\ninduction' n with n ih generalizing k\n[GOAL]\ncase zero\nn k\u271d : \u2115\nhnk\u271d : Odd (n + k\u271d)\nk : \u2115\nhnk : Odd (Nat.zero + k)\n\u22a2 coeff (hermite Nat.zero) k = 0\n[PROOFSTEP]\nrw [Nat.zero_eq, zero_add k] at hnk \n[GOAL]\ncase zero\nn k\u271d : \u2115\nhnk\u271d : Odd (n + k\u271d)\nk : \u2115\nhnk : Odd k\n\u22a2 coeff (hermite Nat.zero) k = 0\n[PROOFSTEP]\nexact coeff_hermite_of_lt hnk.pos\n[GOAL]\ncase succ\nn\u271d k\u271d : \u2115\nhnk\u271d : Odd (n\u271d + k\u271d)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nk : \u2115\nhnk : Odd (Nat.succ n + k)\n\u22a2 coeff (hermite (Nat.succ n)) k = 0\n[PROOFSTEP]\ncases' k with k\n[GOAL]\ncase succ.zero\nn\u271d k : \u2115\nhnk\u271d : Odd (n\u271d + k)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nhnk : Odd (Nat.succ n + Nat.zero)\n\u22a2 coeff (hermite (Nat.succ n)) Nat.zero = 0\n[PROOFSTEP]\nrw [Nat.succ_add_eq_succ_add] at hnk \n[GOAL]\ncase succ.zero\nn\u271d k : \u2115\nhnk\u271d : Odd (n\u271d + k)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nhnk : Odd (n + Nat.succ Nat.zero)\n\u22a2 coeff (hermite (Nat.succ n)) Nat.zero = 0\n[PROOFSTEP]\nrw [coeff_hermite_succ_zero, ih hnk, neg_zero]\n[GOAL]\ncase succ.succ\nn\u271d k\u271d : \u2115\nhnk\u271d : Odd (n\u271d + k\u271d)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nk : \u2115\nhnk : Odd (Nat.succ n + Nat.succ k)\n\u22a2 coeff (hermite (Nat.succ n)) (Nat.succ k) = 0\n[PROOFSTEP]\nrw [coeff_hermite_succ_succ, ih, ih, mul_zero, sub_zero]\n[GOAL]\ncase succ.succ\nn\u271d k\u271d : \u2115\nhnk\u271d : Odd (n\u271d + k\u271d)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nk : \u2115\nhnk : Odd (Nat.succ n + Nat.succ k)\n\u22a2 Odd (n + (k + 2))\n[PROOFSTEP]\nrwa [Nat.succ_add_eq_succ_add] at hnk \n[GOAL]\ncase succ.succ\nn\u271d k\u271d : \u2115\nhnk\u271d : Odd (n\u271d + k\u271d)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nk : \u2115\nhnk : Odd (Nat.succ n + Nat.succ k)\n\u22a2 Odd (n + k)\n[PROOFSTEP]\nrw [(by rw [Nat.succ_add, Nat.add_succ] : n.succ + k.succ = n + k + 2)] at hnk \n[GOAL]\nn\u271d k\u271d : \u2115\nhnk\u271d : Odd (n\u271d + k\u271d)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nk : \u2115\nhnk : Odd (Nat.succ n + Nat.succ k)\n\u22a2 Nat.succ n + Nat.succ k = n + k + 2\n[PROOFSTEP]\nrw [Nat.succ_add, Nat.add_succ]\n[GOAL]\ncase succ.succ\nn\u271d k\u271d : \u2115\nhnk\u271d : Odd (n\u271d + k\u271d)\nn : \u2115\nih : \u2200 {k : \u2115}, Odd (n + k) \u2192 coeff (hermite n) k = 0\nk : \u2115\nhnk : Odd (n + k + 2)\n\u22a2 Odd (n + k)\n[PROOFSTEP]\nexact (Nat.odd_add.mp hnk).mpr even_two\n[GOAL]\nx\u271d : \u2115\n\u22a2 coeff (hermite (2 * 0 + x\u271d)) x\u271d = (-1) ^ 0 * \u2191(2 * 0 - 1)\u203c * \u2191(Nat.choose (2 * 0 + x\u271d) x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 coeff (hermite (2 * (n + 1) + 0)) 0 = (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + 0) 0)\n[PROOFSTEP]\nconvert coeff_hermite_succ_zero (2 * n + 1) using 1\n  -- porting note: ring_nf did not solve the goal on line 165\n[GOAL]\ncase h.e'_3\nn : \u2115\n\u22a2 (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + 0) 0) = -coeff (hermite (2 * n + 1)) 1\n[PROOFSTEP]\nrw [coeff_hermite_explicit n 1, (by rw [Nat.left_distrib, mul_one, Nat.succ_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),\n  Nat.doubleFactorial_add_one, Nat.choose_zero_right, Nat.choose_one_right, pow_succ]\n[GOAL]\nn : \u2115\n\u22a2 2 * (n + 1) - 1 = 2 * n + 1\n[PROOFSTEP]\nrw [Nat.left_distrib, mul_one, Nat.succ_sub_one]\n[GOAL]\ncase h.e'_3\nn : \u2115\n\u22a2 -1 * (-1) ^ n * \u2191((2 * n + 1) * (2 * n - 1)\u203c) * \u21911 = -((-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(2 * n + 1))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_3\nn : \u2115\n\u22a2 -1 * (-1) ^ n * ((2 * \u2191n + 1) * \u2191(2 * n - 1)\u203c) * 1 = -((-1) ^ n * \u2191(2 * n - 1)\u203c * (2 * \u2191n + 1))\n[PROOFSTEP]\nring\n[GOAL]\nn k : \u2115\n\u22a2 coeff (hermite (2 * (n + 1) + (k + 1))) (k + 1) =\n    (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1))\n[PROOFSTEP]\nlet hermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * (2 * n - 1)\u203c * Nat.choose (2 * n + k) k\n[GOAL]\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\n\u22a2 coeff (hermite (2 * (n + 1) + (k + 1))) (k + 1) =\n    (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1))\n[PROOFSTEP]\nhave hermite_explicit_recur :\n  \u2200 n k : \u2115, hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2) :=\n  by\n  intro n k\n  simp only\n    -- Factor out (-1)'s.\n  rw [mul_comm (\u2191k + _ : \u2124), sub_eq_add_neg]\n  nth_rw 3 [neg_eq_neg_one_mul]\n  simp only [mul_assoc, \u2190 mul_add, pow_succ]\n  congr 2\n    -- Factor out double factorials.\n  norm_cast\n    -- porting note: ring_nf did not solve the goal on line 186\n  rw [(by rw [Nat.left_distrib, mul_one, Nat.succ_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one,\n    mul_comm (2 * n + 1)]\n  simp only [mul_assoc, \u2190 mul_add]\n  congr 1\n    -- Match up binomial coefficients using `Nat.choose_succ_right_eq`.\n  rw [(by ring : 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1), (by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),\n    (by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))]\n  rw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel, Int.negSucc_eq]\n    -- porting note: ring could not solve the goal so the lines 195, 198-200 were added.\n  ring_nf\n  simp only [sub_eq_add_neg, \u2190 neg_mul, \u2190 right_distrib _ _ ((-(1 : \u2124)) ^ n), \u2190 neg_add]\n  norm_cast\n  simp only [\u2190 add_assoc, add_comm]\n[GOAL]\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\n\u22a2 \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n[PROOFSTEP]\nintro n k\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n[PROOFSTEP]\nsimp only\n  -- Factor out (-1)'s.\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)) =\n    (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + k) k) -\n      (\u2191k + 2) * ((-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + (k + 2)) (k + 2)))\n[PROOFSTEP]\nrw [mul_comm (\u2191k + _ : \u2124), sub_eq_add_neg]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)) =\n    (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + k) k) +\n      -((-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + (k + 2)) (k + 2)) * (\u2191k + 2))\n[PROOFSTEP]\nnth_rw 3 [neg_eq_neg_one_mul]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)) =\n    (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + k) k) +\n      -((-1 * 1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + (k + 2)) (k + 2)) * (\u2191k + 2))\n[PROOFSTEP]\nsimp only [mul_assoc, \u2190 mul_add, pow_succ]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 -1 * ((-1) ^ n * (\u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)))) =\n    -1 * ((-1) ^ n * (\u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + k) k))) +\n      -((-1 * 1) ^ n * (\u2191(2 * n - 1)\u203c * (\u2191(Nat.choose (2 * n + (k + 2)) (k + 2)) * (\u2191k + 2))))\n[PROOFSTEP]\ncongr 2\n  -- Factor out double factorials.\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 -1 * ((-1) ^ n * (\u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)))) =\n    -1 * ((-1) ^ n * (\u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + k) k))) +\n      -((-1 * 1) ^ n * (\u2191(2 * n - 1)\u203c * (\u2191(Nat.choose (2 * n + (k + 2)) (k + 2)) * (\u2191k + 2))))\n[PROOFSTEP]\nnorm_cast\n  -- porting note: ring_nf did not solve the goal on line 186\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * (n + 1) - 1)\u203c * Nat.choose (2 * (n + 1) + (k + 1)) (k + 1))) =\n    Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * (n + 1) - 1)\u203c * Nat.choose (2 * (n + 1) + k) k)) +\n      -((Int.negSucc 0 * 1) ^ n * \u2191((2 * n - 1)\u203c * (Nat.choose (2 * n + (k + 2)) (k + 2) * (k + 2))))\n[PROOFSTEP]\nrw [(by rw [Nat.left_distrib, mul_one, Nat.succ_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1), Nat.doubleFactorial_add_one,\n  mul_comm (2 * n + 1)]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 2 * (n + 1) - 1 = 2 * n + 1\n[PROOFSTEP]\nrw [Nat.left_distrib, mul_one, Nat.succ_sub_one]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * (2 * n + 1) * Nat.choose (2 * (n + 1) + (k + 1)) (k + 1))) =\n    Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * (2 * n + 1) * Nat.choose (2 * (n + 1) + k) k)) +\n      -((Int.negSucc 0 * 1) ^ n * \u2191((2 * n - 1)\u203c * (Nat.choose (2 * n + (k + 2)) (k + 2) * (k + 2))))\n[PROOFSTEP]\nsimp only [mul_assoc, \u2190 mul_add]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)))) =\n    Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * (n + 1) + k) k))) +\n      -((Int.negSucc 0 * 1) ^ n * \u2191((2 * n - 1)\u203c * (Nat.choose (2 * n + (k + 2)) (k + 2) * (k + 2))))\n[PROOFSTEP]\ncongr 1\n  -- Match up binomial coefficients using `Nat.choose_succ_right_eq`.\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * (n + 1) + (k + 1)) (k + 1)))) =\n    Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * (n + 1) + k) k))) +\n      -((Int.negSucc 0 * 1) ^ n * \u2191((2 * n - 1)\u203c * (Nat.choose (2 * n + (k + 2)) (k + 2) * (k + 2))))\n[PROOFSTEP]\nrw [(by ring : 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1), (by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),\n  (by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1\n[PROOFSTEP]\nring\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 2 * (n + 1) + k = 2 * n + 1 + (k + 1)\n[PROOFSTEP]\nring\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 2 * n + (k + 2) = 2 * n + 1 + (k + 1)\n[PROOFSTEP]\nring\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * n + 1 + (k + 1) + 1) (k + 1)))) =\n    Int.negSucc 0 * (Int.negSucc 0 ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * n + 1 + (k + 1)) k))) +\n      -((Int.negSucc 0 * 1) ^ n * \u2191((2 * n - 1)\u203c * (Nat.choose (2 * n + 1 + (k + 1)) (k + 2) * (k + 2))))\n[PROOFSTEP]\nrw [Nat.choose, Nat.choose_succ_right_eq (2 * n + 1 + (k + 1)) (k + 1), Nat.add_sub_cancel, Int.negSucc_eq]\n  -- porting note: ring could not solve the goal so the lines 195, 198-200 were added.\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 -(\u21910 + 1) *\n      ((-(\u21910 + 1)) ^ n *\n        \u2191((2 * n - 1)\u203c *\n            ((2 * n + 1) * (Nat.choose (2 * n + 1 + (k + 1)) k + Nat.choose (2 * n + 1 + (k + 1)) (k + 1))))) =\n    -(\u21910 + 1) * ((-(\u21910 + 1)) ^ n * \u2191((2 * n - 1)\u203c * ((2 * n + 1) * Nat.choose (2 * n + 1 + (k + 1)) k))) +\n      -((-(\u21910 + 1) * 1) ^ n * \u2191((2 * n - 1)\u203c * (Nat.choose (2 * n + 1 + (k + 1)) (k + 1) * (2 * n + 1))))\n[PROOFSTEP]\nring_nf\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 -(\u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k * 2 + n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k) * 2 +\n              (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k +\n            (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k)) *\n        (-1) ^ n) =\n    -(\u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k * 2 + (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k) * (-1) ^ n) -\n      \u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k) * 2 + (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k)) *\n        (-1) ^ n\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, \u2190 neg_mul, \u2190 right_distrib _ _ ((-(1 : \u2124)) ^ n), \u2190 neg_add]\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 -\u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k * 2 + n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k) * 2 +\n              (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k +\n            (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k)) *\n      (-1) ^ n =\n    -(\u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k * 2 + (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k) +\n          \u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k) * 2 +\n              (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k))) *\n      (-1) ^ n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn\u271d k\u271d : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nn k : \u2115\n\u22a2 -\u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k * 2 + n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k) * 2 +\n              (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k +\n            (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k)) *\n      Int.negSucc 0 ^ n =\n    -\u2191(n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k * 2 + (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) k +\n            (n * (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k) * 2 +\n              (n * 2 - 1)\u203c * Nat.choose (2 + n * 2 + k) (1 + k))) *\n      Int.negSucc 0 ^ n\n[PROOFSTEP]\nsimp only [\u2190 add_assoc, add_comm]\n[GOAL]\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nhermite_explicit_recur :\n  \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n\u22a2 coeff (hermite (2 * (n + 1) + (k + 1))) (k + 1) =\n    (-1) ^ (n + 1) * \u2191(2 * (n + 1) - 1)\u203c * \u2191(Nat.choose (2 * (n + 1) + (k + 1)) (k + 1))\n[PROOFSTEP]\nchange _ = hermite_explicit _ _\n[GOAL]\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nhermite_explicit_recur :\n  \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n\u22a2 coeff (hermite (2 * (n + 1) + (k + 1))) (k + 1) = hermite_explicit (n + 1) (k + 1)\n[PROOFSTEP]\nrw [\u2190 add_assoc, coeff_hermite_succ_succ, hermite_explicit_recur]\n[GOAL]\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nhermite_explicit_recur :\n  \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n\u22a2 coeff (hermite (2 * (n + 1) + k)) k - (\u2191k + 2) * coeff (hermite (2 * (n + 1) + k)) (k + 2) =\n    hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nhermite_explicit_recur :\n  \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n\u22a2 coeff (hermite (2 * (n + 1) + k)) k = hermite_explicit (n + 1) k\n[PROOFSTEP]\nrw [coeff_hermite_explicit (n + 1) k]\n[GOAL]\ncase e_a.e_a\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nhermite_explicit_recur :\n  \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n\u22a2 coeff (hermite (2 * (n + 1) + k)) (k + 2) = hermite_explicit n (k + 2)\n[PROOFSTEP]\nrw [(by ring : 2 * (n + 1) + k = 2 * n + (k + 2)), coeff_hermite_explicit n (k + 2)]\n  -- porting note: Lean 3 worked this out automatically\n[GOAL]\nn k : \u2115\nhermite_explicit : \u2115 \u2192 \u2115 \u2192 \u2124 := fun n k => (-1) ^ n * \u2191(2 * n - 1)\u203c * \u2191(Nat.choose (2 * n + k) k)\nhermite_explicit_recur :\n  \u2200 (n k : \u2115), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (\u2191k + 2) * hermite_explicit n (k + 2)\n\u22a2 2 * (n + 1) + k = 2 * n + (k + 2)\n[PROOFSTEP]\nring\n[GOAL]\nn k : \u2115\nhnk : Even (n + k)\n\u22a2 coeff (hermite n) k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k)\n[PROOFSTEP]\ncases' le_or_lt k n with h_le h_lt\n[GOAL]\ncase inl\nn k : \u2115\nhnk : Even (n + k)\nh_le : k \u2264 n\n\u22a2 coeff (hermite n) k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k)\n[PROOFSTEP]\nrw [Nat.even_add, \u2190 Nat.even_sub h_le] at hnk \n[GOAL]\ncase inl\nn k : \u2115\nhnk : Even (n - k)\nh_le : k \u2264 n\n\u22a2 coeff (hermite n) k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k)\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hnk\n[GOAL]\ncase inl.intro\nn k : \u2115\nh_le : k \u2264 n\nm : \u2115\nhm : n - k = m + m\n\u22a2 coeff (hermite n) k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k)\n[PROOFSTEP]\nrw [(by linarith [by rwa [Nat.sub_eq_iff_eq_add h_le] at hm ] : n = 2 * m + k), Nat.add_sub_cancel,\n  Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit]\n[GOAL]\nn k : \u2115\nh_le : k \u2264 n\nm : \u2115\nhm : n - k = m + m\n\u22a2 n = 2 * m + k\n[PROOFSTEP]\nlinarith [by rwa [Nat.sub_eq_iff_eq_add h_le] at hm ]\n[GOAL]\nn k : \u2115\nh_le : k \u2264 n\nm : \u2115\nhm : n - k = m + m\n\u22a2 ?m.206711\n[PROOFSTEP]\nrwa [Nat.sub_eq_iff_eq_add h_le] at hm \n[GOAL]\ncase inr\nn k : \u2115\nhnk : Even (n + k)\nh_lt : n < k\n\u22a2 coeff (hermite n) k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k)\n[PROOFSTEP]\nsimp [Nat.choose_eq_zero_of_lt h_lt, coeff_hermite_of_lt h_lt]\n[GOAL]\nn k : \u2115\n\u22a2 coeff (hermite n) k = if Even (n + k) then (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn k : \u2115\nh : Even (n + k)\n\u22a2 coeff (hermite n) k = (-1) ^ ((n - k) / 2) * \u2191(n - k - 1)\u203c * \u2191(Nat.choose n k)\ncase neg n k : \u2115 h : \u00acEven (n + k) \u22a2 coeff (hermite n) k = 0\n[PROOFSTEP]\nexact coeff_hermite_of_even_add h\n[GOAL]\ncase neg\nn k : \u2115\nh : \u00acEven (n + k)\n\u22a2 coeff (hermite n) k = 0\n[PROOFSTEP]\nexact coeff_hermite_of_odd_add (Nat.odd_iff_not_even.mpr h)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Hermite.Basic", "llama_tokens": 12641, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696748, "lm_q2_score": 0.6959583187272711, "lm_q1q2_score": 0.5501761514842808}}
{"text": "[GOAL]\nx y : \u211d\nh :\n  ((\u2200 (k : \u2124), x \u2260 (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2200 (l : \u2124), y \u2260 (2 * \u2191l + 1) * \u03c0 / 2) \u2228\n    (\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2203 l, y = (2 * \u2191l + 1) * \u03c0 / 2\n\u22a2 tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)\n[PROOFSTEP]\nsimpa only [\u2190 Complex.ofReal_inj, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_div, Complex.ofReal_mul,\n  Complex.ofReal_tan] using @Complex.tan_add (x : \u2102) (y : \u2102) (by convert h <;> norm_cast)\n[GOAL]\nx y : \u211d\nh :\n  ((\u2200 (k : \u2124), x \u2260 (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2200 (l : \u2124), y \u2260 (2 * \u2191l + 1) * \u03c0 / 2) \u2228\n    (\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2203 l, y = (2 * \u2191l + 1) * \u03c0 / 2\n\u22a2 ((\u2200 (k : \u2124), \u2191x \u2260 (2 * \u2191k + 1) * \u2191\u03c0 / 2) \u2227 \u2200 (l : \u2124), \u2191y \u2260 (2 * \u2191l + 1) * \u2191\u03c0 / 2) \u2228\n    (\u2203 k, \u2191x = (2 * \u2191k + 1) * \u2191\u03c0 / 2) \u2227 \u2203 l, \u2191y = (2 * \u2191l + 1) * \u2191\u03c0 / 2\n[PROOFSTEP]\nconvert h\n[GOAL]\ncase h.e'_1.h.e'_1.h.h.a\nx y : \u211d\nh :\n  ((\u2200 (k : \u2124), x \u2260 (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2200 (l : \u2124), y \u2260 (2 * \u2191l + 1) * \u03c0 / 2) \u2228\n    (\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2203 l, y = (2 * \u2191l + 1) * \u03c0 / 2\na\u271d : \u2124\n\u22a2 \u2191x = (2 * \u2191a\u271d + 1) * \u2191\u03c0 / 2 \u2194 x = (2 * \u2191a\u271d + 1) * \u03c0 / 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_1.h.e'_2.h.h.a\nx y : \u211d\nh :\n  ((\u2200 (k : \u2124), x \u2260 (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2200 (l : \u2124), y \u2260 (2 * \u2191l + 1) * \u03c0 / 2) \u2228\n    (\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2203 l, y = (2 * \u2191l + 1) * \u03c0 / 2\na\u271d : \u2124\n\u22a2 \u2191y = (2 * \u2191a\u271d + 1) * \u2191\u03c0 / 2 \u2194 y = (2 * \u2191a\u271d + 1) * \u03c0 / 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_2.h.a\nx y : \u211d\nh :\n  ((\u2200 (k : \u2124), x \u2260 (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2200 (l : \u2124), y \u2260 (2 * \u2191l + 1) * \u03c0 / 2) \u2228\n    (\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2203 l, y = (2 * \u2191l + 1) * \u03c0 / 2\nx\u271d : \u2124\n\u22a2 \u2191x = (2 * \u2191x\u271d + 1) * \u2191\u03c0 / 2 \u2194 x = (2 * \u2191x\u271d + 1) * \u03c0 / 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h.e'_2.h.e'_2.h.e'_2.h.a\nx y : \u211d\nh :\n  ((\u2200 (k : \u2124), x \u2260 (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2200 (l : \u2124), y \u2260 (2 * \u2191l + 1) * \u03c0 / 2) \u2228\n    (\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2) \u2227 \u2203 l, y = (2 * \u2191l + 1) * \u03c0 / 2\nx\u271d : \u2124\n\u22a2 \u2191y = (2 * \u2191x\u271d + 1) * \u2191\u03c0 / 2 \u2194 y = (2 * \u2191x\u271d + 1) * \u03c0 / 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nx : \u211d\n\u22a2 tan (2 * x) = \u21912 * tan x / (\u21911 - tan x ^ 2)\n[PROOFSTEP]\nhave := @Complex.tan_two_mul x\n[GOAL]\nx : \u211d\nthis : Complex.tan (2 * \u2191x) = 2 * Complex.tan \u2191x / (1 - Complex.tan \u2191x ^ 2)\n\u22a2 tan (2 * x) = \u21912 * tan x / (\u21911 - tan x ^ 2)\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\n\u03b8 : \u211d\n\u22a2 tan \u03b8 \u2260 0 \u2194 \u2200 (k : \u2124), \u03b8 \u2260 \u2191k * \u03c0 / 2\n[PROOFSTEP]\nrw [\u2190 Complex.ofReal_ne_zero, Complex.ofReal_tan, Complex.tan_ne_zero_iff]\n[GOAL]\n\u03b8 : \u211d\n\u22a2 (\u2200 (k : \u2124), \u2191\u03b8 \u2260 \u2191k * \u2191\u03c0 / 2) \u2194 \u2200 (k : \u2124), \u03b8 \u2260 \u2191k * \u03c0 / 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b8 : \u211d\n\u22a2 tan \u03b8 = 0 \u2194 \u2203 k, \u03b8 = \u2191k * \u03c0 / 2\n[PROOFSTEP]\nrw [\u2190 not_iff_not, not_exists, \u2190 Ne, tan_ne_zero_iff]\n[GOAL]\nn : \u2124\n\u22a2 \u2203 k, \u2191n * \u03c0 / 2 = \u2191k * \u03c0 / 2\n[PROOFSTEP]\nuse n\n[GOAL]\n\u22a2 ContinuousOn tan {x | cos x \u2260 0}\n[PROOFSTEP]\nsuffices ContinuousOn (fun x => sin x / cos x) {x | cos x \u2260 0}\n  by\n  have h_eq : (fun x => sin x / cos x) = tan := by ext1 x; rw [tan_eq_sin_div_cos]\n  rwa [h_eq] at this \n[GOAL]\nthis : ContinuousOn (fun x => sin x / cos x) {x | cos x \u2260 0}\n\u22a2 ContinuousOn tan {x | cos x \u2260 0}\n[PROOFSTEP]\nhave h_eq : (fun x => sin x / cos x) = tan := by ext1 x; rw [tan_eq_sin_div_cos]\n[GOAL]\nthis : ContinuousOn (fun x => sin x / cos x) {x | cos x \u2260 0}\n\u22a2 (fun x => sin x / cos x) = tan\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nthis : ContinuousOn (fun x => sin x / cos x) {x | cos x \u2260 0}\nx : \u211d\n\u22a2 sin x / cos x = tan x\n[PROOFSTEP]\nrw [tan_eq_sin_div_cos]\n[GOAL]\nthis : ContinuousOn (fun x => sin x / cos x) {x | cos x \u2260 0}\nh_eq : (fun x => sin x / cos x) = tan\n\u22a2 ContinuousOn tan {x | cos x \u2260 0}\n[PROOFSTEP]\nrwa [h_eq] at this \n[GOAL]\n\u22a2 ContinuousOn (fun x => sin x / cos x) {x | cos x \u2260 0}\n[PROOFSTEP]\nexact continuousOn_sin.div continuousOn_cos fun x => id\n[GOAL]\n\u22a2 ContinuousOn tan (Ioo (-(\u03c0 / 2)) (\u03c0 / 2))\n[PROOFSTEP]\nrefine' ContinuousOn.mono continuousOn_tan fun x => _\n[GOAL]\nx : \u211d\n\u22a2 x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2) \u2192 x \u2208 {x | cos x \u2260 0}\n[PROOFSTEP]\nsimp only [and_imp, mem_Ioo, mem_setOf_eq, Ne.def]\n[GOAL]\nx : \u211d\n\u22a2 -(\u03c0 / 2) < x \u2192 x < \u03c0 / 2 \u2192 \u00accos x = 0\n[PROOFSTEP]\nrw [cos_eq_zero_iff]\n[GOAL]\nx : \u211d\n\u22a2 -(\u03c0 / 2) < x \u2192 x < \u03c0 / 2 \u2192 \u00ac\u2203 k, x = (2 * \u2191k + 1) * \u03c0 / 2\n[PROOFSTEP]\nrintro hx_gt hx_lt \u27e8r, hxr_eq\u27e9\n[GOAL]\ncase intro\nx : \u211d\nhx_gt : -(\u03c0 / 2) < x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\n\u22a2 False\n[PROOFSTEP]\ncases' le_or_lt 0 r with h h\n[GOAL]\ncase intro.inl\nx : \u211d\nhx_gt : -(\u03c0 / 2) < x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : 0 \u2264 r\n\u22a2 False\n[PROOFSTEP]\nrw [lt_iff_not_ge] at hx_lt \n[GOAL]\ncase intro.inl\nx : \u211d\nhx_gt : -(\u03c0 / 2) < x\nhx_lt : \u00acx \u2265 \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : 0 \u2264 r\n\u22a2 False\n[PROOFSTEP]\nrefine' hx_lt _\n[GOAL]\ncase intro.inl\nx : \u211d\nhx_gt : -(\u03c0 / 2) < x\nhx_lt : \u00acx \u2265 \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : 0 \u2264 r\n\u22a2 x \u2265 \u03c0 / 2\n[PROOFSTEP]\nrw [hxr_eq, \u2190 one_mul (\u03c0 / 2), mul_div_assoc, ge_iff_le, mul_le_mul_right (half_pos pi_pos)]\n[GOAL]\ncase intro.inl\nx : \u211d\nhx_gt : -(\u03c0 / 2) < x\nhx_lt : \u00acx \u2265 \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : 0 \u2264 r\n\u22a2 1 \u2264 2 * \u2191r + 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : -(\u03c0 / 2) < x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\n\u22a2 False\n[PROOFSTEP]\nrw [lt_iff_not_ge] at hx_gt \n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\n\u22a2 False\n[PROOFSTEP]\nrefine' hx_gt _\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\n\u22a2 -(\u03c0 / 2) \u2265 x\n[PROOFSTEP]\nrw [hxr_eq, \u2190 one_mul (\u03c0 / 2), mul_div_assoc, ge_iff_le, neg_mul_eq_neg_mul, mul_le_mul_right (half_pos pi_pos)]\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\n\u22a2 2 * \u2191r + 1 \u2264 -1\n[PROOFSTEP]\nhave hr_le : r \u2264 -1 := by rwa [Int.lt_iff_add_one_le, \u2190 le_neg_iff_add_nonpos_right] at h \n[GOAL]\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\n\u22a2 r \u2264 -1\n[PROOFSTEP]\nrwa [Int.lt_iff_add_one_le, \u2190 le_neg_iff_add_nonpos_right] at h \n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\nhr_le : r \u2264 -1\n\u22a2 2 * \u2191r + 1 \u2264 -1\n[PROOFSTEP]\nrw [\u2190 le_sub_iff_add_le, mul_comm, \u2190 le_div_iff]\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\nhr_le : r \u2264 -1\n\u22a2 \u2191r \u2264 (-1 - 1) / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\nhr_le : r \u2264 -1\n\u22a2 \u2191r \u2264 -1\n[PROOFSTEP]\nrw [\u2190 Int.cast_one, \u2190 Int.cast_neg]\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\nhr_le : r \u2264 -1\n\u22a2 \u2191r \u2264 \u2191(-1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.inr\nx : \u211d\nhx_gt : \u00ac-(\u03c0 / 2) \u2265 x\nhx_lt : x < \u03c0 / 2\nr : \u2124\nhxr_eq : x = (2 * \u2191r + 1) * \u03c0 / 2\nh : r < 0\nhr_le : r \u2264 -1\n\u22a2 0 < 2\n[PROOFSTEP]\nexact zero_lt_two\n[GOAL]\nthis : -(\u03c0 / 2) < \u03c0 / 2\n\u22a2 Tendsto (fun x => tan \u2191x) atBot atBot\n[PROOFSTEP]\nrw [tendsto_comp_coe_Ioo_atBot this]\n[GOAL]\nthis : -(\u03c0 / 2) < \u03c0 / 2\n\u22a2 Tendsto tan (\ud835\udcdd[Ioi (-(\u03c0 / 2))] (-(\u03c0 / 2))) atBot\n[PROOFSTEP]\nexact tendsto_tan_neg_pi_div_two\n[GOAL]\nthis : -(\u03c0 / 2) < \u03c0 / 2\n\u22a2 Tendsto (fun x => tan \u2191x) atTop atTop\n[PROOFSTEP]\nrw [tendsto_comp_coe_Ioo_atTop this]\n[GOAL]\nthis : -(\u03c0 / 2) < \u03c0 / 2\n\u22a2 Tendsto tan (\ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2)) atTop\n[PROOFSTEP]\nexact tendsto_tan_pi_div_two\n[GOAL]\nx : \u211d\n\u22a2 cos (arctan x) ^ 2 = \u21911 / (\u21911 + x ^ 2)\n[PROOFSTEP]\nrw_mod_cast [one_div, \u2190 inv_one_add_tan_sq (cos_arctan_pos x).ne', tan_arctan]\n[GOAL]\nx : \u211d\n\u22a2 sin (arctan x) = x / sqrt (\u21911 + x ^ 2)\n[PROOFSTEP]\nrw_mod_cast [\u2190 tan_div_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan]\n[GOAL]\nx : \u211d\n\u22a2 cos (arctan x) = \u21911 / sqrt (\u21911 + x ^ 2)\n[PROOFSTEP]\nrw_mod_cast [one_div, \u2190 inv_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan]\n[GOAL]\nx : \u211d\nh : x \u2208 Ioo (-1) 1\n\u22a2 arcsin x = arctan (x / sqrt (\u21911 - x ^ 2))\n[PROOFSTEP]\nrw_mod_cast [arctan_eq_arcsin, div_pow, sq_sqrt, one_add_div, div_div, \u2190 sqrt_mul, mul_div_cancel', sub_add_cancel,\n  sqrt_one, div_one]\n[GOAL]\ncase hb\nx : \u211d\nh : x \u2208 Ioo (\u2191(Int.negSucc 0)) 1\n\u22a2 1 - x ^ 2 \u2260 0\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase hx\nx : \u211d\nh : x \u2208 Ioo (\u2191(Int.negSucc 0)) 1\n\u22a2 0 \u2264 1 - x ^ 2\n[PROOFSTEP]\nsimp at h \n[GOAL]\nx : \u211d\nh : x \u2208 Ioo (\u2191(Int.negSucc 0)) 1\n\u22a2 1 - x ^ 2 \u2260 0\n[PROOFSTEP]\nsimp at h \n[GOAL]\nx : \u211d\nh : x \u2208 Ioo (\u2191(Int.negSucc 0)) 1\n\u22a2 0 \u2264 1 - x ^ 2\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase hb\nx : \u211d\nh : -1 < x \u2227 x < 1\n\u22a2 1 - x ^ 2 \u2260 0\n[PROOFSTEP]\nnlinarith [h.1, h.2]\n[GOAL]\ncase hx\nx : \u211d\nh : -1 < x \u2227 x < 1\n\u22a2 0 \u2264 1 - x ^ 2\n[PROOFSTEP]\nnlinarith [h.1, h.2]\n[GOAL]\nx : \u211d\nh : -1 < x \u2227 x < 1\n\u22a2 1 - x ^ 2 \u2260 0\n[PROOFSTEP]\nnlinarith [h.1, h.2]\n[GOAL]\nx : \u211d\nh : -1 < x \u2227 x < 1\n\u22a2 0 \u2264 1 - x ^ 2\n[PROOFSTEP]\nnlinarith [h.1, h.2]\n[GOAL]\n\u22a2 arctan 0 = 0\n[PROOFSTEP]\nsimp [arctan_eq_arcsin]\n[GOAL]\nx y : \u211d\nh : tan x = y\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 tan (arctan y) = tan x\n[PROOFSTEP]\nrw [tan_arctan, h]\n[GOAL]\n\u22a2 \u03c0 / 4 \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u22a2 -(\u03c0 / 2) < \u03c0 / 4\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\ncase right\n\u22a2 \u03c0 / 4 < \u03c0 / 2\n[PROOFSTEP]\nlinarith [pi_pos]\n[GOAL]\nx : \u211d\n\u22a2 arctan (-x) = -arctan x\n[PROOFSTEP]\nsimp [arctan_eq_arcsin, neg_div]\n[GOAL]\nx : \u211d\nh : 0 \u2264 x\n\u22a2 arctan x = arccos (sqrt (\u21911 + x ^ 2))\u207b\u00b9\n[PROOFSTEP]\nrw [arctan_eq_arcsin, arccos_eq_arcsin]\n[GOAL]\nx : \u211d\nh : 0 \u2264 x\n\u22a2 arcsin (x / sqrt (\u21911 + x ^ 2)) = arcsin (sqrt (1 - (sqrt (\u21911 + x ^ 2))\u207b\u00b9 ^ 2))\nx : \u211d h : 0 \u2264 x \u22a2 0 \u2264 (sqrt (\u21911 + x ^ 2))\u207b\u00b9\n[PROOFSTEP]\nswap\n[GOAL]\nx : \u211d\nh : 0 \u2264 x\n\u22a2 0 \u2264 (sqrt (\u21911 + x ^ 2))\u207b\u00b9\n[PROOFSTEP]\nexact inv_nonneg.2 (sqrt_nonneg _)\n[GOAL]\nx : \u211d\nh : 0 \u2264 x\n\u22a2 arcsin (x / sqrt (\u21911 + x ^ 2)) = arcsin (sqrt (1 - (sqrt (\u21911 + x ^ 2))\u207b\u00b9 ^ 2))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nx : \u211d\nh : 0 \u2264 x\n\u22a2 x / sqrt (\u21911 + x ^ 2) = sqrt (1 - (sqrt (\u21911 + x ^ 2))\u207b\u00b9 ^ 2)\n[PROOFSTEP]\nrw_mod_cast [\u2190 sqrt_inv, sq_sqrt, \u2190 one_div, one_sub_div, add_sub_cancel', sqrt_div, sqrt_sq h]\n[GOAL]\ncase e_a.hx\nx : \u211d\nh : 0 \u2264 x\n\u22a2 0 \u2264 x ^ 2\ncase e_a\nx : \u211d\nh : 0 \u2264 x\n\u22a2 1 + x ^ 2 \u2260 0\ncase e_a x : \u211d h : 0 \u2264 x \u22a2 0 \u2264 (1 + x ^ 2)\u207b\u00b9\n[PROOFSTEP]\nall_goals positivity\n[GOAL]\ncase e_a.hx\nx : \u211d\nh : 0 \u2264 x\n\u22a2 0 \u2264 x ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase e_a\nx : \u211d\nh : 0 \u2264 x\n\u22a2 1 + x ^ 2 \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase e_a\nx : \u211d\nh : 0 \u2264 x\n\u22a2 0 \u2264 (1 + x ^ 2)\u207b\u00b9\n[PROOFSTEP]\npositivity\n[GOAL]\nx : \u211d\nh : 0 < x\n\u22a2 arccos x = arctan (sqrt (\u21911 - x ^ 2) / x)\n[PROOFSTEP]\nrw [arccos, eq_comm]\n[GOAL]\nx : \u211d\nh : 0 < x\n\u22a2 arctan (sqrt (\u21911 - x ^ 2) / x) = \u03c0 / 2 - arcsin x\n[PROOFSTEP]\nrefine' arctan_eq_of_tan_eq _ \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nx : \u211d\nh : 0 < x\n\u22a2 tan (\u03c0 / 2 - arcsin x) = sqrt (\u21911 - x ^ 2) / x\n[PROOFSTEP]\nrw_mod_cast [tan_pi_div_two_sub, tan_arcsin, inv_div]\n[GOAL]\ncase refine'_2\nx : \u211d\nh : 0 < x\n\u22a2 -(\u03c0 / 2) < \u03c0 / 2 - arcsin x\n[PROOFSTEP]\nlinarith only [arcsin_le_pi_div_two x, pi_pos]\n[GOAL]\ncase refine'_3\nx : \u211d\nh : 0 < x\n\u22a2 \u03c0 / 2 - arcsin x < \u03c0 / 2\n[PROOFSTEP]\nlinarith only [arcsin_pos.2 h]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan", "llama_tokens": 6686, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117940706734, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.5501027568151479}}